audio read-through Writing expressions involving percent increase and decrease

Want more practice with percents and related concepts?

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 .