Lesson 18 ~25 min Unit 1 · Financial Maths +85 XP

Ratio in Context

Scale drawings, mixing solutions, doubling recipes — ratios at work in real life.

Today's hook: A map has a scale of $1:50\,000$. On the map, two towns are 4.5 cm apart. How far apart are they really?

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Think First

warm-up

A map has a scale of $1:50\,000$. On the map, two towns are 4.5 cm apart. How far apart are they really? Jot down your first reaction — then we'll see who's right.

Record your answer in your workbook.

The Big Idea

+5 XP

Ratios appear everywhere: scale drawings (maps, plans), recipes (mixing ingredients), and solutions (cordials, fertilizers). Once you understand the structure, the maths is the same.

A map scale of $1:50\,000$ means 1 cm on the map = $50\,000$ cm in real life (i.e., $500$ m or $0.5$ km). So 4.5 cm on map = $4.5 \times 500$ m $= 2250$ m $= 2.25$ km in real life. A scale ratio is just a multiplier between two distances.

Real distance $= $ Map distance $\times $ Scale denominator

Map scale

$1:n$ means real distance is $n$ times bigger.

Same units

Map ratio is unitless — both sides are lengths.

Apply to recipes

Doubling a recipe = multiplying every quantity by 2.

What You'll Master

objectives

Know

Scale ratios show how a drawing relates to real life
$1:n$ map scale: real distance = $n \times$ map distance
Recipes scale by multiplying every ingredient
Solutions: keep the ratio of ingredients constant

Understand

Why scale ratios are unitless
How to convert between map distance and real distance
Why proportional scaling preserves the ratio

Can Do

Convert map distances using a scale ratio
Scale a recipe up or down
Mix solutions in a given ratio

Words You Need

vocabulary

Scale ratioUnitless ratio showing model to real size, e.g., $1:100$.

Scale factorThe number you multiply by to scale up.

Scale drawingA drawing where lengths are proportional to real lengths.

ConcentrationHow much of one ingredient per total.

DilutionAdding more solvent to lower concentration.

ProportionalQuantities that maintain the same ratio.

Spot the Trap

heads-up

Wrong: "Scale $1:50\,000$ means 1 cm = $50\,000$ km" — NO. Same units. 1 cm = $50\,000$ cm = $500$ m = $0.5$ km.

Right: 1 cm on map = $50\,000$ cm in real life = $0.5$ km. Same units, different size.

Wrong: "Doubling a recipe means double the ratio." — NO. The RATIO stays the same; the quantities double.

Right: Doubling: $3:1$ stays $3:1$; quantities go from $3$ cups, $1$ cup to $6$ cups, $2$ cups.

Scale Drawings and Maps

+5 XP

A scale ratio gives the multiplier between drawing and reality.

A map with scale $1:25\,000$ means 1 cm on the map = $25\,000$ cm in real life = 250 m. If two points are 8 cm apart on the map, real distance is $8 \times 250 = 2000$ m $= 2$ km. Working backwards: a real road of $5$ km would appear on the map as $5000$ m $\div 250$ = $20$ cm.

Real $= $ Map $\times $ Denominator; Map $= $ Real $\div $ Denominator

$1:n$ scale

Real = map $\times n$.

Same units

Convert if needed (cm to km).

Reverse

Map distance = real ÷ denominator.

Recipes and Solutions

+5 XP

Scaling a recipe up or down means multiplying every ingredient by the same factor.

A recipe for 6 cupcakes uses 200 g flour and 100 g sugar (ratio $2:1$). To make 18 cupcakes (3 times more): multiply each ingredient by 3 — flour = 600 g, sugar = 300 g. The ratio $2:1$ is preserved. Mixing a 5\% salt solution: keep salt:water in the right ratio.

Scaled ingredient $= $ Original $\times $ Scale factor

Scale factor

New total ÷ original total.

Multiply everything

Each ingredient by the factor.

Ratio preserved

Always.

Watch Me Solve It · 3 examples

Watch Me Solve It · Map distance

+15 XP per step

PROBLEM

A map has scale $1:50\,000$. Two towns are 4.5 cm apart on the map. Find real distance.

1

Convert scale

1 cm = $50\,000$ cm = $500$ m = $0.5$ km

Real-world equivalent of 1 cm.
2

Multiply by map distance

$4.5 \times 500 = 2250$ m

Or: $4.5 \times 0.5 = 2.25$ km.
3

Express in km

$2250$ m $= 2.25$ km

Final answer.

Answer$2.25$ km

Watch Me Solve It · Recipe scale

+15 XP per step

PROBLEM

A recipe for 4 people uses 320 g flour. How much for 7 people, keeping the same ratio?

1

Scale factor

$\dfrac{7}{4} = 1.75$

Multiplier.
2

Multiply flour

$320 \times 1.75 = 560$ g

For 7 people.
3

Check

Flour per person stays $80$ g (both: $320/4 = 80$, $560/7 = 80$) ✓

Ratio preserved.

Answer$560$ g flour

Watch Me Solve It · Diluting cordial

+15 XP per step

PROBLEM

A cordial mix is 1 part syrup to 7 parts water. How much syrup is needed to make 480 mL of cordial?

1

Total parts

$1 + 7 = 8$

Cordial in parts.
2

One part

$480 \div 8 = 60$ mL

Per part.
3

Syrup

$1 \times 60 = 60$ mL

Syrup is 1 part.

Answer$60$ mL syrup ($420$ mL water)

Common Pitfalls

heads-up

Mixing units in scale

Treating 1 cm : 50,000 km as meaningful.

Fix: Scale is unitless. Same unit on both sides.

Changing the ratio when scaling

Doubling some ingredients but not others.

Fix: Multiply EVERYTHING by the same factor.

Mixing solution wrong

Scaling syrup but not water.

Fix: Total = all parts. Each part scales identically.

Copy Into Your Books

Map Scale

$1:n$ = unitless
Real = map × $n$
Same units both sides

Real-World Examples

$1:25\,000$: 1 cm = 250 m
$1:100$: 1 cm = 1 m
$1:1000$: 1 cm = 10 m

Recipe Scaling

Scale factor = new/old
Multiply all ingredients
Ratio preserved

Solutions

Total parts in ratio
Find 1 part = total/parts
Each ingredient = ratio × 1 part

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Ratio in Context

4 problems

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

1 Map scale $1:100\,000$. 3 cm on map = ? km.

$3 \times 1$ km = $3$ km.$3$ km
2 Recipe doubles: $\tfrac{1}{2}$ cup flour. New?

$1$ cup.$1$ cup
3 Drink mix $1:4$ syrup:water; 200 mL total. Syrup?

5 parts; 1 part = 40 mL; syrup = 40 mL.$40$ mL
4 Plan scale $1:50$. Real wall is 6 m. Plan length?

$6 \div 50 = 0.12$ m $= 12$ cm.$12$ cm

Complete in your workbook.

Quick Check · 5 questions

Scale $1:200$. A 5 cm line on the plan = ? in real life:

+10 XP

Recipe for 4 uses $200$ g sugar. For 6 (same ratio):

+10 XP

A drink is $1$ part juice to $4$ parts water. For $500$ mL:

+10 XP

A plan scale $1:50$ shows a room as 8 cm long. Real length:

+10 XP

A photograph is reduced in size, ratio $5:2$. A $15$ cm side becomes:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. Apply each ratio: (a) Map scale $1:25\,000$. 8 cm on map = ? km. (b) Cordial $1:6$ for $560$ mL. Syrup amount? (c) Recipe for 12 cookies uses $300$ g flour. For 20 cookies?

Answer in your workbook.

Understand Easy 2 MARKS

Q7. A house plan has scale $1:100$. A wall is shown as $4.5$ cm long. (a) What is its real length in metres? (b) A real $7$ m corridor would be how long on the plan?

Answer in your workbook.

Reason Hard 4 MARKS

Q8. A nurse needs to prepare a saline solution in a $9:1$ ratio of water to salt by volume. She has 360 mL of distilled water. (a) How much salt should she add to maintain the ratio? (b) What is the total volume of the solution? (c) Express the salt as a percentage of the total solution. (d) If she only has 30 mL of salt available, how much water should she use?

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — $10$ m.

2. D — $300$ g.

3. B — $100$ mL juice, $400$ mL water.

4. B — $4$ m.

5. B — $6$ cm.

Show Your Working Model Answers

Q6 (3 marks): (a) $8 \times 25\,000 = 200\,000$ cm $= 2$ km [1]. (b) 7 parts; 1 part = $80$ mL; syrup = $80$ mL [1]. (c) Multiplier $= 20/12 = 5/3$; flour $= 300 \times 5/3 = 500$ g [1].

Q7 (2 marks): (a) $4.5 \times 100 = 450$ cm $= 4.5$ m [1]. (b) $7$ m $= 700$ cm; plan $= 700 \div 100 = 7$ cm [1].

Q8 (4 marks): (a) Water = 9 parts = 360 mL; 1 part = 40 mL; salt = 1 part = $40$ mL [1]. (b) Total = $360 + 40 = 400$ mL [1]. (c) $\tfrac{40}{400} \times 100 = 10\%$ [1]. (d) Salt = 1 part = 30 mL; water = 9 parts = $9 \times 30 = 270$ mL [1].

Stretch Challenge · +25 XP, +10 coins

The Cake-Decorating Disaster

A baker wants to make a layer cake. The recipe (for 1 layer) uses 250 g flour, 200 g sugar, and 150 g butter. The cake needs 3 layers, but the baker only has 600 g of sugar. (a) Can she make all 3 layers? (b) If not, what is the maximum number of layers she can make? (c) Using the SAME ratio, how much flour and butter would those layers need?

Reveal solution

(a) 3 layers need $3 \times 200 = 600$ g sugar — exact! YES, she can. (b) Maximum is 3 (limited by sugar). (c) 3 layers: flour $3 \times 250 = 750$ g, butter $3 \times 150 = 450$ g.

Quick Review

Scale = unitless

Same units both sides

$1:n$

Real = map × n

Multiply all

Recipe scaling

Ratio preserved

Always when scaling

Total parts

Sum, then find 1 part

Reverse

Real / denominator = map

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