Lesson 15 ~25 min Unit 1 · Financial Maths +85 XP

Introduction to Ratios

Comparing parts to parts: 3:1 flour to sugar means a recipe story, not a division sum.

Today's hook: A recipe calls for flour and sugar in the ratio $3:1$. How is this different from saying '$3/1$ as much flour as sugar'?

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Think First

warm-up

A recipe calls for flour and sugar in the ratio $3:1$. How is this different from saying '$3/1$ as much flour as sugar'? Jot down your first reaction — then we'll see who's right.

Record your answer in your workbook.

The Big Idea

+5 XP

A ratio $a:b$ compares two quantities of the SAME type (both parts of a whole, both flour vs sugar in cups). The order matters, and the units must match.

A ratio like $3:1$ means “for every 3 parts of one thing, there is 1 part of the other”. It's a comparison of like quantities — both in cups, both in kg, both in students. The colon `:` is read “to”. So $3:1$ = “three to one”. Order matters: $3:1$ is not the same as $1:3$.

$a:b$ means "$a$ parts for every $b$ parts" of the SAME thing

Same units

Both sides of the ratio must have the same units.

Order matters

$3:1$ and $1:3$ describe different things.

Parts, not totals

In $3:1$, the total is $3 + 1 = 4$ parts.

What You'll Master

objectives

Know

A ratio $a:b$ compares same-type quantities
The "$:$" symbol is read "to"
Order matters in a ratio
Total parts = $a + b$ (for 2-part) or $a+b+c$ (for 3-part)

Understand

Why ratios differ from fractions and rates
How $a:b$ relates to fractions $\tfrac{a}{a+b}$
Why mixing units in a ratio breaks meaning

Can Do

Write a ratio from a word problem
Interpret what each part represents
Distinguish ratio from rate or fraction

Words You Need

vocabulary

RatioA comparison of same-type quantities, written $a:b$.

PartsThe numbers in a ratio.

Total partsSum of all parts in the ratio.

AntecedentFirst number in $a:b$.

ConsequentSecond number in $a:b$.

Equivalent ratiosRatios that simplify to the same form, e.g., $2:1 = 4:2$.

Spot the Trap

heads-up

Wrong: "$3:1$ means $3 \div 1 = 3$" — Not in the same sense as fractions. The ratio describes parts, not a division.

Right: Ratio $3:1$ describes parts: every $3$ of one, $1$ of the other.

Wrong: "A class is $\tfrac{3}{1}$ girls to boys" — NO. $\tfrac{3}{1}$ is a fraction $> 1$, but ratios use $a:b$ form.

Right: In a $3:1$ class, total parts $= 4$. So $\tfrac{3}{4}$ girls and $\tfrac{1}{4}$ boys.

What a Ratio Tells You

+5 XP

A ratio gives a recipe. In $3:1$ flour to sugar, every batch has 3 parts flour and 1 part sugar — but a "part" can be any size.

A $3:1$ flour-to-sugar ratio could mean: 3 cups flour, 1 cup sugar (small batch). Or 6 cups flour, 2 cups sugar (double). Or 30 g flour, 10 g sugar (small biscuit). The relationship is the same — what changes is the size of one part. Total parts here are $4$, so flour is $\tfrac{3}{4}$ of the mix.

Total parts $= a + b$; Fraction of first $= \tfrac{a}{a+b}$

Recipe analogy

Scale up or down by changing part size.

Fractions of total

$\tfrac{a}{a+b}, \tfrac{b}{a+b}$.

Same units required

Compare like with like.

Ratios vs Fractions vs Rates

+5 XP

These three look similar but mean different things. Knowing the difference is essential.

Ratio $3:1$ compares same-type things (e.g., cups of flour vs cups of sugar). Fraction $\tfrac{3}{4}$ describes a part OF a whole — e.g., $\tfrac{3}{4}$ of the mix is flour. Rate $\$3/\text{kg}$ compares different units. They're related but not interchangeable.

Ratio $a:b$ vs Fraction $\tfrac{a}{a+b}$ vs Rate (different units)

Ratio

Same units, "$:$" symbol.

Fraction

Part OF a whole.

Rate

Different units, "$/$" symbol.

Watch Me Solve It · 3 examples

Watch Me Solve It · Flour to sugar

+15 XP per step

PROBLEM

A recipe uses flour and sugar in the ratio $3:1$. (a) How is this different from "$3/1$ as much flour as sugar"? (b) What fraction of the total is sugar?

1

(a) Meaning of $3:1$

3 parts flour for every 1 part sugar; total parts = 4

It's a recipe, not a division.
2

Compare to fraction

$\tfrac{3}{1} = 3$ would say flour is 3 TIMES sugar (same idea, but expressed differently)

The relationship is the same; the form is different.
3

(b) Fraction of sugar

$\tfrac{1}{1+3} = \tfrac{1}{4}$

Sugar makes up $\tfrac{1}{4}$ of the mix.

Answer(a) $3:1$ describes parts in a recipe; (b) sugar = $\tfrac{1}{4}$

Watch Me Solve It · Write a ratio

+15 XP per step

PROBLEM

In a class there are 15 girls and 12 boys. Write the ratio of girls to boys in simplest form.

1

Write the raw ratio

$15:12$

Girls : Boys.
2

Find HCF

HCF$(15, 12) = 3$

Both divisible by 3.
3

Simplify

$15 \div 3 : 12 \div 3 = 5:4$

Simplest form.

Answer$5:4$ girls to boys

Watch Me Solve It · Order matters

+15 XP per step

PROBLEM

A drink is mixed using cordial and water in the ratio $1:7$. (a) What is the ratio of water to cordial? (b) What fraction of the drink is cordial?

1

(a) Reverse the ratio

$7:1$

Order matters — water first now.
2

Total parts

$1 + 7 = 8$

Whole drink in parts.
3

(b) Fraction cordial

$\tfrac{1}{8}$

Of the total drink.

Answer(a) $7:1$; (b) $\tfrac{1}{8}$ cordial

Common Pitfalls

heads-up

Mixing units

Writing “5 cm : 2 m” without converting.

Fix: Convert to the same unit. $5$ cm $: 200$ cm $= 5:200 = 1:40$.

Confusing $a:b$ with $\tfrac{a}{b}$

Both useful, different meanings.

Fix: $a:b$ is recipe; $\tfrac{a}{b}$ is how many times bigger; $\tfrac{a}{a+b}$ is fraction of total.

Reversing the order

Writing “boys to girls” when asked for girls to boys.

Fix: Read the question. The order of the words determines the order of numbers.

Copy Into Your Books

Ratio Basics

$a:b$ = $a$ to $b$
Same units
Order matters

Parts and Fractions

Total = $a+b$
Fraction of first = $\tfrac{a}{a+b}$
$3:1 \to \tfrac{3}{4}$ and $\tfrac{1}{4}$

Ratio vs Fraction

Ratio: like to like
Fraction: part of whole
Both express relationships

Ratio vs Rate

Ratio: same units (cups:cups)
Rate: different units (km/h)
Don't confuse

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Introduction to Ratios

4 problems

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

1 8 boys, 6 girls. Ratio of boys to girls in simplest form?

HCF $=2$; $8:6 = 4:3$.$4:3$
2 A drink has 1 part syrup to 5 parts water. What fraction is syrup?

Total $=6$; syrup $= \tfrac{1}{6}$.$\tfrac{1}{6}$
3 Total of all parts in $2:3:5$:

$2+3+5 = 10$.$10$
4 Write the ratio: 20 g salt to 1 kg flour.

Convert: $20$ g : $1000$ g $= 1:50$.$1:50$

Complete in your workbook.

Quick Check · 5 questions

A class of 30 has 18 girls. The ratio of girls to boys is:

+10 XP

In the ratio $4:1$, the total number of parts is:

+10 XP

In $3:5$, what fraction does the first part represent?

+10 XP

Mia mixes paint with 200 mL blue and $50$ mL yellow. The blue:yellow ratio is:

+10 XP

Which of these is NOT a valid ratio?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. For each, write the ratio in simplest form: (a) 24 boys to 36 girls; (b) $400$ mL water to $500$ mL juice; (c) $\$20$ to $\$45$.

Answer in your workbook.

Understand Easy 2 MARKS

Q7. A drink is mixed with $25$ mL cordial and $200$ mL water. (a) What is the ratio of cordial to water in simplest form? (b) What fraction of the drink is cordial?

Answer in your workbook.

Reason Hard 4 MARKS

Q8. A concrete mix uses 2 buckets of cement, 3 buckets of sand, and 4 buckets of gravel. (a) Write the ratio cement:sand:gravel. (b) What fraction of the mix is each ingredient? (c) For a job requiring 18 buckets of concrete total, how many of each ingredient are needed?

Answer in your workbook.

Comprehensive Answers

Quick Check

1. A — $3:2$.

2. D — $5$ parts.

3. B — $\tfrac{3}{8}$.

4. B — $4:1$.

5. C — $\$$ vs apples is rate.

Show Your Working Model Answers

Q6 (3 marks): (a) $24:36 \div 12 = 2:3$ [1]. (b) $400:500 \div 100 = 4:5$ [1]. (c) $20:45 \div 5 = 4:9$ [1].

Q7 (2 marks): (a) $25:200 \div 25 = 1:8$ [1]. (b) Total = 9 parts; cordial $= \tfrac{1}{9}$ [1].

Q8 (4 marks): (a) $2:3:4$ [1]. (b) Total = 9 parts; cement $\tfrac{2}{9}$, sand $\tfrac{3}{9} = \tfrac{1}{3}$, gravel $\tfrac{4}{9}$ [1]. (c) 1 part = $18 \div 9 = 2$ buckets [1]. Cement: $2 \times 2 = 4$. Sand: $3 \times 2 = 6$. Gravel: $4 \times 2 = 8$ [1].

Stretch Challenge · +25 XP, +10 coins

The Three-Way Mix

A drink station mixes three juices: orange, apple, and pineapple in the ratio $5:3:2$. The orange is the cheapest, the apple is twice the price per litre, and the pineapple is three times the price. (a) For a 10-litre mix, how many litres of each? (b) If orange costs $\$2$/L, what is the total cost of the mix? (c) What is the cost per litre of the mixed drink?

Reveal solution

(a) Total 10 parts = 10 L, so each part is 1 L. Orange: $5$ L. Apple: $3$ L. Pineapple: $2$ L. (b) Orange: $5 \times 2 = \$10$. Apple: $3 \times 4 = \$12$. Pineapple: $2 \times 6 = \$12$. Total: $\$34$ for $10$ L. (c) $\$34 \div 10 = \$3.40$/L mixed.

Quick Review

Ratio = $a:b$

Same units, order matters

Total parts

$a + b$ (or $a+b+c$)

Part fraction

$\tfrac{a}{a+b}$

Vs fraction

Ratio is parts, fraction is part-of-whole

Vs rate

Ratio = same unit; rate = diff units

Same units

Convert before forming ratio

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