Scotch Egg Home
Unit 3: Ratios, Rates, and Percents

Topic 3

Percent Increase, Percent Decrease, and Percent Error

Percent compares a change to an original amount.
percent change=new−originaloriginal×100%\text{percent change}=\frac{\text{new} - \text{original}}{\text{original}}\times 100\%
If the result is positive it is a percent increase. If negative it is a percent decrease.
Forward change models:
increase by p%:  new=(1+p)⋅original\text{increase by }p\%:\ \ \text{new}=(1+p)\cdot \text{original}
decrease by p%:  new=(1−p)⋅original\text{decrease by }p\%:\ \ \text{new}=(1-p)\cdot \text{original}
Here pp is written as a decimal. For example 12% means p=0.12p=0.12.
Reverse change models (find the original):
original=new1+pafter an increaseoriginal=new1−pafter a decrease\text{original}=\frac{\text{new}}{1+p}\quad \text{after an increase} \qquad \text{original}=\frac{\text{new}}{1-p}\quad \text{after a decrease}
Percent error measures accuracy of an approximation.
percent error=∣approx−actual∣actual×100%\text{percent error}=\frac{|\text{approx} - \text{actual}|}{\text{actual}}\times 100\%

Core Skills

  • Translate words to the correct formula and identify the original amount.
  • Convert percent to decimal before multiplying.
  • Use growth factor 1±p1\pm p for one step changes.
  • For reverse problems, divide by the factor 1±p1\pm p.
  • Distinguish percent of a number from percentage points.

Example 1: Percent Increase

A jacket price rises from $60 to $75.
75−6060×100%=1560×100%=25%\frac{75-60}{60}\times 100\%=\frac{15}{60}\times 100\%=25\%
Answer: 25 percent increase.

Example 2: Percent Decrease

A population drops from 1{,}250 to 1{,}100.
1100−12501250×100%=−1501250×100%=−12%\frac{1100-1250}{1250}\times 100\%=\frac{-150}{1250}\times 100\%=-12\%
Answer: 12 percent decrease.

Example 3: Forward Calculation With Factor

An item is discounted 30% off the original price $80. Factor =1−0.30=0.70=1-0.30=0.70. New price =0.70⋅80=$56=0.70\cdot 80=\$56.

Example 4: Reverse Calculation After Decrease

After a 20% discount, the sale price is $72. Find the original price. Sale price == (1−0.20)⋅original=0.80⋅original(1-0.20)\cdot \text{original}=0.80\cdot \text{original}. original=720.80=$90\text{original}=\dfrac{72}{0.80}=\$90.

Example 5: Tax or Markup

A store marks up cost by 15%. If the cost is $120, the price is (1+0.15)⋅120=1.15⋅120=$138(1+0.15)\cdot 120=1.15\cdot 120=\$138.

Example 6: Percent Error

A measurement is reported as 9.7 cm, but the actual length is 10.0 cm.
∣9.7−10.0∣10.0×100%=0.310×100%=3%\frac{|9.7-10.0|}{10.0}\times 100\%=\frac{0.3}{10}\times 100\%=3\%
Answer: 3 percent error.

Key Takeaways

  • Change divided by original times 100% gives percent change.
  • Use 1+p1+p for increases and 1−p1-p for decreases.
  • To undo a percent change, divide by the same factor.
  • Percent error uses actual in the denominator and absolute difference in the numerator.