# More Problems Involving Percent Increase and Decrease

Want more practice with percents and related concepts?

- Changing Decimals to Percents
- Changing Percents to Decimals
- Writing Expressions Involving Percent Increase and Decrease
- Calculating Percent Increase and Decrease
- Problems Involving Percent Increase and Decrease

Here, you will practice solving more problems involving percent increase and decrease. You may use a calculator for these exercises.

## Examples

Why? As discussed in Problems Involving Percent Increase and Decrease, a price $\,x\,$ changes to $\,1.19x\,$ after the $\,19\%\,$ increase. After the subsequent $\,30\%\,$ decrease, only $\,70\%\,$ of this remains:

$$\cssId{s17}{(1-0.3)(1.19x)} \cssId{s18}{= (0.7)(1.19)x} \cssId{s19}{= 0.83x} $$The price started at $\,x\,.$ It ended at $\,0.83x\,.$ So, the overall change was a decrease (note that $\,0.83 \lt 1\,$).

How *much* of a decrease was there in going from
$\,x = 1x\,$ to
$\,0.83x\,$?
Answer:

That is, $\,17\%\,$ of $\,x\,$ was ‘lost’ in the process. The combined effect of the back-to-back increase/decrease was a $\,17\%\,$ decrease.

Pause for a moment and appreciate the power in renaming an expression! There are four names for the same expression given above, and each has its strength:

$(1 + 0.4)(1 - 0.4)x$ | This name makes it clear that we're doing a $\,40\%\,$ decrease (the $\,1 - 0.4\,$) and a $\,40\%\,$ increase (the $\,1 + 0.4\,$). |

$(1.4)(0.6)x$ | This name is a whole lot easier to plug into a calculator. |

$0.84x$ | This name, as compared to the original $\,1x\,,$ shows that the overall effect was a decrease. |

$(1 - 0.16)x$ | This name shows that it was a $\,16\%\,$ decrease. |

First approach: Compute new price, then compute percent change:

New price is:
$$\cssId{s86}{(0.3)(1.2)(\$50) = \$18}$$
It was an overall decrease.

The percent decrease is:
$$
\cssId{s89}{\frac{50-18}{50}}
\cssId{s90}{= 0.64}
\cssId{s91}{= 64\%}
$$

Second approach: You don't need the original price at all! Just denote it by $\,x\,$:

$$ \begin{gather} \cssId{s95}{(0.3)(1.2)x} \cssId{s96}{= 0.36x} \cssId{s97}{= (1 - 0.64)x}\cr \cssId{s98}{\text{$64\%\,$ decrease}} \end{gather} $$## Concept Practice

All answers are rounded to two decimal places.