A ratio compares two or more quantities of the same kind, measured in the same units. Ratio is the foundation for rates, map scales, direct and inverse proportion, and similar figures later in Sec 2 and Sec 3, and it turns up in almost every paper as a quick 1–3 mark item.
These notes cover writing and simplifying ratios, ratios with fractions, decimals and mixed units, forming ratios from equations, the unitary method for sharing questions, and changing ratios. Changing ratios is the question type that shows up most often in school tests and in the Sec 3/4 E Math papers.
What these notes cover
- •What is a ratio?
- •Simplest form and equivalent ratios
- •Converting to common units
- •Sharing a total in a given ratio
- •Ratios of three quantities
- •Ratios with fractions and decimals
- •Forming a ratio from an equation
- •Combining two ratios with a common term
- •The unitary method
- •Changing ratios: the three standard types
- •The algebraic method
- •Common mistakes and exam technique
Key definitions
- Ratio
- A comparison of two or more quantities of the same kind, measured in the same units. The ratio of a to b is written a : b and read as "a to b". Order matters: 12 : 16 and 16 : 12 are different ratios.
- Simplest form
- A ratio in which every term is a whole number, the HCF of all the terms is 1, and there are no units. Of all equivalent ratios, exactly one is in simplest form.
- Equivalent ratios
- Ratios obtained by multiplying or dividing every term by the same non-zero number. They describe the same comparison, e.g. 3 : 4 = 6 : 8 = 12 : 16 = 30 : 40.
- Unit (u)
- The unknown multiplier in a ratio. A ratio a : b : c means the three quantities are au, bu and cu for some value of u.
- Unitary method
- Finding the value of one unit, then multiplying to get whatever the question asks for.
Key formulas
Share of a total
A is the share belonging to the ratio term a, and T is the total. For three quantities in the ratio a : b : c, the denominator becomes a + b + c.
Two-fraction shortcut
Comes from multiplying both terms by bd. It does not always give simplest form: (3 ÷ 4) : (5 ÷ 6) = 18 : 20, which still divides by 2 to give 9 : 10.
Cross-over rule
The coefficients cross over, e.g. 7a = 3b gives a : b = 3 : 7. Check by substitution: 7(3) = 21 = 3(7).
Fraction-coefficient rule
Divide both sides by b, e.g. x = (2 ÷ 5)y gives x : y = 2 : 5.
Unitary method
A total matches the sum of all the terms, a difference matches the difference in units, and one part matches that part's term only.
Ratio as an equation
Used in the algebraic method: let the quantities be ak and bk, apply the change, then multiply each numerator by the denominator on the other side.
Worked example
The ratio of the number of stamps Ravi has to the number Mei Ling has is 5 : 3. Ravi gives 18 stamps to Mei Ling, and the ratio becomes 7 : 9. Find the number of stamps Ravi had at first.
- 1The stamps only move from Ravi to Mei Ling, so the total is unchanged.
- 2Before, the total is 5 + 3 = 8 parts; after, the total is 7 + 9 = 16 parts, and the LCM of 8 and 16 is 16.
- 3Equalise the totals: before, 5 : 3 = 10 : 6 (total 16u); after, the ratio stays 7 : 9 (total 16u).
- 4Ravi goes from 10u to 7u, so he gives away 3u: 3u = 18, so u = 6.
- 5Ravi had 10u = 10 × 6 = 60 stamps at first.
- 6Check: at first Mei Ling had 6 × 6 = 36, and 60 : 36 = 5 : 3; afterwards Ravi has 60 − 18 = 42 and Mei Ling has 36 + 18 = 54, and 42 : 54 = 7 : 9.
Watch out: Do not compare Ravi's 5 parts directly with his 7 parts. Those parts are different sizes until the totals are made equal.
Common mistakes
Treating a : b as a fraction of the total
Wrong: Ali and Bala share 840 sweets in the ratio 3 : 4, so Ali = (3 ÷ 4) × 840 = 630.
Right: Add the ratio terms first: Ali = 3 ÷ (3 + 4) × 840 = (3 ÷ 7) × 840 = 360, and Bala = 840 − 360 = 480. Check: 360 : 480 = 3 : 4.
Reversing the cross-over
Wrong: 4a = 5b gives a : b = 4 : 5.
Right: 4 lots of a equal 5 lots of b, so each a is bigger: a : b = 5 : 4. Check with a = 5, b = 4: 4(5) = 20 = 5(4).
Not equalising the unchanged term
Wrong: A : B = 2 : 3; A receives 12 more marbles, B stays the same, and the ratio becomes 5 : 6. "A went from 2 parts to 5 parts, so 3 parts = 12."
Right: B is unchanged, so write the ratios as 4 : 6 and 5 : 6. A goes from 4u to 5u, so 1u = 12.
Questions students ask
How do you write a ratio in simplest form?
Convert all the quantities to the same unit, drop the units, then divide every term by the HCF of all the terms. The final ratio must have whole-number terms with HCF 1 and no units. For example, 1.8 m : 240 cm = 180 : 240 = 3 : 4.
What is the unitary method in ratio questions?
Label each quantity in units, decide how many units the given amount represents, find the value of 1 unit, then multiply to get what is asked. For the ratio 4 : 7 : 9, if R has 35 more than P, then 9 − 4 = 5 units = 35, so 1 unit = 7.
How do you combine two ratios a : b and b : c?
Make the shared term b the same in both ratios using the LCM of its two values, then read off a : b : c. If a : b = 2 : 3 and b : c = 4 : 5, LCM(3, 4) = 12 gives 8 : 12 and 12 : 15, so a : b : c = 8 : 12 : 15, not 2 : 3 : 5.
How do you solve changing ratio questions?
First decide what stays the same: the unchanged quantity if only one changes, the difference if both change by the same amount, or the total if items are transferred. Make the units of that quantity equal in the before and after ratios using the LCM, then compare. If both change by different amounts, let the quantities be ak and bk and solve an equation.
In the full PDF
- •Bar-model diagrams for sharing, difference and changing-ratio questions
- •Further worked examples on mixed units, fractions and decimals, ratios from equations and changing ratios
- •Step-by-step routines for simplifying ratios with fractions, decimals and mixed units
- •The remaining common mistakes, including the cross-multiplication error and adding the same number to both terms
- •The ratio checklist, exam technique routine and 20-second substitution check
- •Section summaries pulling together all five methods in the topic