Percent

Percentage #

Percent simply means out of a hundred.

Percent means out of a hundred

Cut the whole into 100 pieces and color 40 in: $\frac{40}{100}$40/100, written 40%.

Finding a Percent #

Find ten percent first, the rest follows.

Find ten percent first and the rest follows

Split 120 into ten equal parts: 120 ÷ 10 = 12. The lower scale reads 10%.

Three of those parts are 3 × 12 = 36, and the lower scale reads 30%.

Taking a Percent Off #

The part you take away from the whole.

A discount is the part you take away from the whole

Take 2 of the 10 parts off the whole 100%. That is the 20% knocked off.

8 parts are left: 100% − 20% = 80%. That is what you pay.

Try it on $50: 20% of 50 is 10, so you pay 50 − 10 = $40.

Sales Tax #

A percent the till adds on top.

Sales tax is a percent the till adds on top of the price

The shop asks $40, then the till adds tax: 5% of $40 is $2, so you pay $42.

The rate depends on where — and when — you buy: at 8%, the same ball costs $43.20.

Comparing with Percent #

Percents make two scores fair to compare.

Turning both scores into percents makes them fair to compare

Two scores out of different totals. You cannot compare them yet.

Cut each of the 20 into 5: 20 × 5 = 100 pieces, and 15 × 5 = 75 colored.

Do the same to $\frac{16}{25}$16/25: cut each of the 25 into 4, and 16 × 4 = 64 out of 100.

Both are out of 100 now: 75% against 64%, so $\frac{15}{20}$15/20 is the better score.

Percentage Increase #

The whole plus the extra, in one multiply.

Adding a percent on is one multiplication by the whole-plus-extra

A 20% rise keeps all 100% and adds 20% on top.

Try it on 40: 20% of 40 is 8, and 40 + 8 = 48.

48 is 120% of 40. As a multiplier, 120% is $\frac{120}{100}=1.2$120/100 = 1.2, so 40 × 1.2 = 48.

Falls work the same way: down 20% keeps 80%, and one step does it: ×0.8.

Percentage Change #

The difference, out of where you started.

Percentage change compares the difference with where you started

From 40 up to 50: the change is 50 − 40 = 10. But 10 out of what?

Out of where you started: 40 splits into four 10s, so 10 out of 40 is $\frac{1}{4}$1/4.

A quarter of 100 is 25, so $\frac{10}{40}=\frac{25}{100}$10/40 = 25/100 — a 25% change.

From 20% to 25%: up 5 percentage points — and since 5 is a quarter of 20, up 25%.

Reverse Percentages #

Divide by the multiplier to get back.

To undo a percent change, divide by the multiplier that made it

After a 20% rise the price is $60, so $60 is 120% of the old price.

120% of the old price means old × 1.2, and that landed on 60.

Undo the ×1.2 by dividing: 60 ÷ 1.2 = 50, so the old price was $50.

The wrong route: 20% off 60 gives 48. The rise was 20% of 50, not of 60.

Depreciation #

Falling by the same percent every year.

Losing the same percent each year is multiplying by the same number each year

A value that falls 20% keeps 80% — percent off in one step: ×0.8.

A $500 phone loses 20% in its first year: $500 × 0.8 = $400.

Two years means ×0.8 twice, giving $320 — not $500 with $100 taken off twice.

Fractions, Decimals and Percents #

One amount, three ways of writing it.

One amount, three ways of writing it

3 of the 4 pieces are colored in.

Cut into 100 pieces instead and the very same amount reads 0.75.

Or 75 out of 100, which is 75%. The coloring never changed!

Comparing Percents, Fractions and Decimals #

Convert to one language before comparing.

Convert everything to one notation before comparing

Three discounts on one jacket: $\frac{1}{4}$1/4 off, 0.3 off, 27% off. Turn all three into percents.

Lined up as 25, 27 and 30, the sizes are plain: 0.3 is the biggest discount.