Mathematics • Year 8 • Unit 1 • Lesson 15
Introduction to Ratios
Build fluency with ratios: writing them, simplifying them, and reading them as “parts”. One fully-worked example, one guided example with blanks, then eight independent problems from quick reading up to fraction-of-total questions.
1. I do — fully worked example
Read every line. Each step has a short reason so you can see why a ratio is a recipe of parts, not a division sum.
Problem. A class has 15 girls and 12 boys. (a) Write the ratio of girls to boys in simplest form. (b) What fraction of the class is girls?
Step 1 — Write the raw ratio in the order the question asks.
girls : boys = 15 : 12
Reason: the order of the words tells you the order of the numbers. “girls to boys” means girls FIRST.
Step 2 — Find the highest common factor (HCF) of both numbers.
HCF(15, 12) = 3
Reason: simplest form means dividing both sides by the largest number that goes into both.
Step 3 — Divide both sides by the HCF.
15 ÷ 3 : 12 ÷ 3 = 5 : 4
Reason: 5 and 4 share no common factor (other than 1), so 5:4 is simplest form.
Step 4 — Use the ratio to find the fraction of girls.
Total parts = 5 + 4 = 9. Fraction girls = 5 / 9.
Reason: in a ratio a:b, total parts = a + b, and the first group's fraction is a / (a+b).
Answer: (a) 5 : 4 girls to boys; (b) 5/9 of the class is girls.
2. We do — fill in the missing steps
Same shape as Section 1, but with the working faded. Fill in each blank. 4 marks
Problem. A drink is mixed using cordial and water in the ratio 1 : 7. (a) Write the ratio of water to cordial. (b) What fraction of the drink is cordial?
Step 1 — Reverse the order: cordial : water = 1 : 7, so water : cordial = ______ : ______.
Step 2 — Check simplest form: HCF(1, 7) = ______. Already simplest? ______ (Yes / No).
Step 3 — Find the total parts of the drink:
total parts = 1 + 7 = ______
Step 4 — Find the fraction of cordial:
fraction cordial = 1 / ______ = ______
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation (write a ratio, simplify, total parts). The middle two are standard (mix units or three-part ratios). The last two are extension (distinguish ratio from rate from fraction).
Foundation — write and simplify ratios
3.1 Write the ratio of boys to girls in simplest form for a class with 8 boys and 6 girls. 1 mark
3.2 Simplify the ratio 20 : 12 to its simplest form. 1 mark
3.3 A recipe uses flour and sugar in the ratio 3 : 1. What fraction of the mixture is flour? What fraction is sugar? 1 mark
3.4 A box has 10 red beads and 25 blue beads. Write the ratio red : blue in simplest form. 1 mark
Standard — mixed units and three-part ratios
3.5 Write the ratio 5 cm : 2 m in simplest form. (Hint: convert both to the same unit first.) 2 marks
3.6 A fruit salad uses apples, bananas and grapes in the ratio 4 : 3 : 5. (a) What is the total number of parts? (b) What fraction of the salad is bananas? 2 marks
Extension — ratio vs rate vs fraction
3.7 Classify each of these as a RATIO, RATE or FRACTION. (a) 3 boys : 4 girls (b) $\$5$/kg (c) $\tfrac{3}{4}$ of the class is girls (d) 60 km in 1 h. 2 marks
3.8 A class has 30 students. The ratio of girls to boys is 3 : 2. (a) How many parts in total? (b) How many students per part? (c) How many girls and how many boys are in the class? 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (cordial : water = 1 : 7)
Step 1: water : cordial = 7 : 1.
Step 2: HCF(1, 7) = 1. Already simplest? Yes.
Step 3: total parts = 1 + 7 = 8.
Step 4: fraction cordial = 1 / 8 = 1/8.
3.1 — Boys : girls
8 : 6, HCF = 2, simplified = 4 : 3.
3.2 — Simplify 20 : 12
HCF = 4. 20 ÷ 4 : 12 ÷ 4 = 5 : 3.
3.3 — Flour : sugar ratio 3 : 1
Total parts = 4. Flour fraction = 3/4; sugar fraction = 1/4.
3.4 — Red : blue beads
10 : 25, HCF = 5, simplified = 2 : 5.
3.5 — 5 cm : 2 m
Convert 2 m = 200 cm. So 5 : 200 = (HCF 5) = 1 : 40.
3.6 — Fruit salad 4 : 3 : 5
(a) Total parts = 4 + 3 + 5 = 12.
(b) Banana fraction = 3 / 12 = 1/4.
3.7 — Classify each
(a) 3 boys : 4 girls — RATIO (same unit: students).
(b) $\$5$/kg — RATE (different units: $ and kg).
(c) 3/4 of the class is girls — FRACTION (part of a whole).
(d) 60 km in 1 h — RATE (different units: km and h).
3.8 — 30 students, ratio 3 : 2
(a) Total parts = 3 + 2 = 5.
(b) Students per part = 30 ÷ 5 = 6.
(c) Girls = 3 × 6 = 18; Boys = 2 × 6 = 12. (Check: 18 + 12 = 30 ✓.)