070i - Representing and Simplifying Ratios

Lesson 70 — Representing and Simplifying Ratios

Strand: Number | Descriptor: AC9M7N08 | Duration: 45 minutes

Learning Intentions

• To recognise and represent ratios comparing two or more quantities.
• To simplify ratios, including those involving units and fractions.

Success Criteria

I can:

1. Write a ratio from a description or diagram, in the correct order.
2. Explain how a ratio differs from a fraction.
3. Simplify a ratio to lowest terms.
4. Simplify ratios whose quantities carry different units.

Warmup

(6 minutes — counters or diagram, pairs)

A bowl contains red counters and blue counters.

1. Write the comparison of red to blue as two numbers.
2. What fraction of the counters are red?
3. If I remove one red and one blue, is the comparison still “the same”? Test your intuition.
4. What is the simplest way to describe the red-to-blue relationship?

Answers: 1. to ; 2. ; 3. to — not equivalent to (which is ); removing equal amounts changes a ratio; 4. For every red there are blue — the ratio .

Note the distinction opened by Q2: the ratio compares part with part; the fraction compares part with whole. This is today’s central distinction.

Activities

Activity 1 — Explicit Instruction: what a Ratio is (12 min)

Definition. A ratio compares quantities of the same kind, written with a colon: , read ” to “.

Order matters — state this early and often. “The ratio of flour to sugar is ” means three parts flour to one part sugar. Reversing it describes a very different cake.

Ratio versus fraction — the table to build with the class:

	Ratio	Fraction
Compares	part with part	part with whole
In our bowl	red for every blue	of every counters are red
Sum of parts	parts in the whole	denominator is the whole

The bridge: from ratio , the fractions of the whole are and .

Three-part ratios. Comparisons extend naturally: cordial made with concentrate, juice and water in ratio has parts in total.

I do — writing ratios from contexts:

1. A class has girls and boys. Girls to boys .
2. A necklace pattern repeats gold beads, silver. Gold : silver .
3. Concrete is mixed from part cement, parts sand, parts gravel: .

We do: Write each ratio in the stated order.

1. wins to losses.
2. mL of oil to mL of vinegar.
3. A recipe: eggs, g flour, g sugar — eggs to flour to sugar (note: mixed units are fine within a description, but only same-kind quantities can be simplified together).

(Answers: ; ; as counts/grams — comparison across different kinds is recorded but not simplified across kinds.)

Activity 2 — Simplifying Ratios (14 min)

The principle — identical to equivalent fractions. Multiplying or dividing every part of a ratio by the same number gives an equivalent ratio.

I do.

Divide by the HCF (Lesson 7) for one-step simplification.

Units must match before simplifying. Simplify cm to m:

Convert to the smaller unit first, then drop the units — a simplified ratio is unitless.

Fractions and decimals in ratios — clear them first.

We do: Simplify.

4. g : kg

(Answers: ; ; ; ; ; .)

You do:

5. cm : m
6. min : h

(Answers: ; ; ; ; ; ; ; .)

Activity 3 — Inquiry: the Pattern Strip (8 min)

Pairs, coloured squares or grid paper.

A wristband pattern uses red and white beads in the ratio .

1. Draw three different wristbands that fit the ratio, using different total numbers of beads.
2. What totals are possible? What totals are impossible?
3. A wristband has beads in this ratio. How many of each colour?
4. Could a wristband in ratio have exactly beads? Explain.

Socratic scaffolding for Q3–4:

Prompt	Purpose
How many beads in one complete “block” of the pattern?	.
So what totals are possible?	Multiples of :
For beads, how many blocks?	.
So how many red? White?	red, white.
Check.	✓ and ✓
Now Q4: is a multiple of ?	No — so a wristband cannot have exactly beads.
Looking back	The parts of a ratio come in whole blocks. The total must be a multiple of the parts’ sum.

Answers: 2. Multiples of only; 3. red, white; 4. No — is not a multiple of .

Bridging note: Q3 is a first taste of sharing in a ratio, treated fully in Lesson 71.

Checks for Understanding

(5 minutes — exit ticket)

1. A team wins games and loses . Write the win-to-loss ratio in simplest form.
2. Simplify: (a) (b) (c) m : km.
3. Simplify .
4. In a mix of cordial to water, what fraction of the drink is cordial?
5. Reasoning. Explain the difference between “the ratio of cats to dogs is ” and ” of the animals are cats.”

Answers: 1. ; 2. (a) (b) (c) ; 3. : ; 4. ; 5. The ratio compares cats with dogs ( cats per dogs, so of the animals are cats); the fraction compares cats with all animals ( of every animals). They describe different situations.

Common Misconceptions

Misconception	How to pre-empt it
Reading as the fraction of the whole.	The part–part versus part–whole table; revisit with every context.
Reversing the order of a ratio.	Always write the words above the numbers: “flour : sugar” then "".
Simplifying mixed units without converting: from ” cm : m”.	Units must match first. Convert to the smaller unit, then drop units.
Subtracting the same amount from both parts and calling it equivalent.	Warmup Q3 disproves it. Only multiplying or dividing preserves a ratio.
Simplifying only some parts of a three-part ratio.	Every part is divided by the same HCF.
Believing a ratio total can be any number.	The wristband inquiry: totals must be multiples of the parts’ sum.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). The ratio of boys to girls in a class is , and there are students. How many boys?

Answer

per part, so boys (and girls).

E2 (AMC Junior style). Simplify the ratio .

Answer

E3 (Challenge). The ratio simplifies to . Find .

Answer

per part, so .

E4 (Challenge). In a bag, the ratio of red to green marbles is and the ratio of green to blue is . Find the ratio red : green : blue.

Answer

Make the green terms match: and . So red : green : blue .

E5 (Challenge). A photo cm by cm is enlarged so its shorter side becomes cm. Show the sides stay in the same ratio, and find the longer side.

Answer

. Enlargement multiplies both sides by the same factor (), which preserves the ratio. Longer side cm. (Check: ✓)

Homework

1. Write each ratio in the order stated, then simplify: (a) wins to losses (b) adults to children (c) red, blue, green beads.
2. Simplify: (a) (b) (c) (d) .
3. Simplify, converting units first: (a) cm : m (b) g : kg (c) min : h (d) c : 3$.
4. Simplify: (a) (b) (c) (d) .
5. A fruit box holds apples and oranges in the ratio . What fraction of the fruit is (a) apples (b) oranges?
6. A pattern uses black and white tiles in ratio . (a) What totals are possible? (b) A path uses tiles — how many of each colour?
7. The ratio simplifies to . Find .
8. Reasoning. The ratio of sunny to rainy days was . Explain why this does not mean of days were sunny, and give the correct fraction.
9. Reasoning. Explain why adding to both parts of the ratio does not give an equivalent ratio.
10. Challenge. In a school, the ratio of Year 7s to Year 8s is , and Year 8s to Year 9s is . Find the ratio of Year 7s to Year 9s.

Answers: Q1 — (a) (b) (c) . Q2 — (a) (b) (c) (d) . Q3 — (a) (b) (c) (d) . Q4 — (a) (b) (c) (d) . Q5 — (a) (b) . Q6 — (a) multiples of (b) black, white. Q7 — . Q8 — a fraction of the whole cannot exceed ; the ratio is part-to-part. Sunny days were of the total. Q9 — (cross-check: ); only multiplying or dividing preserves proportion. Q10 — make the Year 8 terms match: Y7:Y8 and Y8:Y9 ; so Y7:Y8:Y9 and Y7:Y9 .