Percent Increase or Decrease

```The Mathematics 11
Competency Test
Percent Increase or Decrease
The language of percent is frequently used to indicate the relative degree to which some quantity
changes. So, we often speak of percent increases in pay, or, for that matter, percent decreases
in pay. Similarly, we speak of the percent increase or percent decrease in the number of cases of
some medical condition, or the percent increase or percent decrease in the amount of petroleum
imported by a certain country, and so on.
The basic elements in such language, and the information it represents have already been
described. In this context, we use the terms:
base to refer to the quantity which will change, either increase or decrease. The “base”
quantity often has units. We will sometimes use the term original value to mean the
same thing as the “base.” Also, in practice, the “base” value, being a physical quantity of
stuff, is almost always a positive number.
Amount to refer to the actual amount of change, the amount by which the “base” quantity
either increases or decreases. The “Amount” always has the same units of measurement
as the “base” quantity.
rate is the actual percent value, in this situation representing a percent increase or
decrease. In the arithmetic calculations, the decimal equivalent of the percent increase
or percent decrease is always used.
We’ve previously stated that the fundamental relationship between these three quantities is
Amount = base x rate
(i)
and algebraic rearrangements of this formula. When discussing percent increase or percent
decrease, it is convenient to introduce one new term,
new value, the value of the quantity in question after the increase or decrease has
occurred.
Then
new value = base + amount
(iia)
= base + base x rate
(iib)
= base(1 + rate)
(iic)
These four formulas (and their algebraic rearrangements) cover all possible situations of percent
increase and percent decrease if we note one final convention:
♦ if the “rate” is positive, then
• the “Amount” is positive
• the “new value” is greater than the “base” or “original value”
(That is, a positive “rate” is associated with percent increase.)
♦ if the “rate” is negative then
David W. Sabo (2003)
Percent Increase or Decrease
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•
•
the “Amount” is negative (base x rate in formula (i) is a positive times a negative
and so is negative)
so, “new value” is less than the “base” or “original value”
(That is, a negative “rate” is associated with percent decrease.)
The main source of error in solving percent increase and percent decrease problems is failing to
correctly identify which numbers in the problem match each of these four fundamental terms. In
particular, be very careful to distinguish between the base or original value to which the rate
refers, and the new value of that quantity.
Example 1: A piece of electronic equipment is priced at \$329.50. If the retailer offers a 30%
discount,
(i) What would be the new price of the item, and,
(ii) what would be the total cost of the item when the 14.5% in sales taxes are added on?
solution:
You know how to solve this problem already, so we will focus more on applying the formal
terminology.
(i) the 30% discount on the original prices can be considered to be a 30% decrease (rate = -30%
or -0.30) in the base price or original price of \$329.50. The discounted price is then what we
have called the “new value” above:
discounted price = new value
= (original price)(1 + rate)
= (\$329.50)(1 + [-0.30])
= (\$329.50)(0.70)
= \$230.65
Thus, with the 30% discount, the new price of the item will be \$230.65.
(ii) The 14.5% tax represents a percent increase (rate = +14.5% or 0.145) – your cost (the
amount of money you must actually pay) is an increase of 14.5% on whatever is the stated price
of the item you are buying. So, your final cost here is the “new value” that results when then
14.5% increase is applied to the stated price of the item:
total cost = new value
= (stated price)(1 + rate)
= (\$230.65)(1 + 0.145)
= (\$230.65)(1.145)
= \$264.09
rounded to the nearest cent. Thus, the actual amount of money you would pay for the item
(including sales tax) would be \$264.09.
David W. Sabo (2003)
Percent Increase or Decrease
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NOTE: The importance of correctly associating rate with the base values to which they refer can
be seen even in this example. The work we did above to come up with the final cost of \$264.09
is correct – this is the way the calculation would be done in the store. The sales representative
would first apply the 30% discount to the original price, to obtain the discounted price of \$230.65.
Then, the 14.5% sales tax would be applied to this discounted price, resulting in the final cost to
you of \$264.09.
What you absolutely cannot do is to combine the two rates. You might think that a 30% decrease
followed by a 14.5% increase amounts to an effective rate of change of
-30% + 14.5% = -15.5%
and so, we could just apply this net rate to the original price of the item. However, if we were to
do that, we would get
“supposed final cost” = new value
= (original value)(1 + supposed rate)
= (\$329.50)(1 + [-0.155])
= (\$329.50)(0.845)
= \$278.43
which is incorrect. With this method, you’d end up paying over \$13 more than you should.
The source of the error here is that the two rates in the problem do not refer to the same base
values! The 30% discount is 30% of the original price of \$329.50, whereas the 14.5% sales tax
is 14.5% of the discounted price of \$230.65. This is the reason why you can’t simply add
percents together in the same way that you can add quantities of stuff with the same units
together to get a total. The extra \$13 or so you would pay if the final cost of the item was
calculated in this incorrect way amounts to paying tax on the full \$329.50 instead of the
discounted price of \$230.65.
So, to avoid error in percent problems, always make sure you have correctly identified the base
values to which each percent refers, and also that you correctly distinguish between new values
and base values.
Example 2: Hank buys a piece of equipment, paying a total of \$982.35, which includes 14.5%
sales tax. What was the actual price of the item before tax?
solution:
Obviously, the rate of change here is the 14.5% rate of paying sales tax. This rate is applied to
the original price of the item, which is what we are asked to determine. This results in a new
value, the \$982.35 actually paid. So, starting with
new value = base x (1 + rate)
we can write
\$982.35 = base x (1 + 0.145)
or
\$982.35 = 1.145 x base
David W. Sabo (2003)
Percent Increase or Decrease
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Thus,
base = actual price here
=
\$982.35
= \$857.95
1.145
rounded to the nearest cent. Thus, we conclude that the original stated price of the item was
\$857.95.
NOTE: we can double-check our answer to the last example by computing the total cost of an
item after 14.5% sales tax is added to its labelled price of \$857.95. We should end up with the
value \$982.35, as stated in the problem. So:
assumed original price:
plus 14.5% sales tax
(=0.145 x \$857.95)
\$857.95
total cost
\$982.35
\$124.40
In this type of problem, students very often make the error of subtracting 14.5% of the final cost of
\$982.35 from that final cost, thinking this will give the original price:
\$982.35 – (\$982.35)(0.145) ⇒ \$982.35 - \$142.44 ⇒ \$839.91
But you know this is incorrect, because when you buy an item in a store, the sales tax is always
computed with the original price, whereas this incorrect calculation applies the sales tax to the
final cost which already includes the sales tax.
Example 3: Hank buys a house for \$212,500, and later sells it for \$243,700. By what percent
did the price of the house change from when he bought it to when he sold it?
solution:
Start by sorting out the numbers. We have
original price = “original value” = “base” = \$212,500
and
selling price = “new value” = \$243,700
We are asked to find a rate, the percent by which the house price increased between these two
transactions. There are at least two ways to do this:
(i) the “amount” or increase in the house price
= new value – original value
= \$243,700 - \$212,500 = \$31,200
Then, using
Amount = base x rate
we have
David W. Sabo (2003)
Percent Increase or Decrease
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\$31,200 = \$212,500 x rate
so that
rate =
\$31200
≅ 0.1468
\$212500
rounded to four decimal places. Thus, the house price increased by 14.68%.
(ii) We could also begin with
new value = base x (1 + rate)
Rearranging
1 + rate =
new value
base
so that
rate =
new value
-1
base
Putting the numbers in gives
rate =
\$243700
−1
\$212500
= 1.1468 − 1 = 0.1468
again rounding to four decimal places where necessary. This is, of course, exactly the same
answer of a 14.68% increase obtained with the first method.
Example 4: The Statistics Canada estimate of the population of British Columbia in 2001 was
3,907,738. The population of Newfoundland and Labrador in 2001 was 512,930. If the
population of British Columbia grows by 4.9% over the next five years, and the population of
Newfoundland and Labrador decreases by 7.0% over those same five years, what will be the
populations of the two provinces in 2006?
solution:
This problem asks us to calculate two “new values.” For British Columbia, the population is
stated to have been 3,907,738 in 2001 (the base value). If this population grows or increases by
4.9% over the subsequent five years, then in 2006, we will have a population of
new value = base value x (1 + rate)
= (3,907,738)(1 + 0.049)
= (3,907,738)(1.049)
David W. Sabo (2003)
Percent Increase or Decrease
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= 4,099,217
rounded to the nearest whole number.
For Newfoundland and Labrador, the base value is 512,930 and we use a decrease of 7.0%, so
that the expected population in 2006 will be
new value = base value x (1 + rate)
= (512,930)(1 + [-0.070])
= (512,930)(0.93)
= 477,025
again rounded to the nearest whole number.
David W. Sabo (2003)
Percent Increase or Decrease
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