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Answers
Answer:
243 people out of 400 state that they like dogs. What percent is this?
Solution
243
400
=
0.6075
=
60.75
100
. This is 60.75%.
Notice that the percent can be found from the equivalent decimal by moving the decimal point two places to the right.
EXAMPLE 2
Write each as a percent:
1
4
0.02
2.35
Solutions
1
4
=
0.25
= 25%
0.02 = 2%
2.35 = 235%
PERCENTS
If we have a part that is some percent of a whole, then
percent
=
part
whole
, or equivalently,
part
⋅
whole
=
percent
.
To do the calculations, we write the percent as a decimal.
EXAMPLE 3
The sales tax in a town is 9.4%. How much tax will you pay on a $140 purchase?
Solution
Here, $140 is the whole, and we want to find 9.4% of $140. We start by writing the percent as a decimal by moving the decimal point two places to the left (which is equivalent to dividing by 100). We can then compute: tax = 0.094(140) = $13.16 in tax.
EXAMPLE 4
In the news, you hear “tuition is expected to increase by 7% next year.” If tuition this year was $1200 per quarter, what will it be next year?
Solution
The tuition next year will be the current tuition plus an additional 7%, so it will be 107% of this year’s tuition: $1200(1.07) = $1284.
Alternatively, we could have first calculated 7% of $1200: $1200(0.07) = $84.
Notice this is not the expected tuition for next year (we could only wish). Instead, this is the expected increase, so to calculate the expected tuition, we’ll need to add this change to the previous year’s tuition: $1200 + $84 = $1284.
TRY IT NOW
A TV originally priced at $799 is on sale for 30% off. There is then a 9.2% sales tax. Find the price after including the discount and sales tax.
EXAMPLE 5
The value of a car dropped from $7400 to $6800 over the last year. What percent decrease is this?
Solution
To compute the percent change, we first need to find the dollar value change: $6800 – $7400 = –$600. Often we will take the absolute value of this amount, which is called the absolute change: |–600| = 600.
Since we are computing the decrease relative to the starting value, we compute this percent out of $7400:
\displaystyle\frac{600}{7400}=0.081=
7400
600
=0.081= 8.1% decrease. This is called a relative change.
ABSOLUTE AND RELATIVE CHANGE
Given two quantities,
Absolute change =\displaystyle|\text{ending quantity}-\text{starting quantity}|∣ending quantity−starting quantity∣
Relative change: \displaystyle\frac{\text{absolute change}}{\text{starting quantity}}
starting quantity
absolute change
Absolute change has the same units as the original quantity.
Relative change gives a percent change.
The starting quantity is called the base of the percent change.
The base of a percent is very important. For example, while Nixon was president, it was argued that marijuana was a “gateway” drug, claiming that 80% of marijuana smokers went on to use harder drugs like cocaine. The problem is, this isn’t true. The true claim is that 80% of harder drug users first smoked marijuana. The difference is one of base: 80% of marijuana smokers using hard drugs, vs. 80% of hard drug users having smoked marijuana. These numbers are not equivalent. As it turns out, only one in 2,400 marijuana users actually go on to use harder drugs.[2]
EXAMPLE 6
There are about 75 QFC supermarkets in the United States. Albertsons has about 215 stores. Compare the size of the two companies.
Solution
When we make comparisons, we must ask first whether an absolute or relative comparison. The absolute difference is 215 – 75 = 140. From this, we could say “Albertsons has 140 more stores than QFC.” However, if you wrote this in an article or paper, that number does not mean much. The relative difference may be more meaningful. There are two different relative changes we could calculate, depending on which store we use as the base:
Using QFC as the base,
140
75
=
1.867
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