Number and algebra · Lesson 5 of 7

Ratio and proportion

A ratio is a recipe for parts, not two separate numbers. Find what one part is worth and every share follows.

About 20 min · Interactive · 8 practice questions

Set a ratio and what you know about it. The bar splits into parts, and the readouts show what one part is worth, each share and the whole.

1 part is worth8
Shares24 : 16
Whole40

A ratio is a recipe for parts

Most people read 3:2 as "3 and 2", two fixed numbers. Instead, it is a recipe: 3 parts of one thing for every 2 parts of the other, and a part can be any size at all. 3:2 describes 3 kg and 2 kg, but it also describes 30 students and 20 students, or $240 and $160.

The whole recipe is 3 + 2 = 5 parts. If you know the whole, you can find the size of one part by dividing: 40 ÷ 5 = 8. Every quantity in the problem is then a multiple of 8: the first share is 3 × 8 = 24 and the second is 2 × 8 = 16.

Proportion is the same recipe at a different size. 2:3 and 8:12 are proportional because 8:12 is 2:3 with every part made four times bigger. The parts have changed size; the recipe has not.

The practical consequence: nearly every ratio question on the test is "find one part, then scale". Decide what one part is worth first. The rest is multiplication.

Bar model of the ratio 3 to 2, showing the first share as three fifths of the whole and the second share as two fifths 1 part1 part1 part1 part1 part 3 parts = 3/5 of the whole = 60% 2 parts = 2/5 = 40% the whole = 5 parts = 100% 2/5 compares the second share with the whole. This is what "what fraction of the class" asks for. 2/3 compares the two shares with each other: the second share is two thirds of the first. Different question.
In 3:2, the first thing is 3 parts out of 5, not 3 out of 2. The fraction of the whole always has the total number of parts on the bottom.

The seven ratio methods

Each method is a short fixed recipe. Most of them are the same move, "find one part, then multiply", pointed in a different direction.

1. Simplify a ratio

  1. Make the units match first. Convert the bigger unit into the smaller one.
  2. Find the highest common factor (HCF) of the numbers.
  3. Divide every number by it. Drop the units: a simplified ratio has none.

50 cm : 2 m → 50 : 200 → ÷50 → 1 : 4

2. Share a total in a ratio

  1. Add the parts to get the total number of parts.
  2. Divide the total by that number: this is one part.
  3. Multiply one part by each number in the ratio.
  4. Check the shares add back to the total.

$120 in 5:3:2 → 10 parts → $12 each → $60, $36, $24

3. One share or the difference is known

  1. Ask how many parts the known amount covers. One share is that share's number; a difference is the bigger number minus the smaller.
  2. Divide the known amount by that many parts: one part.
  3. Multiply up for the other share and the whole.

7:4, difference 12 → 7 − 4 = 3 parts → 12 ÷ 3 = 4 → shares 28 and 16, whole 44

4. The unitary method

  1. Divide to find the value of one unit (one pen, one serve, one hour).
  2. Multiply that by the number you want.

6 pens cost $9 → 1 pen $1.50 → 10 pens $15

5. Scale a recipe or mixture

  1. Scale factor = new amount ÷ old amount.
  2. Multiply every ingredient by the same factor. Not just one, and never add.

Serves 4 → serves 10: factor 10 ÷ 4 = 2.5; 300 g flour × 2.5 = 750 g; 2 eggs × 2.5 = 5 eggs

6. Equivalent ratios and the cross-check

  1. a:b = c:d exactly when a × d = b × c (multiply across, the products match).
  2. To find a missing value, work out the scale factor between the known pair, or rearrange the cross-products.

2:3 = 8:? → 8 ÷ 2 = 4, so ? = 3 × 4 = 12. Check: 2 × 12 = 24 and 3 × 8 = 24.

7. Ratio to fraction to percentage

  1. Add the parts for the bottom of the fraction.
  2. The share you want goes on top.
  3. Convert to a percentage if asked.

Girls:boys 3:2 → girls are 3/5 of the class = 60%, not 3/2 and not 2/3

Why the same move keeps working

Every quantity in a ratio problem is "some number of parts" times "the size of one part". Multiplying both parts by the same number keeps the recipe the same, which is why 3:2, 6:4 and 24:16 all describe the same mixture. So whatever you are told, total, one share or a difference, you turn it into "this many parts are worth this much" and divide.

Ratio solver that shows its working

Pick the problem type, enter the numbers

Ratios can be typed as 3:2, 2:3:5 or 3 to 2. The solver shows each step with its reason, not just the answer.

: = : ?

    Scaling a recipe on a double number line

    A recipe for 4 people that needs 300 g of flour is a ratio: 4 serves to 300 g. Scaling it to 10 serves does not mean adding 6 serves' worth of something. It means multiplying every ingredient by 10 ÷ 4 = 2.5.

    A double number line keeps the two quantities locked together. Serves run along the top, flour along the bottom, and any vertical line through both reads off a matching pair. Going down to 1 serve (75 g) and back up is the unitary method drawn as a picture.

    Double number line with serves on top and grams of flour below, showing 4 serves matching 300 grams and 10 serves matching 750 grams serves flour (g) 01410 075300750 × 2.5 × 2.5 ÷ 4 to find one serve Any vertical line gives a matching pair. 4 : 300 and 10 : 750 are the same recipe.
    Both lines are stretched by the same factor, so the ratio of serves to flour never changes. 300 ÷ 4 = 75 g per serve, and 75 × 10 = 750 g.

    Ratio, fraction and percentage table

    The test moves between these three labels without warning: "boys:girls is 2:3" one line, "what percentage are boys" the next. The bottom of the fraction is always the total number of parts.

    RatioFirst part ÷ wholePercentageThink of it as
    1:11/250%Equal shares, half each
    1:21/333.3%Three parts; the small share is one third
    1:31/425%Four parts; cordial 1:3 is a quarter cordial
    2:32/540%Five parts; each part is 20%
    3:23/560%Same five parts, the other way around
    1:41/520%Five parts; one in five
    3:13/475%Four parts; three of them
    2:12/366.7%Three parts; two of them
    3:73/1030%Ten parts; read it straight off
    1:1:21/425%Four parts in total; the first is one of them

    Hide a column, say each answer aloud, then click a striped cell to check it.

    Drill: one part is worth…

    Speed matters in the test. Two short games: in the first, find what one part is worth and then one share. In the second, decide whether two ratios are the same recipe.

    Score 0 of 0 · streak 0

    Common traps

    • Treating 3:2 as 3/2 of the whole. The first share is 3 parts out of 5, so it is 3/5 of the whole, or 60%. The bottom of the fraction is the total number of parts, never the other share.
    • Sharing a difference as if it were the total. "The shares differ by 12" means 12 covers 7 − 4 = 3 parts, not 7 + 4 = 11. Ask which parts the known amount covers before dividing.
    • Simplifying before the units match. 50 cm : 2 m is not 25:1. Convert to the same unit first: 50 : 200 = 1 : 4.
    • Scaling only one ingredient. If the flour is multiplied by 2.5, so are the eggs, the sugar and the butter. A recipe is one ratio with many parts.
    • Adding instead of multiplying. A recipe for 4 scaled to 8 is × 2, not + 4. Doubling the serves doubles every amount; adding 4 to every amount makes a different recipe.
    • Assigning the numbers in the wrong order. "The ratio of boys to girls is 2:3" gives boys 2 parts and girls 3 parts. The order of the words is the order of the numbers.
    • Rounding one part early. If one part is 7.333… and you round it to 7.3, the shares no longer add to the total. Keep the exact value until the last step, or keep it as a fraction.

    ACER-style practice

    Eight questions in the formats the real test uses: multiple choice, type your answer, and true-or-false tables. Questions marked "no calculator" belong to the test's final section, where you cannot use one. Check each answer for a full explanation.

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    What to remember

    • A ratio is a recipe for parts. The whole is the sum of the parts: 3:2 → 5 parts.
    • Share a total: one part = total ÷ number of parts, then multiply by each number in the ratio.
    • One share known: one part = share ÷ its parts. Difference known: one part = difference ÷ (bigger − smaller).
    • Unitary method: find the value of one, then multiply by how many you want.
    • Scale a recipe: factor = new ÷ old, and multiply every ingredient by it.
    • Equivalent ratios: a:b = c:d when a × d = b × c.
    • Ratio to fraction of the whole: that share over the total parts. 3:2 → 3/5 = 60%.

    All questions on this page are original and are not reproduced from ACER material. LANTITE is administered by the Australian Council for Educational Research.