Ratio

Ratio #

Compare two amounts; scale both, nothing changes.

A ratio compares two amounts without saying how big either one is

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2 parts red for every 3 parts blue.

Double both: 2 : 3 = 4 : 6 — equivalent ratios, the very same mix.

Triple instead: 2 : 3 = 6 : 9. Scale both parts alike and the ratio holds.

Part-to-Whole Ratios #

Add the parts and you have the whole.

A ratio names the parts and adding them gives you the whole

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2 parts colored to 3 parts plain — the ratio 2 : 3.

2 + 3 = 5 parts altogether, so the colored share is $\frac{2}{5}$2/5.

Tidying a Ratio #

Divide out what both parts share.

A ratio is tidiest when its two parts share no common factor

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Both parts of 4 : 6 divide by 2: 4 ÷ 2 = 2 and 6 ÷ 2 = 3 — the same mix, 2 : 3.

Three-Part Ratios #

Comparing three quantities at once.

A ratio can compare three quantities at once

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1 part, then 2, then 3. Written 1 : 2 : 3.

1 + 2 + 3 = 6 parts altogether, so the smallest share is $\frac{1}{6}$1/6 of the whole.

Combining Two Ratios #

Two ratios, one shared quantity.

Scale both ratios until their shared quantity matches, then read straight through

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Two ratios, one shared quantity — but B reads 3 parts in one and 4 in the other.

Match B: 3 and 4 first meet at 12, so scale by × 4 and × 3.

With B matched the chain reads straight through: 8 : 12 : 15.

Scale Factor #

Every measurement multiplied by the same number.

A scale factor is the number every measurement gets multiplied by

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A rectangle 2 across and 3 up.

Multiply every side by 3 and the shape keeps exactly the same proportions.

The Unitary Method #

Find one part and every share follows.

Find what one part is worth and every share follows

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20 sweets shared 2 : 3. That is 2 + 3 = 5 equal parts, none counted yet.

20 ÷ 5 puts 4 in every part. Now just count them: 8 in one share, 12 in the other.

Bar Models for Ratio #

Draw the bars and the problem shows itself.

Drawing a ratio as bars turns a word problem into something you can see

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One share is 3 parts, the other 5. Drawn side by side, the gap shows.

Line them up: the gap is 5 − 3 = 2 parts, whatever one part turns out to be worth.

Ratio, Fraction, Percent #

One comparison, three ways to write it.

One comparison can be written as a ratio or a fraction or a percent

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1 part colored to 3 parts plain: the ratio 1 : 3.

That is 1 out of 4 altogether, so the fraction is $\frac{1}{4}$1/4.

And $\frac{1}{4}=\frac{25}{100}$1/4 = 25/100, which is 25%.

Ratios After a Transfer #

The parts move; the total stays put.

A transfer moves parts from one share to the other, and the total stays put

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Ali and Ben share 48 stickers 5 : 3. Ali gives Ben some — now they hold 4 : 4.

Both splits carve the same 48: 8 parts either way, so one part is 48 ÷ 8 = 6.

Ali went from 5 parts to 4, so the transfer is 6 stickers.