AERO Horizon Refresher Course Program 2015
Vision: To become the Premiere Review Centre for Aeronautical Engineering in the Philippines.
Mission: 1. To help Aeronautical Engineering graduates prepare for their board exam by providing quality and
highly effective review program. 2. To promote teamwork to achieve high passing rate in the licensure
examination. 3. Serves as an organization to which aviation professionals in different field, meet to share their
knowledge and experience.
ENGINEERING ECONOMICS
Reviewer: Engr. R. R. Renigen
Basic Economic Environment and Concept
Economics - is the study or science of the production, distribution and consumption of goods and services.
Economy - the cost or profit situation regarding a practical enterprise or project, as in economy studies,
engineering economy, project economy.
Engineering economy:
a. The discipline concerned with economic aspects of engineering; it involves the systematic evaluation of
the costs and benefits of proposed technical projects;
b. The application of engineering or mathematical analysis and synthesis to economic decisions;
c. A body of knowledge and concerned with the evaluation of the worth of commodities and services;
d. The economic analysis of engineering alternatives.
Cost Terminology
Fixed costs - are those unaffected by changes in activity level over a feasible range of operations for the
capacity or capability available. Typical fixed costs include
insurance and taxes on facilities, general
management and administrative salaries, license fees, and interest costs on borrowed capital.
Variable costs - are those associated with an operation that varies in total with the quantity of output or other
measures of activity level. For example, the costs of material and labor used in a product or service are variable
costs – because they vary in total with the number of output units – even though the costs per unit stay the same.
Incremental cost, or incremental revenue - is the additional cost, or revenue, that results from increasing the
output of a system by one (or more) units. Incremental cost is often associated with “go/no go” decisions that
involve a limited change in output or activity level.
Recurring costs - are those that are repetitive and occur when an organization produces similar goods or
services on a continuing basis. Variable costs are also recurring costs, because they repeat with each unit of
output. However, recurring costs are not limited to variable costs. A fixed cost that is paid on a repeatable basis
is a recurring cost. For example, in an organization providing architectural and engineering services, office
space for rental – which is a fixed cost – is also a recurring cost.
Nonrecurring costs – are those that are not repetitive, even though the total expenditure may be cumulative
over a relatively short period of time. Typically, nonrecurring costs involve developing or establishing a
capability or capacity to operate. For example, the purchase cost for real estate upon which a plant will be built
is a nonrecurring cost, as is the cost of constructing the plant itself.
1
Direct Costs – are those that can be reasonably measured and allocated to a specific output or work activity. The
labor and material costs directly associated with a product, service, or construction activity are direct costs. For
example, the materials needed to make a pair of scissors would be a direct cost.
Indirect costs – are those that are difficult to attribute or allocate to a specific output or work activity. The term
normally refers to type of costs that would involve too much effort to allocate directly to a specific output. In
this usage, they are costs allocated through a selected formula (such as, proportional to direct labor hours, direct
labor dollars, or direct material dollars to the outputs or work activities. For example, the costs of common
tools, general supplies, and equipment maintenance in a plant are treated as indirect costs.
Overhead consists of plant operating costs that are not direct labor or direct material costs. Examples of
overhead include electricity, general repairs, property taxes, and supervision.
Standard costs – are representative costs per unit of unit of output that are established in advance of actual
production or service delivery. They are developed from the direct labor hours, materials, and support functions
(with their established costs per unit) planned for the production or delivery process.
Cash cost – a cost that involves payment of cash (and results in a cash flow) to distinguish it from one that does
not involve a cash transaction and is reflected in the accounting system as a noncash cost. This noncash cost is
often referred to as a book cost.
Sunk cost – is one that has occurred in the past and has no relevance to estimates of future costs and revenues
related to an alternative course of action.
Opportunity cost – is the cost of the best rejected (i.e., foregone) opportunity and is often hidden or implied. It
is incurred because of the use of limited resources, such that the opportunity to use those resources to monetary
advantage in an alternative use is foregone. For example, suppose that a project involves the use of vacant
warehouse space presently owned by a company. The cost for that space to the project should be the income or
savings that possible alternative uses of the space may bring to the firm. In other words, the opportunity cost for
the warehouse space should be the income derived from the best alternative use of the space.
Life-Cycle cost –this term refers to a summation of all costs, recurring and nonrecurring, related to a product,
structure, system, or service during its life span.
The General Economic Environment
There are numerous general economic concepts that must be taken into account in engineering studies. In broad
terms, economics deals with the interactions between people and wealth, and engineering is concerned with the
cost-effective use of scientific knowledge to benefit mankind.
Consumer and Producer Goods and Services
Consumer goods and services are those products or services that are directly used by people to satisfy their
wants. Food, clothing, homes, cars, television sets, haircuts, and medial services are examples.
Producer goods and services are used to produce consumer goods and services or other producer goods.
Machine tools, factory building, buses, and farm machinery are example.
Measure of Economic Worth
Goods and services are produced and desired because directly or indirectly they have utility – the power to
satisfy human wants and needs. Thus, they may be used or consumed directly, or they may be used to produce
2
other goods or services that may, in turn, be used directly. Utility is most commonly measured in terms of
value, expressed in some medium of exchange as the price that must be paid to obtain the particular item.
Necessities, Luxuries, and Price Demand
Goods and services may be divided into two types –necessities and luxuries. Necessities are those products or
services that are required to support human life and activities that will be purchased in somewhat the same
quantity even though the price varies considerably. Luxuries are those products or services that are desired by
humans and will be purchase if money is available after the required necessities have been obtained. Obviously,
these terms are relative, because, for most goods and services, what one person considers a necessity may be
considered a luxury by another. For example, a person living in one community may find that an automobile is
a necessity to get to and from work. If the same person lived and worked in a different city, adequate public
transportation might be available, and an automobile would be a luxury. For all goods and services, there is a
relationship between the price that must be paid and the quantity that will be demanded or purchased.
As the selling price per unit (p) is increased, there will be less demand (D) for the product, and as the selling
price is decreased, the demand will increase. The relationship between price and demand can be expressed as
linear function:
p = a − bD
for 0 ≤ D ≤ a/b, a > 0, b > 0
Eqn. 1
where “a” is the intercept on the price axis and “-b” is the slope. Thus, “b” is the amount by which demand
increases for each unit decrease in “p”. Both ”a” and “b” are constants. It follows that,
D=
a−p
b
(b ≠ 0)
Eqn. 2
Demand is the need, want or desire for a product backed by the money to purchase it. In economic analysis,
demand is always based on “willingness and ability to pay” for a product, not merely want or need for the
product.
The demand for a product is inversely proportional to its selling price, i.e., as the selling price is increased, there
will be less demand for the product; and as the selling price is decreased, the demand will increase.
Supply is the amount of a product made available for sale.
If the selling price for a product is high, more producers will be willing to work harder and risk more capital in
order to reap more profit. However, if the selling price for a product declines, capitalists will not produce as
much because of the smaller profit they can obtain for their labor and risk.
Therefore, the relationship between price and supply is that they are directly proportional, i.e., the bigger the
selling price, the more supply; and the smaller the selling price, the less is the supply.
THE LAW OF SUPPLY AND DEMAND
The law of supply and demand may be stated as follows:
“Under conditions of perfect competition, the price at which any given product will be supplied and purchased
is the price that will result in the supply and the demand being equal.”
Competition
Perfect competition occurs in a situation in which any given product supplied by a large number of vendors and
there is no restriction on additional vendors entering the market. Under such conditions, there is assurance of
3
complete freedom on the part of both buyer and seller. Perfect competition may never occur in actual practice,
because of a multitude of factors that impose some degree of limitation upon the actions of buyers or sellers, or
both.
Monopoly is at the opposite pole of perfect competition. A perfect monopoly exists when a unique product or
service is only available from a single vendor and that vendor can prevent the entry of all others into the market.
Under such conditions, the buyer is at the complete mercy of the vendor in terms of the availability and price of
the product. Perfect monopolies rarely occur in practice, because (1) few products are so unique that substitutes
cannot be used satisfactorily, and (2) governmental regulations prohibit monopolies if they are unduly
restrictive.
The Total Revenue Function
The total revenue, TR that will result from a business venture during a given period is the product of the selling
price per unit, p, and the number of units sold, D. Thus
TR = priceXdemand = p( D)
Eqn. 3
If the relationship between price and demand as given in Eqn. 1 is used,
TR = (a − bD) D = aD − bD 2
for 0≤ D ≤ a/b, a > 0, b > 0
Eqn. 4
From calculus the demand, Ď that will produce maximum total revenue can be obtained by solving
d (TR )
= a − 2 bD = 0
dD
Eqn. 5
Thus
a
Ď=
Eqn. 6
2b
To guarantee that Ď maximizes the total revenue, check the second derivative to be sure it is negative.
d 2 (TR )
dD 2
= −2 b
Cost, Volume, and Breakeven Point Relationships
Fixed costs remain constant over a wide range of activities as long as the business does not permanently
discontinue operations, but variable costs vary in total with the volume of output. Thus, at any demand D, total
cost is
CT = C F + C V
Eqn. 7
Where:
C T = total costs
CF = fixed costs
C V = variable costs
4
For the linear relationship assumed here,
C V = (C v )( D)
Eqn. 8
Where:
C v = variable cost per unit
We consider two scenarios for finding breakeven points. In the first scenario, demand is a function of price. The
second scenario assumes that price and demand are independent of each other.
Scenario 1. When total revenue and total cost, as given in Eqns. 7 and 8 are combined, the typical results as a
function of demand are depicted. At breakeven point D’1, total revenue is equal to total cost, and an increase in
demand will result in a profit for the operation. Then at optimal demand, D*, profit is maximized (Eqn. 10). At
breakeven point D’2, total revenue and total cost are again equal, but additional volume will result in an
operating loss instead of a profit. Obviously, the conditions for which breakeven and maximum profit occurs are
our primary interest. First, at any volume (demand), D,
Profit (loss) = total revenue - total costs
(
)(
= aD − bD 2 − C F + C v D
= −C F + (a − C v )D − bD 2
)
for 0 ≤ D ≤ a/b
Eqn. 9
In order for a profit to occur, and to achieve the typical results, two conditions must be met:
1.
(a – Cν) > 0; that is, the price per unit that will result in no demand has to be greater than the variable
cost per unit (this avoids negative demand).
2.
Total revenue (TR) must exceed total cost (CT) for the period involved.
If these conditions are met, we can find the optimal demand at which maximum profit will occur by taking the
2
first derivative of profit = -CF + (a – Cν) D – bD with respect to D and it equals to zero:
d ( profit )
=0
dD
D∗ =
a − Cv
2b
Eqn.10
Where:
D ∗ = optimal value of D that maximizes profit
To ensure that we have maximized profit (rather than minimized it); the sign of the second derivative must be
negative. Checking this, we find that
5
d 2 ( profit )
dD 2
= −2 b
which will be negative for b > 0 (as earlier specified). (Also, recall that in cost minimization problems a positive
signed second derivative is necessary to guarantee a minimum – valued optimal cost solution).
An economic breakeven point for an operation occurs when total revenue equals total cost. Then for
total revenue and total cost, as used in the development of Eqns. 9 and 10 and at any demand D,
Total revenue = total cost
(breakeven point)
aD − bD 2 = C F + C v D
− bD 2 + (a − C v )D − C F = 0
Eqn. 11
Because Eqn. 11 is a quadratic equation with one unknown (D), we can solve for the breakeven points D’1 and
D’2 (the roots of the equation).
D' =
− (a − C v ) ±
(a − C v )2 − 4( − b)(− C F )
Eqn. 12
2( − b)
With the conditions for the profit satisfied (Eqn. 9), the quantity in the brackets of the numerator (the
discriminant) in Eqn. 12 will be greater than zero. This will ensure that D’1 and D’2 have real positive and
unequal values.
D'1 =
− (a − C v ) +
(a − C v )2 − 4( − b)(− C F )
2( − b )
and
D' 2 =
− (a − C v ) −
(a − C v )2 − 4( − b)(− C F )
2( − b )
Scenario 2. When the price per unit (p) for a product or service can be represented more simply as being
independent of demand (versus being a linear function of demand, as assumed in Eqn. 1) and is greater than the
variable cost per unit (Cν), a single breakeven points results. Then under the assumption that demand is
immediately met, total revenue TR = p(D).
Problems:
1. Find the demand, Ď that maximizes total revenue if the equation for price is given by
50, 000 - 200D?
Ans. Ď = 125 units
6
Given:
p = 50,000 – 200D
a = 50,000
b = 200
Required:
Ď
Solution:
a
Ď=
50,000
=
= 125 units
2b
(2)(200)
2. A company produces an electronic time switch that is used in consumer and commercial products made by
several other manufacturing firms. The fixed cost (CF) is PhP73, 000 per month, and the variable cost (Cν) is
PhP83 per unit. The selling price per unit is p = 180 –0.02D, based on Eqn 1. For this situation (a) determine the
optimal volume for this product and confirm that a profit occurs (instead of loss) at this demand; and (b) find
the volumes at which breakeven occurs; that is the range of profitable demand.
Ans. (a) D* = 2,425 units per month (b) D’1 = 932 units per month, D’2 = 3,918 units per month.
Given:
CF = PhP73, 000 per month
Cv = PhP83 per unit
p = 180 – 0.02D
a = 180
b = 0.02
Required:
(a.) D*
(b.) D’1 and D’2
Solution:
(a.) D ∗ =
a − Cv
2b
=
180 − 83
= 2,425 units per month
( 2)(0.02)
7
(b) D'1 =
− (a − C v ) +
(a − C v )2 − 4( − b)(− C F )
2( − b )
(
− 180 − 83
=
)+
(180 − 83)2 − (4)( −0.02)(− 73,000
)
(2)( −0.02)
= 932 units per month
D' 2 =
=
− (a − C v ) −
(a − C v )2 − 4( − b)(− C F )
2( − b )
− (180 − 83) −
(180 − 83)2 − ( 4)( −0.02)(− 73,000)
( 2)( −0.02)
= 3,918 units per month
3. Suppose we know that p = 1,000 – D/5, where p = price in pesos and D = annual demand. The total cost per
2
year can be approximated by CT = 1,000 + 2D
a. Determine the value of D that maximizes profit
b. Show in part (a) that profit has been maximized rather than minimized.
Ans.
(a) D* = 227 units per year
(b)
d 2 ( profit )
dD 2
= − 4 .4
Given:
p = 1,000 – D/5
CT = 1,000 + 2D
2
Required:
. (a) D*
(b)
d 2 ( profit )
dD 2
8
Solution:
(a) Profit = TR - CT
= pD
- CT
2
= (1,000 – D/5) D – (1,000 +2D )
2
= 1,000D – D /5 -1,000 – 2D
2
2
Profit = -11D /5 + 1,000D – 1,000
d ( profit )
=0
dD
-22D/5 + 1,000 = 0
(5)(1,000)
= 227 units per month
22
D∗ =
(b)
d 2 ( profit )
dD 2
=−
22
= − 4 .4
5
4. A company produces circuit boards used to update outdated computer equipment. The fixed cost is PhP42, 000
per month and the variable cost is PhP53 per circuit board. The selling price per unit is p = 150 – 0.02D.
Maximum output of the plant is 4,000 units per month.
a. Determine optimum demand for this product
b. What is the maximum profit per month?
c. What is the company’s range of profitable demand?
Ans. (a) D* = 2,425 circuit boards per month
(b) Max profit = PhP75,612.5 per month
(c) 480 to 4,369 circuit boards per month
Given:
CF = PhP42, 000 per month
Cv = PhP53 per circuit board
p = 150 – 0.02D
a = 150
9
b = 0.02
Required:
. (a) D*
(b) Maximum profit
(a)
D’1 and D’2
Solution:
a − Cv
=
2b
150 − 53
= 2,425 circuit boards per month
( 2)(0.02)
(a)
D∗ =
(b)
Max profit = −C F + (a − C v )D ∗ − bD ∗
2
(
)
= = −42,000 + 150 − 53 (2,425) − (0.02)( 2,425) 2
Max profit = PhP75,612.5 per month
(c)
D' 1 =
− (a − C v ) +
(a − C v )2 − 4( − b)(− C F )
2( − b )
− (150 − 53) +
=
(150 − 53)2 − (4)( −0.02)(− 42,000
(2)( −0.02)
= 480 circuit boards per month
D' 2 =
=
− (a − C v ) −
(a − C v )2 − 4( − b)(− C F )
2( − b )
− (180 − 83) −
(180 − 83)2 − (4)( −0.02)(− 73,000)
( 2)( −0.02)
= 4,369 circuit boards per month
10
)
5. A company has established that the relationship between the sales price for one of its products and the quantity
sold per month is approximately D = 780 – 10p units (D is the demand or quantity sold per month, and p is the
price in pesos). The fixed cost is PhP800 per month, and the variable cost is PhP30 per unit produced. What
number of units, D*, should be produced per month and sold to maximize net profit? What is the maximum
profit per month related to the product?
Ans.
D* = 240 units per month
Maximum profit = PhP4, 960 per month
Given:
D = 780 – 10p
p = 78 – 0.10D
CF = PhP800 per month
Cv = PhP30 per unit
a = 78
b = 0.10
Required:
. (a) D*
(b) Maximum profit
Solution:
a − Cv
=
2b
78 − 30
= 240 units per month
( 2)(0.10)
(a)
D∗ =
(b)
Max profit = −C F + (a − C v )D ∗ − bD ∗
2
= −800 + (78 − 30 )( 240) − (0.10)( 240) 2
= Php4,960 per month
6. A company estimates that as it increases its sales volume by decreasing the selling price of its product, revenue
= aD – bD2 (where D represents the units of demand per month, with 0 ≤ D ≤ a/b). The fixed cost is PhP1,000
per month, and the variable cost is PhP4 per unit. If a = 6 and b = 0.001, determine the sales volume for
maximum profit, and the maximum profit per month.
Ans. D* = 1,000 units per month
Maximum profit = PhP 0 per month
11
Given:
CF = PhP1, 000 per month
Cv = PhP4 per unit
a=6
b = 0.001
Required:
. (a) D*
(b) Maximum profit
Solution:
a − Cv
=
2b
6−4
= 1,000 units per month
( 2)(0.001)
(a)
D∗ =
(b)
Max profit = −C F + (a − C v )D ∗ − bD ∗
2
= = −1,000 + (6 − 4 )(1,000) − (0.001)(1,000) 2
= PhP 0 per month
Principles of Money – Time Relationship
Capital – refers to wealth in the form of money or property that can be used to produce more wealth.
Two basic categories of capital:
1. Equity capital – is that owned by individuals who have invested their money or property in a business
project or venture in the hope of receiving a profit.
2. Debt capital – often called borrowed capital, is obtained from lenders e.g., through the sale of bonds) for
investment.
SIMPLE INTEREST
Suppose a debtor loans money from a creditor. The debtor must pay the creditor the original amount loaned plus
an additional sum called interest. For the debtor, interest is the payment for the use of the borrowed capital
while for the creditor; it is the income from the invested capital.
Simple Interest (denoted as I) is defined as the interest on a loan or principal that is based only on the original
amount of the loan or principal. This means that the interest charges grow in linear function over a period of
time. This is usually used for short-term loans where the period of the loan is measured in days rather than
years. It can be calculated using the following formula:
I = Pin
Eqn. 13
12
Where:
P = principal/loan
I = interest
i = interest rate
n = period
The future amount of the principal may be calculated by adding the interest (I) to the principal (P).
F=P+I
or:
F = P + Pin
Thus,
F = P(1+in)
Eqn. 14
There are two types of simple interest namely, ordinary interest and exact simple interest.
Ordinary simple interest is based on one banker’s year. A banker year is composed of 12 months
of 30 days each which is equivalent to a total of 360 days in a year. The value of n that is used in the preceding
formulas may be calculated as
n=
d
360
Where:
d = number of days the principal was invested
Exact simple interest is based on the exact number of days in a given year. A normal year has 365 days while a
leap year (which occurs once every 4 years) has 366 days. Unlike the ordinary simple interest, the number of
days in a month is based on the actual number of days each month contains in our Gregorian calendar.
To determine the year whether leap year or not, one has to divide the year by 4. If it is exactly divisible by 4, the
year to be leap year otherwise it will be considered just a normal year with 365 days. However, if the year is a
century year (ending with two zeros, e.g. 1700, 1800…), the year must be divided by 400 instead of 4 to
determine the year whether or not a leap year. Hence, year 1600 and year 2000 are leap years.
Under this method of computation of interest, it must be noted that under normal year, the month of February
has 28 days while during leap years it has 29 days. Again, the values of n to be used in the preceding formulas
are as follows:
n=
d
, for normal year
365
Eqn. 16
n=
d
, for leap year
366
Eqn. 17
13
DISCOUNT
Consider the following cases where discount is involved:
CASE 1:
A person has a bond or financial security that is not due yet but payable in some future date, desires to
exchange it into an immediate cash. In the process, he will accept an amount in cash smaller than the face value
of the bond. The difference between the amount he receives in cash (present worth) and the face value of the
bond or financial security (future worth) is known as discount. The process of converting a claim on a future
amount of money in the present is called discounting.
CASE 2:
In a bank loan, a person borrows PhP100 for 1 year with an interest of 10%. The interest as computed is
PhP10. The bank deducts the interest, which is PhP10 from the end of the year. The P10 that was deducted
represent the interest paid in advance in this case, discount also represent the difference between the present
worth (i.e. PhP90) and the future worth (i.e. PhP100).
Discount = Future worth – Present worth
The rate of discount is the discount on one unit of principal per unit time.
d =1−
i=
1
1+ i
Eqn. 18
d
1− d
Eqn. 19
Where:
d = rate of discount
Problems:
1. If you borrow money from your friend with simple interest of 12%, find the present worth of PhP20, 000
which is due at the end of nine months.
Ans. P = PhP18, 348.62
Given:
F = PhP20,000
i = 12% = 0.12
n = 9 months x 1 year/12 months = 0.75 year
14
Required:
P
Solution:
F = P (1+in)
P=
F
20,000
=
= PhP18, 348. 62
1 + in
1 + (0.12)(0.75)
2. Engr. J. Flamiano borrowed money from a bank. He received from the bank Php1, 340 and promised to pay
Php1, 500 at the end of 9 months. Determine the following:
A. Simple interest rate
B. The corresponding discount rate or often referred to as the “banker’s discount”
Ans. A. i = 15.92%
B. d = 13.73%
Given:
P = PhP1,340
F = PhP1,500
n = 9 months x 1 year/12 months = 0.75 year
Required:
A. i
B. d(discount)
Solution:
A. Solving for simple interest rate, i:
F = P (1+in)
1,500
F
−1
−1
1,340
P
i=
=
= 0.1592 or 15.92 %
n
0.75
B. Solving for the discount rate, d:
d =1−
1
=
1+ i
1−
1
= 0.1373 or 13.73 %
1 + 0.1592
3. Engr. Verona buys a television set from a merchant who ask PhP1, 250 at the end of 60 days. She wishes
to pay immediately and the merchant offers to compute the cash price on the assumption that money is
worth 8% simple interest. What is the cash price?
15
Ans. P = PhP1, 233.55
Given:
F = PhP1, 250
i = 8 % = 0.08
n = 60 days x 1 month/30 days x 1 year/12 months = 1/6 yr.
Required:
P
Solution:
P=
F
1,250
=
= PhP1, 233.55
1 + in
1 + (0.08)(1 / 6)
4. Engr. Catalan borrowed money from a loan shark. He received from the loan shark an amount of PhP1, 342
and promised to repay PhP1,500 at the end of 3 quarters. What is the simple interest rate?
Ans. i = 15.69%
Given:
P = PhP1, 342
F = PhP1, 500
n = 3/4 yr.
Required:
i
Solution:
1,500
F
−1
−1
1,342
P
i=
=
= 0.1569 or 15.69 %
n
3/ 4
5. Determine the exact simple interest on PhP5, 000 invested for the period from January 15, 1996 to October
12, 1996, if the rate of interest is 18%.
Ans. I = Php666.39
Given:
P = PhP5, 000
16
i = 18 % = 0.18
Required:
I
Solution:
Solving for the number of days, d money was invested:
January 15 to 31
February
March
April
May
June
July
August
September
October 1 to 12
n=
= 16
= 29 (since 1996 is a leap year)
= 31
= 30
= 31
= 30
= 31
= 31
= 30
= 12
______________
271 days
d
271
=
366 366
I = Pin = (5,000) (0.18) (271/366) = Php666.39
6. The exact simple interest of PhP5, 000 invested from June 21, 1995 to December 25, 1995 is Php100. What
is the rate of interest?
Ans. i = 3.90%
Given:
P = PhP5, 000
I = PhP100
Required:
i
Solution:
Solving for the total number of days, d money was invested:
June 21 to 30
July
August
September
= 09
= 31
= 31
= 30
17
October
November
December 1 to 25
n=
= 31
= 30
= 25
______________
187 days
d
187
=
365 365
I = Pin
i=
I
Pn
=
100
= 0.0390 or 3.90 %
(5,000)(187 / 365)
COMPOUND INTEREST
Compound interest is defined as the interest of loan or principal which is based not only on the original amount
of the loan or principal but the amount of loan or principal plus the previous accumulated interest. This means
that aside from the principal, the interest now earns interest as well. Thus, the interest charges grow
exponentially over a period of time.
Compound interest is frequently used in commercial practice than simple interest, more especially if it is a
longer period which spans for more than a year.
The future amount of the principal may be derived by the following tabulation:
Period
1
2
3
n
Principal
P
P(1 + i)
P(1 + i)2
Interest
Pi
P(1 + i)i
P(1 + i)2i
Total Amount
P + Pi = P(1 + i)
P(1 + i)(1 + i) = P(1 + i)2
P(1 + i)2(1 + i) = P(1 + i)3
P(1 + i)n
The tabulation above shows that the future amount (total amount) is just the value P(1 +i) with an exponent
which is numerically equal to the period.
It is also observed that compound interest is based on the principles of geometric progression and using such
method, the total amount after each period are as follows:
First term,
a1 = P(1 +i)
Eqn. 20
Second term,
a2 = P(1 +i)2
Eqn. 21
Third term,
a3 = P(1 + i)3 and so on.
Eqn. 22
Solving for the common ratio,
a
r= 2
a1
18
P(1 + i )2
r=
P(1 + i )
Eqn. 23
r =1+ i
Using the formula for nth term of a Geometric Progression (G.P.)
a n = a 1 r n −1
a n = P(1 + i )(1 + i ) n −1
a n = P(1 + i ) n
Eqn. 24
a. FUTURE AMOUNT, F:
F = P(1 + i ) n
Eqn. 25
Where:
P = Principal
i = interest per period (in decimal)
n = number of interest periods
(1 + i)n = single payment compound amount factor
b. PRESENT WORTH, P:
P=
F
Eqn. 26
(1 + i )n
Where:
1
(1 + i )n
= single payment present worth factor
For 6% compounded semi-annually for 5 years.
0.06
= 0.03
2
n = (5)(2) = 10
i=
19
For 6% compounded quarterly for 5 years.
0.06
i=
= 0.015
4
n = (5)(4) = 20
For 6% compounded monthly for 5 years.
0.06
i=
= 0.005
12
n = (5)(12) = 60
For 6% compounded bi-monthly for 5 years.
0.06
i=
= 0.01
6
n = (5)(6) = 30
CONTINUOUS COMPOUNDING
The concept of continuous compounding is based on the assumption that cash payments occur once per year but
compounding is continuous throughout the year.
Compounded Continuously
❶ F = Pe rn
Eqn. 27
Where:
e = 2.71828
F = future worth
P = present worth
r = interest rate compounded continuously
n = no. of periods
Effective annual rate
Compounded Continuously
❷Effective rate = e r − 1
❸Compound amount factor = e rn
❹Present worth factor =
1
e rn
The present worth of continuous compounding is
P=
F
e rn
Eqn. 28
20
NOMINAL AND EFFECTIVE RATES OF INTEREST
Rate of interest is the cost of borrowing money. It also refers to the amount earned by a unit principal per unit
time.
There are two types of rates of interest, namely the nominal rate of interest and the effective rate of interest.
Nominal rate of interest is defined as the basic annual rate of interest while effective rate of interest is defined
as the actual or the exact rate of interest earned on the principal during a one-year period.
For example: A principal is invested at 5% compounded quarterly.
In this statement, the nominal rate is 5% while the effective is greater than 5% because of the compounding
which occurs four times a year. The following formula is used to determine the effective rate of interest:
ER = (1 + i )m − 1
Eqn. 29
or:
NR 

ER = 1 +

m 

m
−1
Where:
m = number of interest period per year
i = interest per period =
NR
m
NR = nominal rate of interest
Note:
i = NR if the mode of compounding is annually
Substituting the values of m and i:
4
0.05 

ER = 1 +
 −1
4 

ER = 0.0509
ER = 5.09%
So, the actual interest rate is not just 5% but 5.09%. However, the effective rate and nominal rate are equal if
the mode of compounding is per annum or annually.
21
Problems:
1. What rate in percent, compounded semi-annually is equivalent to 20% compounded annually?
Ans. i = 19.09%
Solution:
2
 i
 0.20 
1 +  = 1 +

1 
 2

1
i = 19.09%
2. What rate in percent, compounded quarterly is equivalent to 17% compounded monthly?
Ans, i = 17.24%
Solution:
4
 i
 0.17 
1 +  = 1 +

12 
 4

12
i = 17.24%
3. What rate in percent, compounded monthly is equivalent to 18% compounded semi-annually?
Ans. i = 17.3%
Solution:
i 

1 + 
 12 
12
 0.18 
= 1 +

2 

2
i = 17.3%
4. Which has the highest effective rate of interest?
①12% compounded monthly.
②12.25% compounded quarterly.
③12.5% compounded semi-annually.
④12.75% compounded annually.
Ans. Highest 1s 12.5% compounded semi-annually
Solution:
①12% compounded monthly
22
12
 0.12 
ER = 1 +
 − 1 = 12.68%
12 

②12.25% compounded quarterly
4
 0.1225 
ER = 1 +
 − 1 = 12.82% %
4 

③12.5% compounded semi-annually
2
 0.125 
ER = 1 +
 − 1 = 12.89%
2 

④12.75% compounded annually
1
 0.1275 
ER = 1 +
 − 1 = 12.75%
1 

5. Which has the second highest effective annual rate of interest?
①12% compounded monthly.
②12.25% compounded quarterly.
③12.5% compounded semi-annually.
④12.75% compounded annually.
Ans.2nd highest 1s 12.25% compounded quarterly
Solution:
①12% compounded monthly
12
 0.12 
ER = 1 +
 − 1 = 12.68%
12 

②12.25% compounded quarterly
4
 0.1225 
ER = 1 +
 − 1 = 12.82% %
4 

③12.5% compounded semi-annually
2
 0.125 
ER = 1 +
 − 1 = 12.89%
2 

④12.75% compounded annually
23
1
 0.1275 
ER = 1 +
 − 1 = 12.75%
1 

6. What is the difference between the total simple interest and the total compound interest on a savings of
PhP100, 000.00 at 7.5% per annum for a period of five years?
Ans. Difference = PhP6,063.00
Given:
P = PhP100, 000.00
i = 0.075
n = 5 years
Required:
Difference
Solution:
For simple interest:
I = Pin
I = (100,000.00)(0.075)(5) = PhP37,500.00
For compound interest:
n
5
F = P (1 + i ) = (100,000)(1 + 0.075) = PhP143,562.93
I = F − P = 143,562.93 − 100,000 = PhP 43,562.93
Difference = 43,562.93 – 37,500 = PhP6,062.93
7. When compounded bi-monthly Php15, 000.00 becomes PhP22, 318.30 after 5 years. What is the nominal rate of
interest?
Ans. NR = 8%
Given:
P = PhP15, 000.00
F =PhP22, 318.30
m=6
N = 5 years
NR NR
i=
=
m
6
mN = (6)(5) = 30
Required:
NR
Solution:
24
NR 

F = P 1 +

m 

mN
 NR 
22,318.30 = (15,000.00)1 +

6 

NR  22,318.30 
=

6
 15,000.00 
1
30
30
−1
 22,318.39  130 
NR = (6) 
 − 1 = 0.0799 = 8%
 15,000.00 



8. An investment of PhP200, 000.00 earns 10% per annum compounded bi-monthly. How much will it earn in
6 years?
Ans. I = PhP162,626.00
Given:
P = PhP200, 000.00
NR = 0.10
N = 6 years
NR 0.10
i=
=
m
6
m=6
mN = 36
Required:
I
Solution:
NR 

F = P 1 +

m 

mN
 0.10 
F = (200,000.00)1 +

6 

F = PhP362,626.00
36
I = F − P = 362,626.00 − 200,000.00 = PhP162,626.00
9. Engr. Rivera invested his money at 16% per annum, net of taxes, computed semi-annually after how many years
will his money doubled?
Ans. N = 4.5 years
25
Given:
F = 2P
i=
NR 0.16
=
m
2
i = 0.16
m=2
Required:
N (no. of years)
Solution:
NR 

F = P 1 +

m 

mN
 0.16 
2 P = P 1 +

2 

 0.16 
2 = 1 +

2 

2N
2N
 0.16 
ln 2 = ln1 +

2 

2N
 0.16 
ln 2 = 2 N ln1 +

2 

ln 2
 0.16 
ln1 +

2 

N = 4.5 years
2N =
10. An investment earns 20% compounded quarterly. After how many years will it quadruple.
Ans. N = 7.27 years
Given:
F = 4P
i=
NR 0.20
=
m
4
26
m=4
Required:
N (no. of years)
Solution:
NR 

F = P 1 +

m 

mN
 0.20 
4 P = P 1 +

4 

 0.20 
4 = 1 +

4 

4N
4N
0.20 

ln 4 = ln1 +

4 

4N
ln 4
 0.20 
4 ln1 +

4 

N = 7.10 years
N=
11. A credit plan charges interest rate of 36% compounded monthly. Find its effective rate.
Ans. ER = 42.58 %
Given:
i=
NR 0.36
=
m
12
m = 12
Required:
ER
Solution:
NR 

ER = 1 +

m 

m
12
0.36 

− 1 = 1 +

12 

− 1 = 0.4258 or 42.58 %
27
12. A master card compounds monthly and charges an interest of 1.5% per month. What is the effective interest
rate per year?
Ans. ER = 19.56%
Given:
i = 1.5 % = 0.015
m = 12
Required:
ER
Solution:
ER = (1 + i )m − 1 = (1 + 0.015)12 − 1 = 0.1956 or 19.56 %
13. Engr. De Leon expects to receive PhP20, 000 in 10 years. If interest is computed at 6% compounded
quarterly, how much is it worth today?
Ans. P = PhP11,025.25
Given:
F = PhP20, 000
i=
NR 0.06
=
m
4
mN = (4)(10) = 40
NR = 0.06
Required:
P
Solution:
NR 

F = P1 +

m 

P=
mN
F
NR 

1 +

m 

mN
=
20,000
0.06 

1 +

4 

40
= PhP11,025.25
28
14. Suppose you borrow PhP8, 000 now, promising to repay the loan principal plus accumulated interest in four
years at i = 10% per year. How much would you repay at the end of four years?
Ans. F = Php11,713
Given:
P =PhP8, 000
i = 10 % = 0.10
n = 4 yrs.
Required:
F
Solution:
F = P(1 + i )n = (8,000)(1 + 0.10 ) = PhP11, 713
4
15. An investor (owner) has an option to purchase a tract of land that will be worth PhP10, 000 in six years. If
the value of the land increases at 8% per year, how much should the investor be willing to pay now for his
property?
Ans. P = PhP6, 302
Given:
F = PhP10, 000
n = 6 yrs.
i = 8 % = 0.08
Required:
P
Solution:
P=
F
(1 + i )n
=
10,000
(1 + 0.08)6
= PhP6, 302
16. How long will it take money to triple itself if invested at 8% compounded annually?
Ans. n = 14.27 years
Given:
F = 3P
29
i = 8 % = 0.08
Required:
n
Solution:
F = P (1 + i )
n
3P = P(1 + i )n
3 = (1 + i )n
ln 3 = ln (1 + i )n
ln 3 = n ln(1 + i )
n=
ln 3
ln 3
=
= 14.27 yrs.
ln(1 + i)
ln(1 + 0.08)
17. If the rate of interest is 7% per annum, compounded continuously, after how many years will a deposit be
tripled?
Ans. n = 15.7 years
Given:
r = 0.07
Required:
n
Solution:
F = Pe rn
3P = Pe rn
3 = e rn
ln 3 = ln e rn
ln 3 = rn ln e
30
n=
ln 3
ln 3
=
= 15.7 years
r ln e (0.07) ln e
18. What nominal interest rate compounded continuously is equivalent to an effective rate of 10% per annum?
Ans. r = 9.53%
Given:
ER = 10%
Required:
r
Solution:
ER = e r − 1
e r = ER + 1
r ln e = ln( ER + 1)
r=
ln( ER + 1) ln(0.10 + 1)
=
= 9.53%
ln e
ln e
19. PhP100, 000.00 is deposited by Engr. Aquino at a nominal rate of 7% compounded annually, for 5 years. What
would be the difference in the sums at the end of 5 years if the interest were compounded continuously?
Ans. Diff. PhP1, 652.00
Given:
P = PhP100, 000.00
i = 0.07
n = 5 years
Required:
Difference
Solution:
For compound interest:
F = P (1 + i ) n
F = (100,000)(1 + 0.07) 5 = PhP140,255.00
31
For compounded continuously:
F = Pe rn
[
]
F = (100,000) e ( 0.07 )(5) = PhP141,907.00
Diff. = 141, 907.00 – 140, 255.00 = PhP1, 652.00
20. An investment earning a nominal annual interest of 10%, compounded continuously, accumulated to a sum of
PhP800, 000.00 at the end of 15 years. What was the original investment?
Ans. P = PhP178, 504.00
Given;
F = 800, 000.00
r = 0.10
n = 15 years
Required:
F = Pe rn
P=
F 800,000
=
= PhP178,504.00
e rn e ( 0.10 )(15)
ANNUITY
Annuity is defined as a series of equal payments occurring at equal interval of time. When an annuity has a
fixed time span, it is known as annuity certain. The following are annuity certain.
1. Ordinary annuity is a type of annuity where the payments are made at the end of each period beginning
from the first period.
Derivation of formula for the sum of ordinary annuity:
Let A be the periodic or uniform payment and assuming only four payments:
a1 = A
a2 = A (1 + i)
a3 = A (1 + i)2
a4 = A (1 + i)3
Annuity is based on the principles of compound interest. Hence, computation of the sum of annuity may be
done using the formulas for geometric progression.
Solving for common ratio:
32
a
r= 2
a1
r=
A(1 + i)
=1+ i
A
Solving for the sum:
(
)
a1 r n − 1
S=
r −1
[
]
[
]
A (1 + i )4 − 1
S=
1+ i −1
S=
A (1 + i )4 − 1
i
a. SUM OF ORDINARY ANNUITY:
F=
[
]
A (1 + i )n − 1
i
Eqn. 30
Where:
i = interest per period
n = number of periods
A = uniform payment
(1 + i )n
−1
i
= uniform series compound amount factor
b. PRESENT WORTH OF ORDINARY ANNUITY:
Using compound sentence formula:
P=
F
(1 + i )n
33
But,
F=
[
]
[
]
A (1 + i )n − 1
i
Therefore:
P=
A (1 + i )n − 1
Eqn. 31
i(1 + i ) n
Where:
[
] = uniform series present worth factor
A (1 + i )n − 1
i(1 + i ) n
2. Annuity due is a type of annuity where the payments are made at the beginning of each period starting from
the first period.
3. Deferred annuity is a type of annuity where the first payment does not begin until some later date in the
cash flow.
When an annuity does not have a fixed time span but continues indefinitely, then it is referred to as a perpetuity.
The sum of a perpetuity is an infinite value.
PRESENT WORTH OF PERPETUITY:
P=
A
i
Where:
A = uniform payment
i = interest per period
Problems:
1. An enterprising student is planning to have personal savings totaling Php1, 000,000 when she retires at age
65. She is now 20 years old. If the annual interest rate will average 7% over the next 45 years on her savings
account, what equal end-of-year amount must she save to accomplish her goal?
Ans. A = Php3, 500
Given:
F = Php1, 000, 000
i = 7 % = 0.07
34
n = 45 yrs.
Required:
A
Solution:
F=
[
]
A (1 + i )n − 1
i




i
0.07
A = F
 = (1,000,000) 
 = Php3, 500
n
45
 (1 + i ) − 1
 (1 + 0.07 ) − 1
2. Suppose that you have Php10, 000 cash today and can invest it at 8 % compound interest each year. How
many years will it take you to become a millionaire?
Ans. n = 60 years
Given:
P = P10, 000
i = 8 % = 0.08
F = Php1, 000, 000
Required:
n
Solution:
F = P(1 + i )n
(1 + i )n
=
F
P
ln (1 + i )n = ln
F
P
n ln (1 + i ) = ln
F
P
 1,000,000 
F
ln

 10,000 
P
n=
=
= 59.84 yrs.
ln(1 + i )
ln (1 + 0.08)
ln
35
3. If Php500.00 is invested at the end of each year for 6 years, at an annual interest rate of 7%, what is the total
peso amount available upon the deposit of the sixth payment?
Ans. F = Php3, 576.64
Given:
A = Php500
n = 6 yrs.
i = 7 % = 0.07
Required:
F
Solution:
F=
[
]
A (1 + i )n − 1
i
=
[
]
(500) (1 + 0.07 )6 − 1
= Php3, 576.64
0.07
4. How much money must you invest today in order to withdraw Php1, 000 per year for 10 years if the interest
rate is 12%?
Ans. P = Php5, 650.22
Given:
A = Php1, 000
n = 10 yrs.
i = 12 % = 0.12
Required:
P
Solution:
P=
[
] = (1,000)[(1 + 0.12)10 − 1] = Php5, 650.22
A (1 + i )n − 1
i(1 + i ) n
(0.12)(1 + 0.12)10
5. J. Dela Cruz borrowed Php50, 000.00 from Social Security System, in the form of calamity loan, with interest
at 8% compounded quarterly payable in equal quarterly installments for 10 years. Find the quarterly
payments.
Ans. A = Php1, 827.79
36
Given:
P = Php50, 000.00
i=
NR 0.08
=
m
4
mN = (4)(10) = 40
NR = 0.08
n = mN = (4)(10)
Required:
A (quarterly payments)
Solution:
P=
A=
A=
[
]
A (1 + i )n − 1
i(1 + i ) n
[
P i(1 + i )n
]
(1 + i ) n − 1
mN 
 NR 
NR 

P
1 +

m 
 m 

NR 

1 +

m 

mN
−1
( 4)(10) 
 0.08 
0.08 

(50,000) (
) 1 +

4 
 4 

A=
= Php1, 827.79
( 4)(10)
0.08 

−1
1 +

4 

6. Engr. Pineda, a self- employed engineer wants to get a lump sum of 5 million pesos when he retires at the end of
25 years. How much in pesos, should he deposit every end of 3 months in a fund that give an interest of 10%
compounded quarterly to satisfy his desire?
Ans. A = PhP11, 559.39
Given:
F = PhP5, 000, 000.00
i = 0.10/4 = 0.025
37
n = (25)(4) = 100
Required:
A
Solution:
[
]
A (1 + i )n − 1
F=
i
A=
(5,000,000)(0.025)
Fi
=
= PhP11,559.39
n
(1 + i ) − 1
(1 + 0.025) 100 − 1
7. Engr. Aguinaldo wants her newly born son a lump sum of one million pesos when he starts college at the end of
17 years. To achieve this she deposits at the end of every month a certain amount in a fund that pays an interest
rate of 12% compounded monthly. How much is the monthly deposit?
Ans. A = PhP1, 512.15
Given:
F = PhP1, 000, 000.00
n = (17)(12) = 204
i = 0.12/12 = 0.01
Required:
A
Solution:
A=
(1,000,000)(0.01)
Fi
=
= PhP1,512.15
(1 + i ) n − 1
(1 + 0.01) 204 − 1
8. Engr. Enriquez creates an educational plan for her newly born daughter by depositing PhP1, 000.00 every end of
the month in a fund that pays a rate of 9% compounded monthly. How much lump sum could she withdraw from
the fund when her daughter enters college at the age of 17?
Ans. F = PhP478, 918.00
Given:
n = (17)(12) 204
i = 0.09/12 = 0.0075
A = PhP1, 000.00
Required:
F
Solution:
F=
[
]
A (1 + i )n − 1
i
38
F=
[
(1,000) (1 + 0.0075)
0.0075
204
] = PhP478,918.00
−1
Depreciation Concepts and Terminology
Depreciation is the decrease in value of physical properties with the passage of time and use. More specifically,
depreciation is an accounting concept that establishes an annual deduction against before-tax income such that
the effect of time and use on an asset’s value can be reflected in a firm’s financial statements. Annual
depreciation deductions are intended to “match” the yearly fraction of value used by an asset in the production
of income over the asset’s actual economic life. The actual amount of depreciation can never be established
until the asset is retired from service. Because depreciation is a noncash cost that affects income taxes, we must
consider it properly when making after-tax engineering economy studies.
Depreciable property is property for which depreciation is allowed under federal, state, or municipal income tax
laws and regulations. To determine if depreciation deductions can be taken, the classification of various types of
property must be understood. In general, property is depreciable if it meets the following basic requirements:
1. It must be used in business or held to produce income.
2. It must have a determinable useful life, and the life must be longer than one year.
3. It must be something that wears out, decays, gets used up, becomes obsolete, or loses value from natural
causes.
4. It is not inventory, stock in trade, or investment property.
Depreciable property is classified as either tangible or intangible. Tangible property can be seen or touched,
and it includes two main types called personal property and real property. Personal property includes assets
such as machinery, vehicles, equipment, furniture, and similar items. In contrast, real property is land and
generally anything that is erected on, or attached to land. Land itself, however, is not depreciable because it
does not have a determinable life.
Intangible property is personal property as a copyright, patent, or franchise. Engineering projects rarely include
this class of property.
A company can begin to depreciate it owns when the property is placed in service for use in the business or for
the production of income. Property is considered to be placed in service when it is ready and available for a
specific use, even if it is not actually used yet. Depreciation stops either when the cost of placing it in service
has been recovered or it is retired from service.
Depreciation Methods and Related Time Periods
The depreciation methods permitted under the Internal Revenue Code have changed with time. In general, the
following summary indicates the primary methods used for property placed in service during three distinct time
periods.
The primary methods used were straight-line (SL), declining balance (DB), and sum-of-the-digits
(SYD). We will refer to these methods, collectively, as the classical or historical methods of depreciation.
39
Adjusted (cost) basis – the original cost basis of the asset, adjusted by allowable increases or decreases, is used
to compute depreciation and depletion deductions. For example, the cost of any improvement to a capital asset
with a useful life greater than one year increases the original cost basis, and a casualty or theft loss decreases it.
If the basis is altered, the depreciation deduction may need to be adjusted.
Basis, or cost basis – the initial cost of acquiring an asset (purchase price plus any sales taxes), including
transportation expenses and other normal costs of making the asset serviceable for its intended use. This amount
is also called the unadjusted cost basis.
Book value (BV) – the worth of a depreciable property as shown on the accounting records of a company. It is
the original cost basis of the property, including any adjustments, less all allowable depreciation or depletion
deductions. It thus represents the amount of capital that remains invested in the property and must be recovered
in the future through the accounting process. The BV of a property may not be a useful measure of its market
value.
Market value (MV) – the amount that will be paid by a willing seller for a property where each has equal
advantage and is under no compulsion to buy or sell. The MV approximates the present value of what will be
received through ownership of the property, including the time value of money (or profit).
Recovery period – the number of years over which the basis of a property is recovered through the accounting
process. For the classical methods of depreciation, this period is normally the useful life. Under the Modified
Accelerated Cost Recovery System (MACRS), this period is the property class for the General Depreciation
System (GDS), and it is the class life for the Alternative Depreciation System (ADS).
Recovery rate – a percentage (expressed in decimal form) for each year of the MACRS recovery period that is
utilized to compute an annual depreciation deduction.
Salvage value – the estimated value of a property at the end of its useful life. It is the expected selling price of a
property when the asset can no longer be used productively by its owner. The term net salvage value is used
when the owner will incur expenses in disposing of the property, and these cash outflows must be deducted
from the cash inflows to obtain a final net SV. When the classical methods of depreciation are applied, an
estimated salvage value is initially established and used in the depreciation calculations. Under MACRS, the SV
of depreciable property is defined to be zero.
Useful life – the expected (estimated) period of time that a property will be used in a trade or business or to
produce income. It is not how long the property will last but how long the owner expects to productively use it.
Useful life is sometimes referred to as depreciable life. Actual useful life of an asset, however, may be different
than its depreciable life.
MULTIPLE CHOICE QUESTIONS
1. The paper currency issued by the central Bank, which forms part of the country’s money supply.
A.
B.
C.
D.
Bank note *
Check
Coupon
T-bills
2. Reduction in the level of national income and output usually accompanied by the fall in the general price level.
A. Deflation *
B. Depreciation
40
C. Devaluation
D. Inflation
3. It is a series of equal payments occurring at equal interval of time.
A Amortization
B. Annuity *
C. Dept
D. Deposit
4. The place where buyers and sellers come together.
A. Business
B. Buy and sell section
C. Market *
D. Recreation center
5. A market whereby there is only one buyer of an item for which there are no goods substitute.
A. Monopoly
B. Monopsony *
C. Oligopoly
D. Oligopsony
6. It is a series of equal payments occurring at equal interval of time where the first payment is made after several
periods, after the beginning of the payment.
A. Annuity due
B. Deferred annuity *
C. Ordinary annuity
D. Perpetuity
7. The total income equals the total operating cost.
A. Balanced sheet
B. Break even – no gain no loss *
C. Check and balance
D. In-place value
8. Kind of obligation which has no condition attached.
A. Analytic
B. Gratuitous *
C. Private
D. Pure
9. Direct labor costs incurred in the factory and direct material costs are the costs of all materials that go into
production. The sum of these two direct costs is known as
A. GS and A expenses
B. Operating and maintenance costs
C. O and M costs
41
D. Prime cost *
10. An index of short term paying ability is called
A. acid-test ratio *
B. current ratio
C. profit margin ratio
D. receivable turn-over
11. Artificial expenses that spreads the purchase price of an asset or another property over a number of years.
A. Amnesty
B. Bond
C. Depreciation *
D. Sinking fund
12. Estimated value at the end of the useful life.
A. Book value
B. Fair value
C. Market value
D. Salvage value *
13. Consists of the actual counting or determination of the actual quantity of the materials on hand as of a given
date.
A. Material count
B. Material update
C. Physical inventory *
D. Technological assessment
14. Additional information of prospective bidders on contact documents issued prior to bidding date.
A. Bid bulletin *
B. Delict
C. Escalatory
D. Technological assessment
15. A series of uniform accounts over an infinite period of time.
A. Annuity
B. Depreciation
C. Inflation
D. Perpetuity *
16. The quantity of a certain commodity that is offered for sale at a certain price at a given place and time.
A. Demand
B. Goods
C. Stocks
D. Supply *
42
17. Work-in process is classified as
A. an asset *
B. an expenses
C. An owner’s equity
D. a liability
18. What is the highest position in the corporation?
A. Board of Directors
B. Chairman of the Board *
C. President
D. Stockholders
19. Type of ownership in business where individuals exercise and enjoy the right in their own interest.
A. Equitable
B. Public
C. Private *
D. Pure
20. Decrease in the value of a physical property due to the passage of time.
A. Depletion
B. Depreciation *
C. Inflation
D. Recession
21. An association of two or more individuals for the purpose of operating a business as co-owners for profit.
A. Company
B. Corporation
C. Partnership *
D. Sole proprietorship
22. We may classify an interest rate, which specifies the actual rate of interest on the principal for one year as
A. effective rate *
B. exact interest rate
C. nominal rate
D. rate of return
23. It is defined to be the capacity of a commodity to satisfy human want.
A. Discount
B. Luxury *
C. Necessity
D. Utility
43
24. It is the amount which a willing buyer will pay to a willing seller for a property where each has equal advantage
and is under no compulsion to buy or sell.
A. Book value
B. Fair value
C. Market value *
D. Salvage value
25. This occurs in a situation where a commodity or service is supplied by a number of vendors and there is nothing
to prevent additional vendors entering the market.
A. Elastic demand
B. Monopoly
C. Oligopoly
D. Perfect competition *
26. These are products or services that are desired by human and will be purchased if money is available after the
required necessities have been obtained.
A. Luxuries *
B. Necessities
C. Product goods and services
D. Utilities
27. These are products or services that are required to support human life and activities that will be purchased in
somewhat the same quantity even though the price varies considerably.
A. Luxuries
B. Necessities *
C. Product goods and services
D. Utilities
28. A condition where only few individuals produce a certain product and that any action of one will lead to almost
the same action of the others.
A. Monopoly
B. Oligopoly *
C. Perfect competition
D. Semi-monopoly
29. Grand total of the assets and operational capability of a corporation.
A. Authorized capital *
B. Investment
C. Money market
D. Subscribed capital
30. The worth of the property equals to the original cost less depreciation.
A. Book value *
B. Face value
44
C. Market value
D. Scrap value
31. Money paid for the use of borrowed capital.
A. Credit
B. Interest *
C. Discount
D. Profit
32. Liquid assets such as cash and other assets that can be converted quickly into cash, such as accounts receivable
and merchandise are called
A. current assets *
B. Fixed assets
C. total assets
D. None of the above
33. The length of time which the property may be operated at a point.
A. Economic life *
B. Operating life
C. Physical life
D. All of the above
34. The provision in the contract that indicates the possible adjustment of material cost and labor cost.
A. Contingency clause
B. Escalatory clause *
C. Main clause
D. Secondary clause
35. The present worth of all depreciation over the economic life of the item is called
A. book value
B. capital recovery
C. depreciation recovery *
D. sinking fund
36. Gross profit, sales less cost of goods sold, as percentage of sale is called
A. gross margin *
B. net income
C. profit margin
D. rate of return
37. Worth of the property as shown in the accounting records of an enterprise.
A. Book value *
B. Fair value
C. Market value
45
D. Salvage value
38. Those funds that are required to make the enterprise or project a going concern.
A. Current accounts
B. Initial investment
C. Substantial capital
D. Working capital *
39. A market situation where there is only one seller with many buyer.
A. Monopoly *
B. Monopsony
C. Oligopoly
D. Oligopsony
40. A market situation where there are few sellers and few buyers.
A. Bilateral oligopoly *
B. Bilateral oligopsony
C. Oligopoly
D. Oligopsony
41. A market situation where there is one seller and one buyer.
A. Bilateral monopoly *
B. Bilateral monopsony
C. Monopoly
D. Monopsony
42. A market situation where there are only two buyers with many sellers.
A. Duopoly
B. Duopsony *
C. Oligopoly
D. OLigopsony
43. The cumulative effect of elapsed time on the money value of an event, based on the earning power of equivalent
invested funds capital should or will earn.
A. Interest rate
B. Present worth factor
C. Time value of money *
D. Yield
44. Defined as the future value minus the present value.
A. Capital
B. Discount *
C. Interest
D. Rate of return
46
45. The flow back of profit plus depreciation from a given project is called
A. cash flow *
B. capital recovery
C. earning value
D. economic return
46. The profit derived from a project or business enterprise without consideration of obligations to financial
contributors or claims of other based on profit.
A. Earning value
B. Economic return *
C. Expected yield
D. Yield
47. The payment for the use of borrowed money is called
A. interest *
B. loan
C. maturity value
D. principal
48. The interest rate at which the present work of the cash flow on a project is zero of the interest earned by an
investment.
A. Effective rate
B. Nominal rate
C. Rate of return *
D. Yield
49. The ratio of the interest payment to the principal for a given unit of time and usually expressed as a percentage
of the principal.
A. Interest
B. Interest rate *
C. Investment
D. All of the above
50. The true value of interest rate computed by equations for compound interest for a 1 year period is known as
A. effective interest *
B. expected return
C. interest
D. nominal interest
51. The intangible item of value from the exclusive right of a company to provide a specific product or service in a
stated region of the country.
A. Book value
B. Franchise value *
47
C. Goodwill value
D. Market value
52. The recorded current value of an asset is known as
A. book value *
B. present value
C. salvage value
D. scrap value
53. Scrap value of an asset is sometimes known as
A. book value
B. future value
C. replacement value
D. salvage value *
54. Sometimes called second hand value.
A. Book value
B. Going value
C. Salvage value *
D. Scrap value
55. An intangible value which is actually operating concern has due to its operation.
A. Book value
B. Fair value
C. Going value *
D. Goodwill value
56. The value which a disinterested third party, different from the buyer and seller, will determine in order to
establish a price acceptable to both parties.
A. Fair value *
B. Franchise value
C. Goodwill value
D. Market value
57. A type of annuity where the payments are made at the end of each payment period starting from the first period.
A. Annuity due
B. Deferred annuity
C. Ordinary annuity *
D. Perpetuity
58. It is a series of equal payments occurring at equal intervals of time where the first payment is made after several
periods, after the beginning of the payment.
A. Deferred annuity *
B. Delayed annuity
48
C. Progressive annuity
D. Simple annuity
59. A type of annuity where the payments are made at the start of each period, beginning from the first period.
A. Annuity due *
B. Deferred annuity
C. Ordinary annuity
D. Perpetuity
60. Which is NOT an essential element of an ordinary annuity?
A. the amounts of all payments are equal.
B. The payments are made at equal interval of time.
C. The first payment is made at the beginning of each period. *
D. Compound interest is paid on all amounts in the annuity.
61. “A” is a periodic payment and “i” is the interest rate, then present worth of a perpetuity =
A. Ai
B. Ain
C. An/i
D. A/i *
62. A mathematical expression also known as the present value of an annuity of one called
A. demand factor
B. load factor
C. present worth factor*
D. sinking fund factor
63. As applied to a capitalized asset, the distribution of the initial cost by a periodic changes to operation as in
depreciation or the reduction of a dept by either periodic or irregular prearranged program is called
A. amortization*
B. annuity
C. annuity factor
D. capital recovery
64. The reduction of the value of an asset due to constant use and passage of time.
A. Book value
B. Depletion
C. Depreciation *
D. Scrap value
65. A method of computing depreciation in which the annual charge is a fixed percentage of the depreciated book
value at the beginning of the year to which the depreciation applies.
A. Declining balance method *
B. Sinking fund method
49
C. Straight line method
D. SYD method
66. A method of depreciation whereby the amount to recover is spread uniformly over the estimated life of the asset
in terms of the periods or units of output.
A. Declining balance method
B. Sinking fund method
C. Straight line method *
D SYD method
67. Which of the following depreciation methods cannot have a salvage value of zero?
A. Declining balance method *
B. Sinking fund method
C. Straight line method
D. SYD method
68. A method of depreciation where a fixed sum of money is regularly deposited at compound interest in a real or
imaginary fund in order to accumulate an amount equal to the total depreciation of an asset at the end of the
asset’s estimated life.
A. Declining balance method
B. Sinking fund method *
C. Straight line method
D. SYD method
69. The function of interest rate and time that determines the cumulative amount of a sinking fund resulting from
specific periodic deposits.
A. Capacity factor
B. Demand factor
C. Present worth factor
D. Sinking fund factor *
70. The first cost of any property includes
A. the original purchase price and freight and transportation charges
B. installation expenses
C. initial taxes and permits fee
D. all of the above *
71. In SYD method, the sum of years digit is calculated using which formula with n = number of useful years of the
equipment.
A. n(n-1)
2
B. n(n+1) *
2
C. n(n+1)
50
D. n(n-1)
72. Capitalized cost of any property is equal to the
A. annual cost
B. first cost + cost of perpetual maintenance *
C. first cost + interest of the first cost
D. first cost + salvage value
73. The lessening of the value of an asset due to the decrease in the quantity available (referring to the natural
resources, coal, oil, etc).
A. Depletion *
B. depreciation
C. Incremental cost
D. Depreciation
74. Is the simplest form of business organization.
A. Corporation
B. Enterprise
C. Partnership
D. Sole proprietorship *
75. An association of two or more persons for a purpose of engaging in a profitable business.
A. Corporation
B. Enterprise
C. Partnership *
D. Sole proprietorship
76. A distinct legal entity which can practically transact any business transaction which a real person could do.
A. Corporation *
B. Enterprise
C. Partnership
D. Sole proprietorship
77. Double taxation is a disadvantage of which business organization?
A. Corporation *
B. Enterprise
C. Partnership
D. Sole proprietorship
78. Which is NOT a type of business organization?
A. Corporation
B. Enterprise *
C. Partnership
D. Sole proprietorship
51
79. What is the minimum number of incorporators in order that be organized?
A. 3
B. 5 *
C. 10
D. 7
80. In case of bankruptcy of a partnership,
A. the partners are not liable for the liabilities of the partnership.
B. the partnership assets (excluding the partners’ personal assets) only will be used to pay the liabilities.
C. the partners personal assets are attached to the debt of the partnership. *
D. the partners may sell stock to generate additional capital.
81. Which is TRUE about partnership?
A. It has a perpetual life.
B. It will be dissolved if one of the partners ceases to be connected with the partnership. *
C. It can be handed down from one generation of partners to another.
D. Its capitalization must be equal for each partner.
82. Which is TRUE about corporation?
A. It is not the best form of business organization.
B. The minimum number of incorporators to start a corporation is three.
C. Its life is dependent on the lives of the incorporators.
D. The stockholders of the corporation are only liable to the extent of their investments. *
83. Represent ownership, and enjoys certain preferences than ordinary stock.
A. Authorized capital stock
B. Common stock
C. Incorporator’s stock
D. Preferred stock *
84. Represent the ownership of stockholders who have a residual claim on the assets of the corporation after all
other claims have been settled.
A. Authorized capital stock
B. Common stock *
C. Incorporator’s stock
D. Preferred stock
85. The amount of company’s profits that the board of directors of the corporation decides to distribute to ordinary
shareholders.
A. Dividend *
B. Par value
C. Return
D. Share stock
52
86. A certificate of indebtness of a corporation usually for a period not less than 10 years and guaranteed by a
mortgage on certain assets of the corporation.
A. Bond *
B. Common stock
C. Preferred stock
D. T-bill
87. A form of fixed-interest security issued by central or local government, companies, banks or other institutions.
They are usually a form of long-term security, buy may be irredeemable, secured or unsecured.
A. Bonds *
B. Certificate of deposit
C. T-bills
D. All of these
88. A type of bond where the corporation pledges securities which it owns (i.e., stocks, bonds of its subsidiaries).
A. Collateral trust bond *
B. Coupon bond
C. Mortgage bond
D. Registered bond
89. A type of bond which does not have security except a promise to pay by the issuing corporation.
A. Collateral trust bond
B. Debenture bond *
C. Mortgage bond
D. Registered bond
90. A type of bond issued jointly by two or more corporations.
A. Collateral trust bond
B. Debenture bond
C. Joint bond *
D. Registered bond
91. A type of bond whose guaranty is in lien on railroad equipments.
A. Debenture bond
B. Equipment obligations bond *
C. Infrastructure bond
D. Registered bond
92. If the security of the bond is a mortgage on certain specified asset of a corporation, this bond is classified as
A. coupon bond
B. joint bond
C. mortgage bond *
D. registered bond
53
93. A type of bond where the corporation’s owners name are recorded and the interest is paid periodically to the
owners with their asking for it.
A. Incorporated bond
B. Preferred bond
C. Registered bond *
D. All of these
94. Bond to which are attached coupons indicating the interest due and the date when such interest is to be paid.
A. Collateral trust bond
B. Coupon bond *
C. Mortgage bond
D. Registered bond
95. An amount of money invested at 12% interest per annum will double in approximately
A. 4 years
B. 5 years
C. 6 years *
D. 7 years
96. The 72 rule of thumb is use to determine
A. how many years money will triple
B. how many years of money will double *
C. how many years to amass 1 million
D. how many years to quadruple the money
97. To triple the principal, one must use
A. derivatives
B. implicit functions
C. logarithms *
D. integration
98. A currency traded in a foreign exchange market for which the demand is consistently high in relation to its
supply
A. Certificate of deposit
B. Hard currency *
C. Money market
D. Treasury bill
99. Everything a company owns and which has a money value is classified as an asset. Which of the following is
classified as an asset?
A. Fixed assets
B. Intangible assets
C. Trade investments
D. All of these *
54
100. Which an example of an intangible asset?
A. Cash
B. Furnitures
C. Investment in subsidiary companies
D. Patents *
101. Lands, buildings, plant and machinery are examples of
A. current assets
B. fixed assets *
C. intangible assets
D. trade investments
102. An increase in the value of a capital asset is called
A. capital expenditure
B. capital gain*
C. capital stock
D. profit
103. The reduction in the money value of a capital asset is called
A. capital expenditure
B. capital loss *
C. deficit
D. loss
104. It is a negotiable claim issued by a bank in lieu of a term deposit.
A. Bond
B. Capital gain
C. Certificate of deposit *
D. Time deposit
105. Any particular raw material or primary product (e.g. cloth, wood, flour, coffee…) is called
A. commodity *
B. necessity
C. stock
D. utility
106. It denotes the fall in the exchange rate of one currency in terms of others. The term usually applies to floating
exchange rates.
A. Currency appreciation
B. Currency devaluation
C. Currency depreciation *
D. Currency float
55
107. The deliberate lowering of the price of a nation’s currency in terms of the accepted standard (Gold, American
dollar or the British pound).
A. Currency appreciation
B. Currency devaluation *
C. Currency depreciation
D. Currency float
108. The residual value of a company’s assets after all outside liabilities (shareholders excluded) have been allowed
for.
A. Dividend
B. Equity *
C. Par value
D. Return
109. A saving which takes place because goods are not available for consumption rather than the consumer really
want to save.
A. Compulsory saving
B. Consumer saving
C. Forced saving *
D. All of these
110. A document that shows proof of legal ownership of a financial security.
A. Bank note
B. Bond
C. Check
D. Coupon *
111. Defined as the capacity of commodity to satisfy human want.
A. Discount
B. Luxuries
C. Necessity
D. Utility *
112. It is the profit obtained by selling stocks at a higher price than its original purchase price.
A. Capital gain *
B. Debenture
C. Goodwill
D. Internal rate of return
113. The quantity of a certain commodity that is offered for sale at a certain price at a given time and place.
A. Demand
B. Market
C. Supply *
D. Utility
56
114. The quantity of a certain commodity that is bought at a certai8n price at a given time and place.
A. Demand *
B. Market
C. Supply
D. Utility
115. “When one of the factors of production is fixed in quantity or is difficult to increase, increasing the other
factors of production will result in a less than proportionate increase in output”.
A. Law of demand
B. Law of diminishing return *
C. Law of supply
D. Law of supply and demand
116. “When free competition exists, the price of a product will be that value where supply is equal to the demand”.
A. Law of demand
B. Law of diminishing return
C. Law of supply
D. Law of supply and demand *
117. An accounting term that represents an inventory account adjustment.
A. Cost of good sold *
B. Overhead
C. Payback
D. Variance
118. The simplest economic order quantity (EOQ) model is based on which of the following assumptions?
A. Shortages are not allowed.
B. Demand is constant with respect to time.
C. Reordering is instantaneous. The time between order placement and receipt is zero.
D. All of the choices *
119. In economics, a “short-term” transaction usually has a lifetime of
A. 3 months or less
B. 1 year or less
C. 5 years or less *
D. 10 years or less
120. In the cash flow, expenses incurred before time = 0 is called
A. disbursements
B. first costs
C. receipts
D. sunk costs *
57
121. An imaginary cost representing what will not be received if a particular strategy is rejected.
A. Initial cost
B. Opportunity cost *
C. Replacement cost
D. Sunk cost
122. In replacement studies, the existing process or piece of equipment is known as
A. asset
B. challenger
C. defender *
D. liability
123. In replacement studies, the new process or piece of equipment being considered for purchase is known as
A. asset
B. challenger *
C. defender
D. liability
124. ______ means that the cost of the asset is divided into equal or unequal parts, and only one of these parts is
taken as an expense each year.
A. Artificial expense
B. Capitalizing the asset *
C. Depreciating the asset
D. Expensing the asset
125. Indicate the CORRECT statement about depreciation.
A. The depreciation is not the same each year in straight line method.
B. The declining balance method can be used even if the salvage value is zero.
C. The sum-of-years’ digit method (SYD), the digits 1 to (n + 1) is summed.
D. Double declining balance depreciation is independent of the salvage value. *
126. An artificial deductible operating expense designed to compensate mining organizations for decreasing
mineral reserves.
A. Deflation
B. Depletion *
C. Inflation
D. Reflation
127. The change in cost per unit variable change is known as
A. fixed cost
B. incremental cost *
C. semi-variable cost
D. sunk cost
58
128. What type of cost increases step-wise?
A. Direct labor cost
B. Operating and maintenance cost
C. Semi-variable cost *
D. Supervision cost
129. Which of the following is NOT a variable cost?
A. Cost of miscellaneous supplies
B. Income taxes
C. Insurance cost *
D. Payroll benefit costs
130. Which of the following is NOT a fixed cost?
A. Depreciation expenses
B. Janitorial service expenses
C. Rent
D. Supervision costs *
131. The annual costs that are incurred due to the functioning of a piece of equipment is known as
A. General, selling and administrative expenses
B. Operating and maintenance costs *
C. Prime cost
D. Total cost
132. The sum of the direct labor cost and the direct material cost is known as
A. indirect manufacturing expenses
B. marketing cost
C. prime cost *
D. total cost
133. Research and development costs and administrative expenses are added to the factory cost to give the
___________ of the product.
A. manufacturing cost *
B. marketing cost
C. prime cost
D. total cost
134. The sum of the prime cost and the indirect manufacturing cost is known as
A. factory cost *
B. manufacturing cost
C. research and development cost
D. total cost
135. The manufacturing cost plus selling expenses or marketing expanses equals
59
A. administrative cost
B. indirect production cost
C. miscellaneous cost
D. total cost *
136. Which of the following is NOT a direct labor expense?
A. Assembly
B. Inspection
C. Supervision *
D. Testing
137. All are administrative expense EXCEPT:
A. Accounting
B. Data processing
C. Marketing *
D. Office supplies
138. On of the following is NOT a selling or marketing expense. Which one?
A. Advertising
B. Commission
C. Insurance *
D. Transportation
139. Research and development expenses includes all EXCEPT one. Which one?
A. Drafting
B. Laboratory *
C. Prototype
D. Testing
140. Which is not a factory overhead expense?
A. Expediting
B. Pension, medical, vacation benefits
C. Quality control and inspection
D. Testing *
141. Bookkeeping consists of two steps, namely recording the transactions and categorization of transactions.
Where are the transactions (receipts and disbursements) recorded?
A. Columnar
B. Journal *
C. Ledger
D. Statement of account
142. The following are ledger accounts EXCEPT:
A. Asset accounts
60
B. Bank accounts *
C. Liability accounts
D. Owner’s equity accounts
143. The journal and the ledger together are known simply as _____________ of the company.
A. accounting system
B. balance sheet
C. bookkeeping system
D. the books *
144. The basic accounting equation is
A. Assets = Liability + Owner’s equity *
B. Liability = Assets + Owner’s equity
C. Owner’s equity = Assets + Liability
D. Owner’s equity = Liability – Assets
145. The ability to convert assets to cash quickly is known as
A. insolvency
B. leverage
C. liquidity *
D. solvency
146. The ability to meet debts as they become due is known as
A. insolvency
B. leverage
C. liquidity
D. solvency *
147. What is considered as an index of short-term paying ability?
A. Acid test ratio
B. Current ratio *
C. Gross margin
D. Return of investment
148. An acid test ratio is a ratio of
A. gross profit to net ratio
B. net income before taxes to net sales
C. net income to owner’s equity
D. quick assets to current liabilities *
149. The ratio of the net income to the owner’s equity is known as
A. gross margin
B. price-earning ratio
C. profit margin ratio
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D. return of investment *
150. Payback period is the ratio of
A. cost of goods sold to average cost of inventory on hand
B. gross profit to net sales
C. initial investment to net annual profit *
D. net income before taxes to net sale
151. A secondary book of accounts, the information of which is obtained from the journal.
A. Balance sheet
B. Ledger *
C. Trial balance
D. Worksheet
152. The present worth of cost associated with an asset for an infinite period of time is referred to as
A. annual cost
B. capitalized cost *
C. increment cost
D. operating cost
153. A stock of a product which is held by a trade or government as a means of regulating the price of that product.
A. Buffer stock *
B. Hoard stock
C. Stock pile
D. Withheld stock
154. A negotiable claim issued by a bank in lieu of a term deposit is called
A. Certificate of deposit *
B. Cheque
C. Currency
D. T-bills
155. A form of business firm which is owned and run by a group of individuals for their mutual benefit.
A. Cooperative *
B. Corporation
C. Enterprise
D. Partnership
156. A document which shows the legal ownership of financial security and entitled to payments thereon.
A. Bond
B. Consol
C. Contract
D. Coupon *
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157. A government bond which have an indefinite life rather than a specific maturity.
A. Debenture
B. Consol *
C. Coupon
D. T-bill
158. Refers to the order quantity that minimizes the inventory cost per unit time.
A. Economic order quantity *
B. Private order quantity
C. Public order quantity
D. Social order quantity
159. What is referred to as an individual who organizes factors of production to undertake a venture with a view to
profit?
A. Agent
B. Commissioner
C. Entrepreneur *
D. Salesman
160. The money that is inactive and does not contribute to productive effort in an economy is known as
A. frozen asset
B. hard money
C. idle money *
D. soft money
161. Find the simple interest on Php8,000 at 5.5 % simple interest for 4 years and 9 months.
Given:
P = Php8, 000
i = 5.5 % = 0.055
n = 4 years and 9 months = 4.75 years
Required:
I
Solution:
I = Pin = (8,000) (0.055) (4.75) = Php2,090
162. The interest on Php10, 500 is Php590.625, invested for 9 months. What is the rate of interest?
Given:
P = Php10,500
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I = Php590.625
n = 9 months x 1 yr. / 12 months = 0.75 yr.
Required:
i
Solution:
I = Pin
i=
I
590.625
=
= 0.075 or 7.5 %
Pn
(10,500)(0.75)
163. A principal earns interest of Php3, 166.67 for 2 years and 6 months at a simple interest rate of 12.67 %. Find the
principal.
Given:
I = Php3, 166.67
n = 2.5 years
i = 12.67 % = 0.1267
Required:
P
Solution:
I = Pin
P=
I
3,166.67
=
= Php1, 000.00
in
(0.1267)( 2.5)
164. How many years will it take Php10, 000 to earn Php1, 500 is invested at 8.5 %, simple interest?
Given:
P = Php10, 000
I = Php1, 500
i = 8.5 % = 0.085
Required:
n
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Solution:
I = Pin
n=
I
1,500
=
= 1.7647 yrs.
Pi
(10,000)(0.085)
165. Discount Php18, 500 for 5.5 years at 3.5 % simple interest.
Given:
F = Php18, 500
n = 5.5 yrs.
i = 3.5 % = 0.035
Required:
P
Solution:
P=
F
18,500
=
= Php15, 513.63
1 + in
1 + (0.0350(5.5)
166. What is the present value of a loan whose maturity value is Php30, 000 at 12 % from August 9,
2003 to December 25, 2003? (Use Banker’s rule).
Given:
F = Php30, 000
i = 12 % = 0.12
Required:
P
Solution:
Solving for the total number of days, d:
August 9 to 31
= 22
65
September
October
November
December 1 to 25
n=
= 30
= 31
= 30
= 25
______________
138 days
d
138
=
360 360
30,000
= Php28, 680.69
 138 
1 + (0.12)

 360 
167. Accumulate Php4, 000 for 30 years at 6 % compounded monthly.
P=
F
=
1 + in
Given:
P = Php4, 000
i = 6 % /12 = 0.06 / 12
n= 30 years x 12 months / 1 year = 360 months
Required:
F
Solution:
 0.06 
F = P(1 + i )n = ( 4,000)1 +


12 
360
= Php24, 090.30
168. Find the present value of Php2, 500 due in 10 years if money is worth 16 % compounded quarterly.
Given:
F = Php2, 500
i = 16 % /4 = 0.16 / 4
n= 10 years x 4 months / 1 year = 40 months
Required:
P
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Solution:
 0.16 
P = F(1 + i )− n = ( 2,500)1 +

4 

− 40
= Php520.72
169. If money is worth 7.5 % compounded monthly, find the discount if Php200, 000 is discounted for
30 years.
Given:
F = Php200, 000
i = 7.5 % /12 = 0.075 / 12
n= 30 years x 12 months / 1 year = 360 months
Required:
P
Solution:
 0.075 
P = F(1 + i )− n = ( 200,000)1 +

12 

− 360
= Php21, 227.97
170. What rate compounded quarterly will Php200 amount to Php350 in 4 years?
Given:
F = Php350
P = Php200
n= 4 years x 4 months / 1 year = 16 months
Required:
i
Solution:
F = P(1 + i )n
F
= (1 + i )n
P
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ln
F
= n ln (1 + i )
P
F
ln(1 + i) = P
n
ln
 F
 ln 
 P
 n 



1 + i = e
 F
 ln 
 P
 n 


 −1
i = e
 350 
 ln

 200 
 16 


i
 −1
= e
4
 350 
 ln

 200 
 16 


 − 1)
i = ( 4)(e 
i = 0.1424 or 14.24 %
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