# Writing expressions involving percent increase and decrease

Want more practice with percents and related concepts?

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

Recall that whenever you see the percent symbol, $\,\%\,,$ you can trade it in for a multiplier of $\,\frac{1}{100}\,.$ (Indeed, per-cent means per-one-hundred.)

For example, $\,20\%\,$ goes by all these names:

$$ \begin{align} \cssId{s11}{20\%} \ \ &\cssId{s12}{= \ \ 20\cdot\frac{1}{100}} \ \ \cssId{s13}{= \ \ \frac{20}{100}}\cr\cr &\cssId{s14}{= \ \ \frac{2}{10}} \ \ \cssId{s15}{= \ \ \frac{1}{5}} \ \ \cssId{s16}{= \ \ 0.2} \end{align} $$In particular, note that $\,100\% = 100\cdot\frac{1}{100} = 1\,,$ so $\,100\%\,$ is just another name for the number $\,1\,.$

Also recall that it's easy to go from percents to decimals: just move the decimal point two places to the left. For example:

$$ \cssId{s22}{20\%} \cssId{s23}{= 20.\%} \cssId{s24}{= 0.20} $$It's good style to put a zero in the ones place (i.e., write $\ 0.20\ ,$ not $\ .20\ $).

To change from decimals to percents, just move the decimal point two places to the right. For example:

$$ \cssId{s30}{0.2} \cssId{s31}{= 0.20} \cssId{s32}{= 20.\%} \cssId{s33}{= 20\%} $$The ‘Puddle Dipper’ memory device may be useful to you:

PuDdLe:
to change from Percents to Decimals,
move the decimal point two places to the Left.

DiPpeR:
to change from Decimals to Percents,
move the decimal point two places to the Right.

## Examples

Here, you will practice writing expressions involving percent increase and decrease, and related concepts.

Another name for the expression ‘$\,20\%\text{ of } x\,$’ is: $0.2x$

Why? The mathematical word ‘of ’ indicates multiplication, so:

$$ \begin{align} \cssId{s46}{(20\%\text{ of } x)}\ &\cssId{s47}{= (20\%)(x)}\cr &\cssId{s48}{= (0.2)(x)}\cr &\cssId{s49}{= 0.2x} \end{align} $$Another name for the expression ‘$\,100\%\text{ of } x\,$’ is: $x$

Another name for the expression ‘$\,300\%\text{ of } x\,$’ is: $3x$

If $\,x\,$ increases by $\,20\%\,,$ then the new amount is:

$$ \cssId{s55}{x + 0.2x} \cssId{s56}{= 1x + 0.2x} \cssId{s57}{= 1.2x} $$If $\,x\,$ has a $\,20\%\,$ increase, then the new amount is: $1.2x$

If $\,x\,$ increases by $\,47\%\,,$ then the new amount is:

$$ \cssId{s61}{x + 0.47x} \cssId{s62}{= 1.47x} $$If $\,x\,$ decreases by $\,30\%\,,$ then the new amount is:

$$ \cssId{s64}{x - 0.3x} \cssId{s65}{= 1x - 0.3x} \cssId{s66}{= 0.7x} $$If $\,x\,$ has a $\,30\%\,$ decrease, then the new amount is: $0.7x$

If $\,x\,$ increases by $\,100\%\,,$ then the new amount is:

$$ \cssId{s70}{x + x} \cssId{s71}{= 1x + 1x} \cssId{s72}{= 2x} $$If $\,x\,$ increases by $\,182\%\,,$ then the new amount is:

$$ \cssId{s74}{x + 1.82x = 2.82x} $$If $\,x\,$ increases by $\,200\%\,,$ then the new amount is:

$$ \cssId{s76}{x + 2x = 3x} $$If $\,x\,$ doubles, then the new amount is: $2x$

If $\,x\,$ triples, then the new amount is: $3x$

If $\,x\,$ quadruples, then the new amount is: $4x$

If $\,x\,$ is halved, then the new amount is: $\displaystyle\frac{1}{2}x = 0.5x$

## Practice

Answers must be input in decimal form to be recognized as correct. Also, you must exhibit good style by putting a zero in the ones place, as needed. For example, input 0.5x , not (say) .5x or 1/2x .