Lesson 16 ~25 min Unit 1 · Financial Maths +85 XP

Simplifying Ratios

Reduce a ratio to its lowest terms — like simplifying fractions. Handle mixed units. Three-part ratios too.

Today's hook: A concrete mix is 2 parts cement, 3 parts sand, 4 parts gravel. What fraction is cement?

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Think First

warm-up

A concrete mix is 2 parts cement, 3 parts sand, 4 parts gravel. What fraction is cement? Jot down your first reaction — then we'll see who's right.

Record your answer in your workbook.

The Big Idea

+5 XP

Simplify a ratio by dividing all parts by the same number, ideally the HCF. Three-part ratios work the same way. Mixed units? Convert first.

$12:18$ becomes $2:3$ after dividing by HCF$=6$. Ratios with three parts like $6:9:15$ become $2:3:5$ (divide by HCF $=3$). Mixed units: $5$ cm $: 2$ m must first be converted to $5$ cm $: 200$ cm, then simplified to $1:40$.

Simplify $a:b$ by dividing both by HCF$(a,b)$

Find the HCF

Highest Common Factor of all parts.

Divide them all

Each part divided by HCF.

Convert units first

If parts are in different units, convert.

What You'll Master

objectives

Know

Ratios simplify by dividing all parts by HCF
Three-part ratios use HCF of all three
Mixed units must be converted to the same unit first
Simplest form: parts share no common factor > 1

Understand

Why simplifying a ratio doesn't change what it describes
How simplifying works the same for 2-part or n-part ratios
Why mixed-unit ratios are nonsense until converted

Can Do

Simplify any ratio to lowest terms
Handle 3-part ratios
Convert mixed-unit ratios before simplifying

Words You Need

vocabulary

HCFHighest Common Factor — the biggest number dividing all parts.

Simplest formLowest terms — no common factor > 1 remaining.

Three-part ratio$a:b:c$, e.g., 2:3:4 for cement:sand:gravel.

Unit conversionChanging units so both sides match (cm $\to$ m, etc.).

Equivalent ratiosDifferent appearances, same relationship: $2:3 = 4:6 = 10:15$.

Cross-multiplicationMethod to check if two ratios are equivalent.

Spot the Trap

heads-up

Wrong: "$\$2 : 50$ cents = $2:50 = 1:25$" — NO. Convert first! $\$2 = 200$ cents; $200:50 = 4:1$.

Right: $\$2 = 200$ cents. Then $200:50 = 4:1$.

Wrong: "$12:18:24$ simplifies to $4:6:8$ (÷ by 3 partially)" — NO. Find HCF of ALL three: HCF $= 6$. Simplify to $2:3:4$.

Right: HCF of $12, 18, 24$ is $6$. $12:18:24 \div 6 = 2:3:4$.

Simplifying Two-Part Ratios

+5 XP

Find the HCF of both numbers. Divide both by the HCF. Done.

$\,16:24$. HCF$(16, 24) = 8$. Divide both by 8: $2:3$. That's the simplest form. Try another: $\,45:60$. HCF $= 15$. $\,45 \div 15 : 60 \div 15 = 3:4$. Each step is just like simplifying a fraction.

$a:b = \dfrac{a}{HCF}:\dfrac{b}{HCF}$

Find HCF

Biggest number dividing both.

Divide both

Each part by HCF.

Verify

Simplified ratio has no common factor.

Three-Part Ratios and Mixed Units

+5 XP

Three-part ratios: HCF of ALL three parts. Mixed units: convert first.

Three-part: $30:45:75$. HCF$(30, 45, 75) = 15$. Simplified: $2:3:5$. Mixed units: $400$ g $: 1$ kg. Convert: $400$ g $: 1000$ g. Now simplify: HCF $= 200$, ratio $= 2:5$.

Three-part: $a:b:c \div HCF(a,b,c)$. Mixed units: convert FIRST.

HCF of all parts

Same logic, just more numbers.

Convert units

Both sides in the SAME unit before simplifying.

Choose sensible unit

cm or g for small, m or kg for big.

Watch Me Solve It · 3 examples

Watch Me Solve It · Concrete fraction

+15 XP per step

PROBLEM

A concrete mix is 2 parts cement, 3 parts sand, 4 parts gravel. What fraction is cement?

1

Total parts

$2 + 3 + 4 = 9$

Sum of all parts.
2

Fraction of cement

Part/total $= \tfrac{2}{9}$

Direct ratio-to-fraction conversion.
3

Check the simplest form

Ratio $2:3:4$ has no common factor (HCF $=1$). Already simplest.

Final answer.

Answer$\tfrac{2}{9}$ of the mix

Watch Me Solve It · Simplify $24:36$

+15 XP per step

PROBLEM

Simplify $24:36$ to its simplest form.

1

Find HCF

HCF$(24, 36)$: factors of 24 = 1,2,3,4,6,8,12,24; factors of 36 = 1,2,3,4,6,9,12,18,36

HCF = 12.
2

Divide both

$24 \div 12 : 36 \div 12 = 2:3$

Simplified.
3

Check

Is $2:3$ in simplest form? Yes — no common factor > 1

Done.

Answer$2:3$

Watch Me Solve It · Mixed units

+15 XP per step

PROBLEM

Simplify $30$ minutes $: 2$ hours.

1

Convert to same unit

$2$ hours $= 120$ minutes

Both now in minutes.
2

Write the ratio

$30 : 120$

Same units.
3

Simplify

HCF $= 30$. $30:120 \div 30 = 1:4$

Simplest form.

Answer$1:4$

Common Pitfalls

heads-up

Forgetting to convert units

Ratio like $5$ cm $: 2$ m treated as $5:2$. Nonsense.

Fix: $2$ m $= 200$ cm. $5:200 = 1:40$.

Partial simplification

$12:18:24 \to 4:6:8$ (÷ by 3) but not finished.

Fix: Keep going until no common factor > 1. $\to 2:3:4$.

Wrong HCF for 3 parts

Using HCF of just 2 of the 3 parts.

Fix: HCF must divide ALL parts. Check each.

Copy Into Your Books

Simplifying Steps

Find HCF of all parts
Divide each part by HCF
Result has no common factor

Two-Part Examples

$12:18 = 2:3$ (HCF 6)
$24:36 = 2:3$ (HCF 12)
$45:60 = 3:4$ (HCF 15)

Three-Part Examples

$6:9:15 = 2:3:5$ (HCF 3)
$10:20:30 = 1:2:3$ (HCF 10)
Same logic, just more

Mixed Units

Convert before simplifying
$5$ cm $: 2$ m $= 1:40$
$\$2 : 50$c $= 4:1$

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Simplifying Ratios

4 problems

Four drill problems to sharpen your skills. Work each, then reveal the answer.

1 Simplify $20:30$.

HCF = 10. $2:3$.$2:3$
2 Simplify $9:27:36$.

HCF = 9. $1:3:4$.$1:3:4$
3 Simplify $\$3 : 50$ cents.

$\$3 = 300$ cents. $300:50 = 6:1$.$6:1$
4 Simplify $40$ min $: 2$ h.

$2$ h $= 120$ min. $40:120 = 1:3$.$1:3$

Complete in your workbook.

Quick Check · 5 questions

$30:45$ in simplest form is:

+10 XP

Simplify $\$1.50 : 25$ cents.

+10 XP

Simplify $8:12:20$.

+10 XP

$500$ mL $: 2$ L in simplest form:

+10 XP

A class has 12 girls and 18 boys. The ratio of girls to total students is:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. Simplify each: (a) $36:54$; (b) $12$ kg $: 0.4$ kg; (c) $10:25:40$.

Answer in your workbook.

Understand Easy 2 MARKS

Q7. A drink is mixed using $250$ mL syrup and $1.25$ L water. Find the syrup-to-water ratio in simplest form.

Answer in your workbook.

Reason Hard 4 MARKS

Q8. A salad recipe calls for $200$ g spinach, $400$ g tomatoes, and $300$ g chicken. (a) Write the ratio spinach:tomato:chicken in simplest form. (b) What fraction of the salad is each ingredient? (c) If you want $1800$ g of salad total (same proportions), how much of each do you need?

Answer in your workbook.

Comprehensive Answers

Quick Check

1. C — $2:3$.

2. B — $6:1$.

3. B — $2:3:5$.

4. A — $1:4$.

5. C — $2:5$.

Show Your Working Model Answers

Q6 (3 marks): (a) HCF $=18$; $36:54 \div 18 = 2:3$ [1]. (b) $12000$ g : $400$ g; HCF $=400$; ratio $= 30:1$ [1]. (c) HCF $= 5$; $10:25:40 \div 5 = 2:5:8$ [1].

Q7 (2 marks): $1.25$ L $= 1250$ mL [1]. $250:1250 \div 250 = 1:5$ [1].

Q8 (4 marks): (a) $200:400:300 \div 100 = 2:4:3$ [1]. (b) Total 9 parts; spinach $\tfrac{2}{9}$, tomato $\tfrac{4}{9}$, chicken $\tfrac{3}{9} = \tfrac{1}{3}$ [1]. (c) 1 part = $1800 \div 9 = 200$ g; spinach $400$ g, tomato $800$ g, chicken $600$ g [2].

Stretch Challenge · +25 XP, +10 coins

The Compound Ratio

In a school, the ratio of boys to girls is $5:7$. Among the boys, the ratio of Year 7 to Year 8 to Year 9 is $3:4:5$. (a) What fraction of the whole school are Year 8 boys? (b) If there are 720 students in the school, how many Year 9 boys are there?

Reveal solution

(a) Boys = $\tfrac{5}{12}$ of school. Of those, Y8 = $\tfrac{4}{12}$. Y8 boys = $\tfrac{5}{12} \times \tfrac{4}{12} = \tfrac{20}{144} = \tfrac{5}{36}$ of the school. (b) Y9 boys = $\tfrac{5}{12} \times \tfrac{5}{12} \times 720 = \tfrac{25}{144} \times 720 = 125$ Y9 boys.

Quick Review

HCF

Divide all parts by it

Same units

Convert first if mixed

3-part ratios

HCF of all three

Simplest form

No common factor > 1

Equivalent

$2:3 = 4:6 = 10:15$

Check

Multiply back to verify

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