Section PF.2 – Absolute and Relative Change
Oftentimes, percentages are used to compare how a quantity has changed over time. When making this type of comparison, we use absolute change and relative change.
| Absolute and Relative Change |
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Given two quantities, Absolute change = ending quantity – starting quantity Absolute change has the same units as the original quantity. Relative change: [latex]\frac{\text{absolute change}}{\text{starting quantity}}[/latex] (multiply relative change by 100 to write as a %) Relative change gives a percent change. If the relative change is positive this is a percent increase. If the relative change is negative this is a percent decrease.
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| Example 1 |
| The value of a car dropped from $7,400.00 to $6,800.00 over the last year. What percent decrease is this? |
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To compute the percent change, we first need to find the dollar value change: $6,800-$7,400 = -$600.00. The absolute change is -600.00. Since we are computing the decrease relative to the starting value, we compute this percent out of $7,400: [latex]\frac{-600}{7,400}[/latex] = -0.081 = -8.1% The value of -8.1% is called a relative change. The fact that it is negative indicates that the value of the car decreased over the last year. |
| Example 2 |
| There are about 75 QFC supermarkets in the U.S. Albertsons has about 215 stores. Compare the size of the two companies by finding the absolute change and the relative change |
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When we make comparisons, we must ask first whether we want 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, [latex]\frac{140}{75}[/latex] = 1.867 This tells us Albertsons is 186.7% larger than QFC. Using Albertsons as the base, [latex]\frac{140}{215}[/latex] = 0.651 This tells us QFC is 65.1% smaller than Albertsons. Notice both of these are showing percent differences. We could also calculate the size of Albertsons relative to QFC: [latex]\frac{215}{75}[/latex], which tells us Albertsons is 2.867 times the size of QFC. Likewise, we could calculate the size of QFC relative to Albertsons: [latex]\frac{75}{215}[/latex] = 0.349, which tells us that QFC is 34.9% of the size of Albertsons. |
| Example 3 |
| Suppose a stock drops in value by 60% one week, then increases in value the next week by 75%. Is the value higher or lower than where it started? |
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To answer this question, suppose the value started at $100. After one week, the value dropped by 60%: $100 – $100(0.60) = $100 – $60 = $40.00. In the next week, notice the base of the percent has changed to the new value, $40. Computing the 75% increase: $40 + $40(0.75) = $40 + $30 = $70.00. In the end, the stock is still $30.00 lower than the original value, which is [latex]\frac{$30}{$100}[/latex] = 30% lower, than it started. |
| You Try PF.2.A |
| The U.S. federal debt at the end of 2001 was $5.77 trillion, and grew to $6.20 trillion by the end of 2002. At the end of 2005 it was $7.91 trillion, and grew to $8.45 trillion by the end of 20062. Calculate the absolute and relative increase for 2001-2002 and 2005-2006 (round to two decimal places if needed). Which year saw a larger increase in federal debt? |
| Example 4 |
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A Seattle Times article on high school graduation rates reported “The number of schools graduating 60 percent or fewer students in four years – sometimes referred to as “dropout factories” – decreased by 17 during that time period. The number of kids attending schools with such low graduation rates was cut in half.” a) Is the “decrease by 17” number a useful comparison? b) Considering the last sentence, can we conclude that the number of “dropout factories” was originally 34? |
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a) This number is hard to evaluate, since we have no basis for judging whether this is a large or small change. If the number of “dropout factories” dropped from 20 to 3, that’d be a very significant change, but if the number dropped from 217 to 200, that would be less of an improvement. b) The last sentence provides relative change which helps put the first sentence in perspective. We can estimate that the number of “dropout factories” was probably previously around 34 if the schools are about the same size. However, it’s possible that students simply moved schools rather than the school improving, so that estimate might not be fully accurate. |
| Example 5 |
| A politician’s support increases from 40% of voters to 50% of voters. Describe the change. |
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When talking about a change of quantities that are already measured in percentages, we must be careful in how we describe the change. We could describe this using an absolute change: 50% – 40% = 10%. Notice that since the original quantities were percentages, this change also has the units of percent. In this case, it is best to describe this as an increase of 10 percentage points. In contrast, we could compute the percent change: [latex]\frac{10%}{40%}[/latex] = 0.25 = 25% increase. This is the relative change, and we’d say the politician’s support has increased by 25%. |
Section PF.2 – Answers to You Try Problems
PF.2.A
2001-2002: [Absolute Change = $0.43 trillion] / [Relative Change = 7.45% increase]
2005-2006: [Absolute Change = $0.54 trillion] / [Relative Change = 6.83% increase]
2005-2006 saw a larger absolute increase, but a smaller relative increase.
