072i - Solving Ratio Problems

Lesson 72 — Solving Problems Involving Ratios

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

Learning Intentions

• To solve multi-step problems involving ratios, including maps, scales and rates.
• To connect ratios with fractions and percentages in problems.

Success Criteria

I can:

1. Use a map scale to convert between map and real distances.
2. Combine ratio work with fraction and percentage reasoning.
3. Solve problems where a ratio changes.
4. Set out multi-step ratio solutions clearly.

Warmup

(6 minutes — retrieval mix, mini whiteboards)

1. Share 902 : 7$.
2. In the ratio , what fraction of the whole is the first part?
3. Simplify .
4. The ratio simplifies to . Find the missing number.

Answers: 1. 20$70\tfrac385 : 315$.

Activities

Activity 1 — Explicit Instruction: Map Scales (14 min)

A map scale is a ratio. A scale of means every cm on the map represents cm in the real world.

Unit conversion is the working muscle here (Lesson 18): cm m km.

I do — map to real. On a map, two towns are cm apart. Find the real distance.

I do — real to map. A km walking trail is drawn on the same map. How long is the line?

The direction check: map to real multiplies (real things are bigger); real to map divides. Estimate first — a m line would be absurd on a page.

I do — finding the scale. On a plan, a wall that is really m long is drawn cm long. Find the scale.

We do:

1. Scale ; map distance cm. Real distance?
2. Scale ; a room is m long. Plan length?
3. A cm map line represents km. Find the scale.

(Answers: cm km; cm; .)

Activity 2 — Mixed Ratio Problems (14 min)

Pairs. Full working and sentence answers.

Problem 1 — Ratio and percentage. In a school of students, the junior : senior ratio is . (a) How many juniors? (b) What percentage of the school are seniors?

Problem 2 — The changing ratio. A team’s win : loss ratio is after games (no draws). They then win their next games. What is the new ratio?

Problem 3 — Ratio of a remainder. Nadia spends of her pay on rent. The rest is split between savings and spending in ratio . Her pay is 1200$. Find each amount.

Problem 4 — Best buy via ratio. Brand A mixes cordial (syrup : water); Brand B mixes . Which drink tastes stronger?

Socratic scaffolding for Problem 4:

Prompt	Purpose
Understand: what does “stronger” mean here?	A higher proportion of syrup in the drink.
The ratios have different totals. What now?	Convert each to a fraction of the whole.
Brand A.	syrup.
Brand B.	syrup.
Compare and .	Common denominator : vs . B is stronger.
Alternative route?	Scale to the same syrup: vs . Same syrup, less water in B — stronger ✓
Looking back	Two valid methods; the scaling route avoided fractions entirely.

Answers:

1. (a) one part ; juniors (b) seniors , which is .
2. Before: wins, losses. After: (already in lowest terms).
3. Rent 400$8003:2$480$320$.
4. Brand B (see scaffolding).

Activity 3 — Inquiry: the Gear Ratio (6 min)

Pairs. Ratios as rates of turning.

On a bicycle, the front cog has teeth and the rear cog has .

1. Write the teeth ratio front : rear in simplest form.
2. When the pedals turn once, how many times does the rear wheel turn?
3. If the rider pedals at turns per minute, how fast does the rear wheel spin?

Socratic scaffolding:

Prompt	Purpose
Simplify the ratio.	.
The chain moves tooth by tooth. One pedal turn moves how many teeth?	teeth of chain.
How many rear turns is that?	turns.
So the ratio is the answer?	Yes — a gear turns the wheel times per pedal stroke.
At pedal turns/min?	wheel turns per minute.
Looking back	Gearing is a ratio acting on a rate — the same multiplication, applied per minute.

Answers: 1. ; 2. times; 3. turns per minute.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. On a map, a path is cm. Find the real distance in km.
2. A real distance of km appears on a map. How long is the line?
3. In a group, the child : adult ratio is and there are people. How many adults? What percentage is that?
4. Cordial A is mixed ; cordial B is mixed . Which is stronger?
5. Reasoning. Explain why converting ratios to fractions of the whole makes them comparable.

Answers: 1. cm km; 2. cm; 3. one part ; adults ; 4. A: ; B: — B is stronger (fifteenths are smaller than fourteenths); 5. Ratios with different totals cannot be compared part-to-part; fractions of the whole put both on the same base, like percentages.

Common Misconceptions

Misconception	How to pre-empt it
Multiplying when converting real to map.	The direction check: real distances are big, map distances small. Estimate before calculating.
Dropping unit conversions in scale work.	Convert to centimetres first, every time, then back at the end.
Comparing ratios part-to-part when totals differ.	Problem 4 — convert to fractions of the whole, or scale to a common part.
Updating only one side of a changing ratio.	Problem 2 — recompute both actual counts before re-forming the ratio.
Applying a ratio to the whole when it applies to a remainder.	Problem 3 — draw a bar model: rent first, then split what is left.
Treating a gear ratio as teeth added rather than a turning factor.	The chain-tooth count in the inquiry grounds it physically.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A map has scale . Two cities are cm apart on the map. What is the real distance?

Answer

cm km.

E2 (AMC Junior style). The ratio of red to green apples in a crate is . After red apples are removed, the ratio is . How many green apples are there?

Answer

Green unchanged: let green ; red goes from to , and gives , so . Green . (Check: red ; ✓)

E3 (Challenge). A photo cm wide is enlarged in the ratio (new : old). Find the new width, and the old height if the new height is cm.

Answer

New width cm. Old height cm.

E4 (Challenge). Two gears mesh: one has teeth and turns at rpm; the other has teeth. How fast does it turn?

Answer

Teeth passed per minute: . The larger gear turns rpm. (More teeth means slower — an inverse relationship.)

E5 (Challenge). The ratio of Ana’s age to Ben’s is now. In years it will be . Find their present ages.

Answer

Let the ages be and . Then :

Ana is , Ben is . (Check in years: ✓)

Homework

1. Scale . Find the real distance (km) for map lengths: (a) cm (b) cm (c) cm.
2. Scale . Find the map length for real distances: (a) km (b) km (c) m.
3. A plan shows a m wall as cm. Find the scale.
4. In a cinema of people, the adult : child ratio is . (a) How many children? (b) What percentage are adults?
5. A team’s win : loss ratio is after games. They lose the next . Find the new ratio.
6. Tom spends of his money on a game. The rest he splits between savings and snacks in ratio . He started with 150$. Find each amount.
7. Cordial X is mixed ; cordial Y is mixed . Which is stronger? Show your method.
8. A front cog has teeth, the rear has . (a) Simplify the gear ratio. (b) How many wheel turns per pedal turn? (c) At pedal turns/min, find the wheel rate.
9. Reasoning. Explain why a scale of makes cm represent exactly m.
10. Challenge. The ratio of two numbers is . If is added to each, the ratio becomes . Find the original numbers.

Answers: Q1 — (a) km (b) km (c) km. Q2 — (a) cm (b) cm (c) cm. Q3 — . Q4 — (a) one part ; children (b) . Q5 — . Q6 — game 60$905:1$75$15\tfrac17 = \tfrac{2}{14}\tfrac{2}{15}3:1324050,000= 500\tfrac{4k+9}{7k+9} = \tfrac2312k + 27 = 14k + 18k = 4.51831.52:3 \to$ different constants.)