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 ✓.)