CHAPTER 2: EXPONENTIAL AND
NATURAL LOGARITHM FUNCTIONS
WITH APPLICATIONS TO BUSINESS
Section 2.2.a. Percentages
In mathematics we have three main ways at looking at a part or portion of
a number. We can consider a fraction such as a seventh, 17 , two-thirds, 23 , or
one and a fifth, 1 51 . Another approach is to use decimals such as 0.2, 0.514
or 3.14. In business however the preferred means of writing or expressing
parts of a number is to use percentages. You will often hear expressions
such as ‘Inflation rose by 1.5% over the month of August’ or ‘Shares on
Wall Street fell by 1.2% yesterday’ . The word percentage literally means ‘per
cent’ which means per hundredth. By measuring quantities in percentages
we are measuring them in their hundredth parts. So when we talk of r% of
a quantity we mean the fraction r/100 of it.
Therefore
1
50
= of it;
100
2
75
3
75% of a price means
= of it;
100
4
5
1
125
= = 1 of it.
125% of a quantity means
100
4
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Example: Determine
50% of a quantity means
(a) 30% of 15
(b) 90% of 200
(c) 120% of 310.
Solution: For calculation purposes we will usually convert our percentages
to decimals.
(a) 30% of 15 is
30
× 15 = 0.3 × 15 = 4.5.
100
(b) 90% of 200 is
90
× 200 = 0.9 × 200 = 180.
100
(c) 120% of 310 is
120
× 310 = 1.2 × 310 = 372.
100
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Exercise: Determine
(a) 20% of 18
(b) 85% of 430
(c) 350% of 1610.
Example:
(a) A deposit increases from e1,500 to e1,950. Express the increase as a
percentage of the original value.
(b) The price of a Microword share was $14.10 at the beginning of August.
Over the month of August the price rose by 3%. What is the value of
a Microword share by close of trading on the 31st of August?
(c) In a sale all prices are reduced by 15%. What is the sale price of a
Gocci Handbag which had an original price of e710?
Solution:
(a) The increase in the deposit is 1, 950 − 1, 500 = 450. As a fraction of
the original this is
450
= 0.3.
1500
This is the same as 30 hundredths and therefore there is a 30% increase.
(b) The percentage 3% is the same as
3
= 0.03.
100
So the rise in price is
0.03 × 14.10 = 0.423.
Hence the new price is e14.10+e0.42=e14.52.
(c) As a fraction, 15% is the same as
15
= 0.15.
100
Hence the fall in price is
0.15 × 710 = 106.5.
Thus the new price is
710 − 106.5 = 603.5
or e603.50.
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Scaling Factor
In the previous example our calculations were performed in two steps. The
value of the rise or fall was calculated and added to original to obtain the
answer. Let us now observe that the entire calculation can be done in one
step. This will not only lead to quicker calculations but will allow us to
investigate more difficult problems.
Let us suppose that the price of a particular good is eP and that it will rise
by r%. The new price of the good will be the original P plus r% of P . In
other words the new price of the good is
r
r
P =P 1+
.
100
100
P + r%P = P +
r
We will call 1 +
100
the scaling factor.
Let us reconsider some of our previous examples.
Example. The price of a Microword share was $14.10 at the beginning of
August. Over the month of August the price rose by 3%. What is the value
of a Microword share by close of trading on the 31st of August?
Solution. The original price of the share is P = $14.10. The change in the
price of a share is r = 3%. The scaling factor is
103
3
=
= 1.03.
1+
100
100
Thus the new share price is (14.10)(1.03) =e14.52.
Example. The production cost of a new desktop computer is e590. If VAT
is charged at 21% what is the cost of the desktop to the customer?
Solution. The scaling factor is
21
= 1.21.
100
1+
Therefore the cost of the desktop to the customer is (1.21)(590) = 713.9,
that is, e713.90.
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Exercise:
(a) On the first of January 2000 the population of a village is 7, 200. If the
annual rise in the population is 7%, what is the population at midnight
on the 31st of December?
(b) The GDP of a country was 94 billion euros five years ago. If it increased
by 5% over the last five years what is the current GDP of the country?
We can consider a decrease of r% as an increase of −r%. Thus if the price
or value of a good decreases by r% then we multiply by a scaling factor of
−r
r
= 1−
.
100
100
1+
Example: Last year the average daily usage of a motorway was 21, 000 vehicles. The introduction of a toll six months ago has lead to a decrease of
10% in the average daily usage of the motorway. What is the current average
daily usage?
Solution. The scaling factor is
1−
10
= 0.9.
100
Therefore the current average daily usage of the motorway (0.9)(21, 000) =
18, 900 vehicles.
Examples: At the start of trading yesterday morning the value of a NewLife
Insurance share was listed on the Irish Stock Market at e29.20. If their value
fell by 1.3% during the day what was the price at close of trading?
Solution. The scaling factor is
1.3
= 0.987.
1−
100
Therefore the stock price at the close of trading was (0.987)(29.20) = 28.8204
or e28.82.
Exercise:
(a) A company has an annual electricity bill of e1,230,000. By increasing
its insulation it can reduce its electricity cost by 6%. What is the new
annual cost of its electricity?
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(b) As part of a promotional weekend The Great Islands Hotel reduced its
prices by 23%. If the cost of a weekend is usually e420 what is the
promotional price?
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