Introduction to Everyday Math
Now In this section we are going to discuss along with the application of some other operations of arithmetic
like Ratio, Proportion, Rates, Percentage, which are very helpful to solve everyday problems and business
related problems.
Concept of Ratio
Ratio:
Sometimes, we have to deal with the quantities whose comparison is required. Suppose in a class of
45-students, 15 of the students are girls, 30 of the students are boys, and then we can compare the number of
boys with number of girls in two different ways.
1. There are 15 more boys than girls in the class. In this case we are comparing the number of boys and
number of girls by finding their difference.
2. The number of boys is twice the number of girls in the class. Here, we are comparing the number of
boys and number of girls by finding a fraction consisting of the number of boys over the number of
girls. i.e., fraction is
“Ratio is fraction of two quantities of the same kind”
In other words
“Ratio is the comparison of two homogeneous quantities” or ” Ratio is the division of two
quantities and having the same units”
It is denoted by
or
(read as “a ratio b”) or
The ratio has no unit. It is a numerical quantity which indicates how many times one quantity is greater than
the other.
e.g. The boy : girl, i.e. 30 : 15 or 2 : 1 indicates that the number of boys is twice the number of girls.
e.g. The girl : boy, i.e. 1 : 2 or indicates that there are half girls than boys.
Example:
Find the ratio of
(1) 50g to 200g
(2) 700g to 1kg
Solution:
1. The ratio of 50g to 200g can be found in two ways.
50 : 200
=
or
or
50 : 200
(divided by 50)
=
or
1:4
=1 : 4
Hence, ratio of 50g to 200g is 1 to 4.
2. 700g and 1kg have different units. First, we express them in the same units we know that
700g : 1kg
=700g : 1000g (since 1kg=1000g)
=700 : 1000 (divided by 100)
=7 : 10
Hence, ratio of 700g to 1kg is 7 to 10.
Example:
Suppose three men receive their profits as 4000, 3000 and 1000. Find the ratio between their profits.
Solution:
The ratio of their share (profit) is
4000 : 3000 : 1000
=4:3:1
(divided by 1000)
Example:
Simplify the ratio of 3.5 : 2.5
Solution:
We are given that
3.5 : 2.5
=
= 35 : 25
=7:5
(multiply by 10)
(Dividing by 5)
Example:
Express in the form of
(1) 25, 45
(2) 30min, 1hour
Solution:
and fraction
1. 25, 45
= 25 : 45
=5 : 9
(Divided by 5)
And in fraction form we have
2. 30min, 1hour
30min : 1hour
(since 60min=1hour)
=
h : 1h
(1min=
=
:1
(30min=
hour)
hour= h)
=1:2
(Multiply by 2)
And in fraction form we have
Uses of Ratio and Continued Ratio
Uses of Ratio:
Ratio is used for calculating continued ratio, proportion, rates, percentage as well as continued
proportion. Ratio can also be used for dividing the profit or amount between two or more people.
For this we have the following process:
1. First, we find sum of given ratios.
2. Share can be divided by the following formula.
Share of a person= (given amount) (Ratio/Sum of ratios)
Continued Ratio:
So far, we have learnt the method of comparing two quantities of the same kind. But there may be the
situation when we have to compare more than two quantities, which are in a continued ratio.
e.g. Suppose that Rs. 74000 are to be divided among three friends A, B, C such that
A : B = 4 : 5 and B : C = 3 : 2
Sum of ratio = 12 + 15 + 10 = 37
Share of A =
x 74000 = 12 x 2000 = 24000
Share of B =
x 74000 = 15 x 2000 = 30000
Share of C =
x 74000 = 10 x 2000 = 20000
Example:
Divide 60000 in ratio 5 : 7
Solution:
Let A and B be two persons having ratio of share 5 : 7
Sum of ratios = 5 + 7 = 12
Share of A =
x 60000 = 25000
Share of B =
x 60000 = 35000
Example:
Three partners invested Rs. 12500, 9000 and 7500 respectively. If the total profit earned Rs. 5800,
how much profit each partner will receive.
Solution:
Let A, B and C be three partners
A invested = 12500
B invested = 9000
C invested = 7500
Profit = 5800
Ratio among their investments is
Sum of ratios = 25 + 18 + 15 = 58
Share of A =
x 5800 = 2500
Share of B =
x 5800 = 1800
Share of C =
x 5800 = 1500
Example:
A profit of Rs. 8920 is to be divided among four persons in the ratios 5 : 9 : 11 : 15 respectively. How
much does partner get?
Solution:
Given Profit = 8920
Ratio between Share = 5 : 9 : 11 : 15
Sum of ratios = 5 + 9 + 11 + 15 = 40
1st Partner's Share =
x 8920 = 1115
2nd Partner's Share =
x 8920 = 2007
3rd Partner's Share =
x 8920 = 2453
4th Partner's Share =
x 8920 = 3345
Examples Uses of Ratio and Continued Ratio
Example:
A, B and C starts business from 5000 for 1 year, 6000 for 9 months and 3000 for 7 months. They
earned profit 3375. Find share of each.
Solution:
Let total profit = 3375
Ratio of Investments
Ratio between their periods
Ratio of investments with respect to time
Sum of ratios = 20 + 18 + 7 = 45
Share of A =
x 3375 = 1500
Share of B =
x 3375 = 1350
Share of C =
x 3375 = 525
Example:
Three partners invested Rs. 16000, 14500 and 10500 respective. The profit was distributed among the
partners. If the third partner got Rs. 273 as profit, what was (a) total profit (b) profit of 1st and 2nd partner?
Solution:
Let total profit = P
Let A, B, C be three partners.
Ratio between their partner investments
Sum of ratio = 32 + 29 + 21 = 82
Profit receive by third partner =
273 =
xP
xP
Total Profit = 1066
Share of 1st partner =
x 1066 = 416
Share of 2nd partner =
x 1066 = 377
Example:
A servant receives house rent allowance
of his basic pay. Medical allowance
of his basic pay.
If he receives Rs. 450 as medical allowance, find his total basic pay an his house rent allowance.
Solution:
Let basic pay =
Let amount received as medical allowance = 450
But medical allowance =
(basic pay)
=
= 9000 Rs
House rent allowance =
x 9000 = 900 Rs
Increase and Decrease in Ratio
Decrease and Increase in Ratio:
If the no. of teachers in a college is increased from 50 to 60. Then the ratio of new staff and old staff
is
i.e.
We say that, no. of teachers has been increased in ratio 6 : 5. In other words, no. of new staff is
no. of old staff.
Hence
Rule:
To increase a no. “ ” we multiply
by an improper fraction.
Similarly to decrease a no. “ ” we multiply
e.g. (1) Increase Rs. 20 in ratio 6 : 5
New value =
e.g. (2) Decrease 56 in ratio 7 : 8
New value =
Concept of Rate
by an proper fraction.
times the
Rate:
Before defining rate, first we consider some examples.
1. If one dozen eggs cost 24 Rs., what is the cost of 1 egg?
(Ans). One dozen eggs cost = 24 Rs
Cost of 1 egg = 24/12=2Rs/egg
1. A car travels 540 km consumed 60 liters petrol. Find distance travelled in liter.
(Ans). In 60 liters distance travelled= 540km
In 1 liter distance travelled = 540/60=9km/liter
1. A boy works 5-hour and paid 120 Rs. Find his pay for one hour.
(Ans) In 5 hours boy gets = 120 Rs.
In 1 hour boy gets = 120/5=24Rs/hour
In each of the above examples, the two quantities that is the quantity in numerator and the quantity in
denominator have different units, whereas in ratio both quantities have same units.
“The division of two quantities having different unit, is called RATE”
In the above examples, we have
1. Rate = 24/12 = 2Rs/egg
2. Rate = 540/60 = 9km/liter
3. Rate = 120/5 = 24Rs/hour
Example:
A car uses 40 liters of petrol to travel 320 km. How far can is travel for 32 liters?
Solution:
In 40 liters car traveled = 320km
In 1 liter car traveled = 320/40 = 6km/liter
In 32 liter car traveled = (8km/liter)(32liter) = 256km
Concept of Proportion
Proportion:
“A statement of equality of two ratios is called proportion”
Four numbers a, b, c, d are said to be in proportion when the ratio first two a and b is equal to ratio of last two
c and d.
i.e.
or
a:b=c:d
e.g.
or
2:3=6:9
Some authors used the notation for proportion as a : b :: c : d, but this notation is not preferred now.
Here
or
a:b=c:d
If four numbers are in proportion, then we can also derived some other proportion from it.
Let
be the given proportion, then
(1)
(2)
(3)
(4)
(5)
are called the Derived Proportions.
Here
or
a : b = c : d, then
The numbers a and d are called extremes of proportion, and the numbers b and c are called means of
proportion.
Hence Product of extremes = Product of Means
To solve proportion, we use above principal, A single term in the proportion is called proportional.
“a” is the 1st proportional.
“b” is the 2nd proportional.
“c” is the 3rd proportional.
“d” is the 4th proportional.
Example:
Find the 3rd proportional in 2 : 3 =
Solution:
Let
2 : 3 = : 15
i.e.
: 15
3x
3
= 2 x 15
(by the principle of proportion)
= 30
Example:
Find the missing value in
Solution:
Let
: 8 = 9 : 12
12
=9x8
12
= 72
: 8 = 9 : 12
(by the principle of proportion)
Example:
Find the 1st proportional in 18, 8 and 6.
Solution:
Let be the 1st proportional
: 18 = 8 : 6
x 6 = 18 x 8
=
= 24
Example:
Find the 2nd proportional in 4, 20, 30.
Solution:
Let be the 2nd proportional
4 : =20 : 30
x 20 = 4 x 30
=
=6
Direct Proportion
Types of Proportion:
There are four types of proportion.
1.
2.
3.
4.
Direct Proportion
Inverse Proportion
Compound Proportion
Continued Proportion
Direct Proportion:
Suppose the price of one piece of soap is 20 Rs.
If a person wants to buy one dozen pieces of soap, then he has to pay 240 Rs. If he wants to buy two dozen
pieces of soap, he has to pay 480 Rs and so on.
We can easily see that if the person buys more pieces, he has to pay more or he has to pay less if he buys less
pieces.
That is, as pieces of soap are increased total price also increased, conversely, if pieces of soap are
decreased total price also decreased. In such situation, we say that pieces and price are directly related.
In other words, If increase in one quantity causes increase in other quantity or decrease in one quantity causes
decrease in other quantity, then we say that they are related directly (They are direct proportion).
If and are in direct proportion, then division of and will be constant.
i.e.
In the above example, we see that
each ratio is the same.
Hence, if we are dealing with quantities, which are related directly, (which are in direct proportion),
then we shall use the follow rule.
24 x 240 = 12 x 480
In general
Principle of Direct Proportion
Example:
If 30 dozens of eggs cost 300 Rs. Find the cost of 5-dozens of eggs.
Solution:
Let be the required price of 5 dozens eggs
Since quantities are in direct proportion, so we use the above principle.
x 30 = 5 x 300
= 50 Rs.
Example:
A car travel 81 km in 4.5 liters. How far will it goes by 20 liters of petrol.
Solution:
Let be required required distance travelled by car in 20 liters.
Since quantities are related directly, so by the above principle
4.5 x
= 20 x 81
= 360 km
Inverse Proportion
Inverse Proportion:
Suppose that 20 men build a house in 6-days. If men are increased to 30 then they take 4-days to build
the same house. If men become 40, they take 2-days to build the house.
i.e.
It can be seen that as the no. of men is increased, the time taken to build the house is decreased in the
same ratio.
In other words,
If increased in one quantity causes decrease in other quantity or decrease in one quantity, then we say
that both quantities are inversely related.
More explicitly,
If two quantities and are in inverse proportion, then their product will be constant.
i.e.
where = constant
In the above example, we see that
20 x 6 = 120
30 x 4 = 120
40 x 3 = 120
Shows each product is constant or same.
Therefore, if we are dealing with quantities, which are related inversely, the we can use the following rule.
20 x 6 = 30 x 4
In general,
Example:
Four pipes can fill a tank in 70 minutes. How long will it take to fill the tank by 7 pipes?
Solution:
By the principle of inverse proportion, we have
4 x 70 = 7 x
= 40 minutes
Example:
Thirty-five workers can build a house in 16-days. How many days will 28 workers working at the
same rate take to build the same house?
Solution:
By the principle of inverse proportion, we have
28 x = 35 x 16
= 20 days
Example:
Imran brought 40 toys each cost Rs.14. How many toys Imran can buy at Rs.8 each from the same
amount?
Solution:
By the principle of inverse proportion, we have
14 x 40 = 8 x
= 70 toys
Compound Proportion
“The proportion involving two or more quantities is called Compound Proportion”
for Solving Compound Proportions
Rules
CASE-1
If quantity 1 and quantity 2 are directly related and quantity 2 and quantity 3 are also directly related.
Then we use the following rule
CASE-2
If quantity 1 and quantity 2 are directly related and quantity 2 and quantity 3 are inversely related.
Then we use the following rule
CASE-3
If quantity 1 and quantity 2 are inversely related and quantity 2 and quantity 3 are directly related.
The we use the following rule
CASE-4
If quantity 1 and quantity 2 are inversely related and quantity 2 and quantity 3 are also inversely
related.
Then we use the following rule
Example:
195 men working 10 hour a day can finish a job in 20 days. How many men employed to finish the
job in 15 days if they work 13 hours a day.
Solution:
Let be the no. of men required
20 x 10 x 195 = 15 x 13 x
= 200 men
Example:
A soap factory makes 100 units in 9 days with help of 20 machines. How many units can be made in
12 days with the help of 18 machines?
Solution:
20 x 9 x
= 600 x 18 x 12
= 720 units
Continued Proportion
Two or more quantities are said to be in continued proportion if 1st is related to 2nd , 2nd is related to
3rd , 3rd is related to 4th and so on.
i.e. if a, b, c, d, e are in continued proportion then
Example:
Three persons A, B and C earned a profit of 70000 Rs in a business. Their share in profit is as follows
A:B=4:2
B : C = 10 : 5
Find share of each person.
Solution:
Total amount of profit = 70000
Ratio of A, B and C is
Sum of ratio = 40 + 20 + 10 = 70
Share of A =
x 70000 = 40000
Share of B =
x 70000 = 20000
Share of C =
x 70000 = 10000
Example:
Divide 2562 among the three friends Waqas, Usman and Shakeel. Such that ratio between their shares
as
Waqas : Usman = 4 : 5
Usman : Shakeel = 8 : 10
Solution:
Let amount to be divide = 2562
Ratio of their share
Sum of ratios = 32 + 40 + 50 + 122
Share of Waqas =
x 2562 = 672
Share of Usman =
Share of Shakeel =
Concept of Percentages
x 2562 = 840
x 2562 = 1052
Comparison of fractions is not an easy task, especially when the two fractions have different
denominators.
For example; if you are asked which of the fractions
compare, whether
is greater than or less than
and
is greater than the other. i.e. we want to
.
Since, denominators of
and
are different.
To compare these fraction, first we make their denominators same
Now, we have two fraction as
Since numerator of
,
with same denominator.
is greater than numerator of
.
is greater than
But comparison becomes more easier if common denominator is 100.
The fraction with denominator 100, is called a percentage, denoted by a% or a/100. The sign % is called
percent.
e.g.,
= 3%,
= 5%
The term percent is a short form of the Latin word “Per Centum” that means “Out of Hundred”.
e.g. In a paper of math, out of total marks 50, Waqas got 35 marks, Usman got 43 marks and Shakeel
got marks 32.
Waqas got 35 out of 50 marks
i.e.
Usman got 43 out of 50
i.e.
Shakeel got 32 out of 50
i.e.
Examples:
15% =
(replace % by 1/100)
48% =
Example:
Express in decimal
(1)
(2)
(3)
Changing a Fraction into Percentage:
We can change a fraction into percentage by multiplying the fraction by 100% and simplify it, if
possible.
Examples:
Express as percentages
Expressing One Quantity as Percentage of Another Number
In a school, 56 out of 70 teachers are female. What percentage of the teachers are females? What
percentage of them is male? The fraction of female teachers in the school is 56/70 changing this fraction to
percentage we have
x 100% = 80%
80% of the teachers are female and percentage of male teachers (100%-80%) =20%
In general to express one quantity “ ” as a percentage of the other quantity “ ”, we have
1. Write the fraction
2. Multiply fraction
by 100% to convert into percentage.
Example:
108 students out of 150 are passed in Math and 96 out of 160 are passed in English. Find percentage
of passed students.
Solution:
Percentage of passed in Math =
x 100% = 72%
Percentage of passed in English =
Finding a Percentage of a Number:
In order to find percentage (
x 100% = 60%
) of another number “ ”. We have the following method.
1. Multiply
with the number “ ” i.e.
2. Simplify it, if possible.
Example:
If 75% of the students in a class of 40 passed a Mathematics test. How many of them failed?
Solution:
Total Students = 40
Percentage of passed students = 75%
No. of students who passed the test = 75% of 40 students =
No. of failed students = 40 – 30 = 10
Example:
Find (1) 25% of 21.60 (2)
Solution:
% x 1.60
x 40 = 30
o
25% of 21.60 =
o
% x 1.60 =
Percentage Changing
x 21.60 = 5.40
% x 1.60 =
x
=
= 0.60
The change in the value of an item can be expressed as a percentage increase or decrease of the
original value.
An increase of 5% in the salary of a men who earns Rs.500 per month. That means, there is an
increase of Rs.5 For every hundred (100). i.e. after increase, Rs.100 becomes 105.
Therefore,
New Salary =
x Original Salary =
x 500 = 525
On the other hand, a decrease of 5% in his salary means that for every Rs.100 in the original salary, there is a
decrease of Rs.5 i.e. each Rs.100 becomes Rs.95
i.e.
New Salary =
x Original Salary =
In general,
If an amount “ ” as increased by
If an amount “ ” as decreased by
x 500 = 475
Profit and Loss
Manufacturer produces goods at a certain cost. If goods are sold at a higher price than the cost price,
then the manufacturer makes a profit or gain. But, if for some reason, the manufacturer sells goods at lower
price than the cost price, then he suffers a loss.
Thus Profit = Selling Price – Cost price
Loss = Cost Price – Selling Price
Suppose, a person bought a book of Rs.100 and sold it for Rs.120. He makes a profit of Rs.20
i.e. Profit = 120 – 100 = Rs.20
Similarly if a person bought a machine for Rs.500 and sold it for Rs.400, then he make a loss of Rs.100
i.e. Loss = 500 – 400 = Rs.100
Percentage Profit and Percentage Loss:
For the purpose of comparison, we usually express the actual profits or loss as a percentage of cost
price.
e.g. a shopkeeper sold an article for Rs.60 which costs Rs.50 and another article sold for Rs.110 which cost
Rs.100. In each case, shopkeeper makes a profit of Rs.10. It seems that in both cases profit is equally gained.
But in percentage profit, we have
Percentage Profit =
x 100% = 20%
Percentage Profit =
x 100% = 10%
Hence, in first case he made a more profit than in 2nd case, this is better and accurate method for comparison.
Example:
A bag costing Rs.28 sold for Rs.35. Find percentage profit.
Solution:
Let
Cost Price = Rs.28
Selling Price = Rs.35
Profit = Selling Price – Cost Price = 35 – 28 = Rs.7
Percentage Profit =
x 100% = 25%
Example:
A machine costing Rs.60 and sold for Rs.50. Find percentage loss.
Solution:
Let
Cost Price = Rs.60
Sold Price = Rs.50
Loss = Cost Price – Sold Price = 60 – 50 = Rs.10
Percentage Loss =
x 100% =
%
Example:
Let a bookseller gain 30% by selling a book for Rs.65. Find the cost price of the book.
Solution:
Let
be the cost price
Profit = 30% of =
But Profit = Selling Price – Cost Price
Concept of Discount
It is usually seen that the retailers cannot sell defective items, old items, etc at the retail-selling price. If these
items are sold at lower price, then it is called sale price.
The difference between marked price (selling price) and the sale price, is called discount.
In other words, a reduction in the price of an article for payment in cash is called discount.
Hence
Discount is usually given as a percentage of the original price.
Example:
A watch priced at Rs.160 is sold for Rs.140. Find percentage discount.
Solution:
Let
Marked Price = Rs.160
Sale Price = Rs.140
Discount = Marked Price – Sale Price = 160 – 140 = Rs.20
Percentage Discount =
x 100 =
Example:
The marked price of a machine is Rs.600. A discount of 6% is given during a sale. What is the sale price of
the washing machine.
Solution:
Let
Marked Price = Rs.600
Discount Rate = 6%
Discount = 6% of the marked price =
x 600 = Rs.36
Sale Price = Marked Price – Discount = 600 – 36 = Rs.564
Concept of Commission
A commission is the payment, an agent gets for selling or buying something on behalf of
another person.
It is usually written as a percentage of cost price or selling price.
Example:
An agent got 2% commission on selling a plot for 1000000. Find his commission.
Solution:
Let
Sale Price = 1000000
Commission Rate = 2%
Amount of Commission = 2% x 1000000 =
x 1000000 = 20000
Example:
An agent received Rs.6000 as commission for selling a house. If his commission rate is 2%.
What is the selling price?
Solution:
Let
Selling Price =
Commission Rate = 2%
Commission = (Selling Price)(Rate)
6000 =
x