Lesson 17 ~25 min Unit 1 · Financial Maths +85 XP

Dividing a Quantity in a Given Ratio

Splitting $\$480$ in the ratio $1:2:3$ — find each share quickly with the unitary method.

Today's hook: Three friends win $\$480$ and split it in ratio $1:2:3$. Who gets what?

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Think First

warm-up

Three friends win $\$480$ and split it in ratio $1:2:3$. Who gets what? Jot down your first reaction — then we'll see who's right.

Record your answer in your workbook.

The Big Idea

+5 XP

To divide a quantity in a ratio: find the total parts, find the value of one part, then multiply for each share. Always check the shares add up to the original total.

Split $\$480$ in the ratio $1:2:3$. Total parts $= 1+2+3 = 6$. Value of 1 part $= 480 \div 6 = \$80$. Shares: $1 \times 80 = \$80$, $2 \times 80 = \$160$, $3 \times 80 = \$240$. Check: $80 + 160 + 240 = \$480$ ✓.

Share = (parts share) $\times \dfrac{\text{Total}}{\text{Sum of parts}}$

Sum of parts

Add all the numbers in the ratio.

Value of 1 part

Total ÷ sum of parts.

Multiply each

Each ratio number × 1-part value.

What You'll Master

objectives

Know

Find total parts: add all ratio numbers
Value of 1 part = Total ÷ Total Parts
Each share = ratio number × value of 1 part
Sum of shares = original total (check this!)

Understand

Why the unitary method works for ratio division
How the “parts” concept ties to fractions of total
The use of a sum check to catch errors

Can Do

Divide any quantity in a 2-part or 3-part ratio
Verify the sum of shares equals the total
Apply ratio division to money, weight, time

Words You Need

vocabulary

ShareOne person's portion in a ratio division.

Total partsSum of all numbers in the ratio.

Value of one partTotal quantity divided by total parts.

Unitary methodFind one first, then scale.

Sum checkVerifying shares add up to the original total.

DistributionSharing out in proportion to ratio.

Spot the Trap

heads-up

Wrong: "Split $\$480$ in $1:2:3$ = $\$160$ each (just divide by 3)" — NO. The ratio means UNEQUAL shares.

Right: Total parts $1+2+3 = 6$. Each part $= \$80$. Shares: $\$80, \$160, \$240$.

Wrong: "$\$480$ in $1:2:3$ = $\$1, \$2, \$3$" — NO. Find the value of 1 part first.

Right: 1 part = $\$480 / 6 = \$80$. Then $1 \times 80 = \$80$, $2 \times 80 = \$160$, $3 \times 80 = \$240$.

The Unitary Method for Ratios

+5 XP

Just like rates: find ONE part first, then scale up.

Divide $\$600$ in the ratio $2:3$. Total parts $= 2+3=5$. Value of 1 part $= 600 \div 5 = \$120$. First share: $2 \times 120 = \$240$. Second: $3 \times 120 = \$360$. Sum check: $240 + 360 = \$600$ ✓.

1 part $= \dfrac{\text{Total}}{\text{Sum of parts}}$

Sum of parts

Add ratio numbers.

One-part value

Total ÷ sum.

Multiply

Each ratio × one-part value.

Three-Part Division

+5 XP

Same method, three shares. Watch the arithmetic.

Split $\$540$ in $2:3:4$. Sum of parts $= 9$. Value of 1 part $= 540/9 = \$60$. Shares: $2 \times 60 = \$120$, $3 \times 60 = \$180$, $4 \times 60 = \$240$. Check: $120 + 180 + 240 = \$540$ ✓.

Share$_i = $ ratio$_i \times \dfrac{\text{Total}}{a+b+c}$

Same logic

Just one more share.

Sum check is gold

All shares should total back to the original.

Use it for everything

Money, weight, time, areas.

Watch Me Solve It · 3 examples

Watch Me Solve It · The lottery win

+15 XP per step

PROBLEM

Three friends share $\$480$ in the ratio $1:2:3$. Who gets what?

1

Total parts

$1+2+3=6$

Add all parts.
2

One part value

$480 \div 6 = \$80$

Total/parts.
3

Three shares

$1\times 80 = \$80$; $2 \times 80 = \$160$; $3 \times 80 = \$240$

Multiply each.

Answer$\$80, \$160, \$240$ (check: sum = $\$480$ ✓)

Watch Me Solve It · Concrete mix

+15 XP per step

PROBLEM

A 36 kg bag of concrete ingredients is mixed in ratio $2:3:4$ (cement:sand:gravel). How much of each?

1

Sum of parts

$2+3+4=9$

Total.
2

Mass per part

$36 \div 9 = 4$ kg

Each unit.
3

Multiply

Cement $= 2 \times 4 = 8$ kg; Sand $= 3 \times 4 = 12$ kg; Gravel $= 4 \times 4 = 16$ kg

Each ingredient.

Answer$8$ kg, $12$ kg, $16$ kg (check: $36$ kg ✓)

Watch Me Solve It · Reverse problem

+15 XP per step

PROBLEM

Two people share a sum of money in ratio $4:5$. The smaller share is $\$160$. How much did they share in total?

1

Find value of 1 part

$160 \div 4 = \$40$

Smaller share is 4 parts.
2

Total parts

$4+5 = 9$

Sum.
3

Total amount

$9 \times \$40 = \$360$

Or: bigger share $= 5 \times 40 = \$200$; total $= 160 + 200 = \$360$.

Answer$\$360$

Common Pitfalls

heads-up

Splitting equally

Treating $1:2:3$ as three equal shares.

Fix: Ratio shares are UNEQUAL. Use ratio numbers.

Multiplying by total parts

Using $\$480 \times 3$ instead of $\div$.

Fix: Find 1 part = total ÷ parts, then multiply.

Forgetting the sum check

Submitting wrong shares that don't add up to total.

Fix: ALWAYS verify shares sum to original total.

Copy Into Your Books

Method

Sum all ratio parts
1 part = total ÷ sum
Each share = ratio × 1-part value

Example

$\$600$ in $2:3$ → parts = 5, 1 part = $\$120$
Shares: $\$240, \$360$
Sum check: $\$600$ ✓

Three-Part

$\$540$ in $2:3:4$ → parts = 9, 1 part = $\$60$
Shares: $\$120, \$180, \$240$
Sum check: $\$540$ ✓

Reverse Problem

Given one share, find total
1 part = given share / number of parts
Total = 1 part × sum of parts

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Dividing a Quantity in a Given Ratio

4 problems

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

1 Split $\$60$ in ratio $1:2$.

1 part $= 20$; shares $\$20, \$40$.$\$20, \$40$
2 Split $\$210$ in ratio $2:3:2$.

7 parts; 1 part $= 30$; shares $\$60, \$90, \$60$.$\$60, \$90, \$60$
3 Split 24 m in ratio $3:5$.

8 parts; 1 part $=3$ m; shares $9$ m, $15$ m.$9$ m, $15$ m
4 Two friends share $\$140$ in ratio $3:4$. Larger share?

7 parts; 1 part $=20$; larger $= 4 \times 20 = \$80$.$\$80$

Complete in your workbook.

Quick Check · 5 questions

Split $\$300$ in ratio $2:3$. Shares are:

+10 XP

Split $\$540$ in ratio $1:2:3$. Smallest share:

+10 XP

In a ratio split $3:4:5$ totaling $48$ kg, the middle share is:

+10 XP

Lucia and Asha split $\$120$ in ratio $7:5$. Lucia's share is:

+10 XP

Two friends share a pizza in ratio $3:2$. The smaller share is 4 slices. Total slices:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. Divide each according to the ratio: (a) $\$700$ in $3:4$; (b) $45$ kg in $1:2:6$; (c) $\$360$ in $5:3:2$.

Answer in your workbook.

Understand Easy 2 MARKS

Q7. Mia, Jay, and Tahlia bought a $\$96$ gift for their friend. They contributed in the ratio $3:4:5$. How much did each pay?

Answer in your workbook.

Reason Hard 4 MARKS

Q8. Three siblings split a $\$420$ inheritance in the ratio $2:3:5$. (a) Calculate each share. (b) The oldest sibling thinks her share should be EQUAL to the others — what would she receive if it were equal? (c) Explain why the ratio split is fair given that the oldest contributed more to the original earnings.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. C — $\$120, \$180$.

2. B — $\$90$.

3. B — $16$ kg.

4. C — $\$70$.

5. C — $10$ slices.

Show Your Working Model Answers

Q6 (3 marks): (a) 7 parts; 1 part = $\$100$; $\$300, \$400$ [1]. (b) 9 parts; 1 part = 5 kg; $5, 10, 30$ kg [1]. (c) 10 parts; 1 part = $\$36$; $\$180, \$108, \$72$ [1].

Q7 (2 marks): 12 parts; 1 part = $\$8$ [1]. Mia: $\$24$, Jay: $\$32$, Tahlia: $\$40$ [1].

Q8 (4 marks): (a) 10 parts; 1 part = $\$42$. Shares: $\$84, \$126, \$210$ [2]. (b) Equal would be $\$140$ each [1]. (c) Ratio rewards proportional contribution — oldest gets more because she put in more. If shares were equal, the others would gain at her expense. This is a value judgement, not maths [1].

Stretch Challenge · +25 XP, +10 coins

The Recipe Scaler

A bakery recipe for 12 muffins uses 240 g flour, 80 g sugar, and 60 g butter (ratio $4:\tfrac{4}{3}:1$, or equivalently $12:4:3$). The bakery wants to make 30 muffins. (a) What's the multiplier? (b) How much of each ingredient (in grams) is needed? (c) What total weight of mix? (d) Express the original ingredients in their simplest ratio form.

Reveal solution

(a) Multiplier $= 30/12 = 2.5$. (b) Flour $= 240 \times 2.5 = 600$ g. Sugar $= 80 \times 2.5 = 200$ g. Butter $= 60 \times 2.5 = 150$ g. (c) Total $= 950$ g. (d) Simplest ratio: $240:80:60$, HCF $= 20$, so $12:4:3$ (which is the simplest 3-part form).

Quick Review

Sum of parts

Add all ratio numbers

1 part

Total ÷ sum of parts

Each share

Ratio × 1-part value

Sum check

Shares add to total

Reverse

Given share → 1 part → total

Same logic

2-part, 3-part, any-part

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