5.5 PERCENT CHANGE
16. The bookstore at the college earns 20% as commission on everything sold in the store. Last
month the commission was $36,800. What was the amount of the total sales from the bookstore
last month?
17. Reggie, a college student, has a part-time job and earns 60% of his tuition cost by working at that
job. If his total tuition this year is $8,000, how much does he have to earn?
18. Mary, a college fashion student works part-time as a model. Her earnings usually cover 85% of
her school supplies expenses. If her supplies this year are expected to cost $3,500, how much
should she earn?
5.5
PERCENT CHANGE
At times a quantity or amount may increase or decrease in value. It is important to know how to
calculate the increase or decrease as a percent. This is called the percent change and is written as %C.
Percent change is expressed as the amount of change (increase or decrease) from the initial value and
is expressed as:
%C =
Amount of Change
× 100%
Initial Value
%C =
Final Value − Initial Value
× 100%
Initial Value
If Vi is the Initial Value and Vf if the Final Value,
%C =
V f − Vi
Vi
× 100%
CALCULATING THE PERCENT CHANGE
GIVEN THE INITIAL VALUE AND FINAL VALUE
Percent increase or decrease is calculated using the formula: %C =
V f − Vi
× 100%
Vi
If the answer is positive, then it is a percent increase whereas if it is negative, then it is a percent
decrease.
EXAMPLE 5.5A - CALCULATING THE PERCENT INCREASE
A math textbook increased in price from $90 to $99. By what percent did the price increase?
SOLUTION
Vi = $90 (Initial Value of the textbook)
Vf = $99 (Final Value of the textbook)
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CHAPTER 5 | PERCENTS AND PERCENT CHANGES
V f − Vi
%C =
Vi
× 100%
=
99 − 90
× 100%
90
Substitute the values
=
9
× 100%
90
Simplify
=
1
× 100%
10
Simplify
= 0.1× 100%
Simplify
=10%
Therefore, the price of the math textbook increased by 10%.
EXAMPLE 5.5B - CALCULATING THE PERCENT DECREASE
The price of shares of a company decreased from $24 to $20 per share from last month. By what
percent did the price of shares of the company change from last month?
SOLUTION
Vi = $24
Vf = $20
%C =
V f − Vi
Vi
× 100%
=
20 − 24
× 100%
24
Substitute the values
=
−4
× 100%
24
Simplify
= −0.166666... × 100%
= −16.67%
Simplify
Simplify
Therefore, the price of shares of the company decreased by 16.67% from last month.
EXAMPLE 5.5C - CALCULATING THE FINAL VALUE GIVEN THE INITIAL VALUE AND PERCENT CHANGE
If a $15 hourly rate of pay is increased by 20%, find the new hourly rate.
SOLUTION
Vi = $15, %C = 20%, Vf =?
80
5.5 PERCENT CHANGE
%C
%C ==
20%
20% ==
0.2
0. 2 =
V f − Vi
Vi
V f − 15
15
× 100%
Formula
× 100%
Substitute values
V f − 15
Eliminate the percent
15
15(0.2) = Vf − 15
Use Direct Method to rearrange
3 = Vf − 15
3 + 15 = Vf − 15 + 15
Add 15 to both sides
18 = Vf
Vf = 18
Rearrange
Therefore, the new hourly rate is $18.00.
CALCULATING THE INITIAL VALUE, GIVEN THE
FINAL VALUE AND PERCENT CHANGE
The final value can be calculated by rearranging the percent change formula, as illustrated in the
following example:
EXAMPLE 5.5D - CALCULATING THE INITIAL VALUE GIVEN THE FINAL VALUE AND PERCENT CHANGE
What amount decreased by 50% is $1,000?
SOLUTION
Vi = ? , %C = – 50%, Vf = $1,000
%C
%C ==
−50% ==
−50%
−0.5
−0.5 ==
V f − Vi
Vi
× 100%
1, 000 − Vi
× 100%
Vi
Substitute values
1, 000 − Vi
Vi
Eliminate the percent
Vi(−0.5) = 1,000 − Vi
Use Direct Method to rearrange
−0.5Vi = 1,000 − Vi
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CHAPTER 5 | PERCENTS AND PERCENT CHANGES
−0.5Vi + Vi = 1,000 − Vi + Vi
Add Vi to both sides
0.5Vi = 1,000
Vi 11, 000
00.5.5Vi
, 000
===
00.5.5
00.5.5
Vi = 2,000
Divide both sides by 0.5
Simplify
Therefore, $2,000 decreased by 50% is $1,000.
EXERCISE 5.5
Answers to the odd numbered problems are available online
In the following problems, round your answers to 2 decimal places:
1.
An English text costing $90 was marked up at the college bookstore by 20%. What was the final
price?
2.
A French text costing $50 was marked up at the college book store by 25%. What was the final
price?
3.
A $400 snowboard was on sale for 40% off. What was the final sale price?
4.
A used Math textbook costing $100 was marked down by 45% by a student who wanted to sell
it. What was the final price?
5.
Tim has a student loan and it increased last semester from $1,000 to $3,000. What was the
percent change?
6.
Tuition this year increased from $4,000 to $4,500. What was the percent change?
7.
Samuel bought a used car. Last year it was valued at $14,000 but this year it had depreciated
(went down in price) to $12,000. What was the percent change in the cost of the car?
8.
This year, a company earned $500,000. Last year the same company earned $550,000. What was
the percent change in the company’s earnings?
9.
A T-shirt costing $20 was marked up by 35%. What was the final price? (Do not include tax.)
10. A $150 pair of hiking boots was marked up by 40%. What was the final price? (Do not include
tax.)
11. If the final cost of a pair of skates after 13% tax is $100.00, what was the price before tax?
12. College students had a pizza party and the final cost was $2,789 after 8% tax. What was the cost
before tax?
13. The final price of a backpack is $54.67 after a store-wide sale of 40% off. What was the original
cost of the backpack (without tax)?
14. A business student bought gold stock for $3,000 after the stock had decreased by 45%. What was
the gold stock worth before the decrease?
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