073i - Consolidation Ratios

Lesson 73 — Problem Solving and Consolidation: Ratios

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

Learning Intentions

• To consolidate representing, simplifying, scaling and sharing with ratios.
• To select and justify a strategy for unfamiliar ratio problems.

Success Criteria

I can:

1. Move fluently between ratios, fractions and percentages.
2. Choose the right method: scaling, unitary, parts, or fraction-of-whole.
3. Solve unfamiliar and multi-step ratio problems.
4. Check answers against both the total and the ratio.

Warmup

(6 minutes — the four methods, pairs)

Here are four questions. Match each to the best method — scaling, unitary, parts, or fraction-of-whole — then solve.

1. Share 563 : 4$.
2. The ratio is ; the first part is . Find the second.
3. Ratio of cats to dogs is . What fraction of the animals are cats?
4. A recipe ratio becomes .

Answers: 1. Parts — 24$32756\tfrac25\times 412$.

Activities

Activity 1 — Mixed Skills Circuit (14 min)

Stations or graded worksheet spanning Lessons 70–72.

Station A — simplify.

4. g : kg

Station B — scale and share.

6. Share 1352 : 3 : 4$.
7. Two numbers in ratio differ by . Find both.
8. Flour : butter ; a baker uses g of butter. How much flour?

Station C — maps and conversions.

9. Scale ; map length cm. Real distance in km?
10. Real distance km on a map. Map length?
11. A m boat is drawn cm long. Find the scale.

Station D — ratio meets fraction and percentage.

12. In a choir, sopranos : altos . What percentage are altos?
13. A paddock holds sheep and goats in ratio ; there are animals. How many goats?
14. Ana and Ben share a bill in ratio . Ana pays 18$ more than Ben. Find the bill.

(Answers: 1. ; 2. ; 3. ; 4. ; 5. ; 6. ; 7. and ; 8. g; 9. km; 10. cm; 11. ; 12. ; 13. one part ; goats; 14. parts , one part , bill 72$.)

Activity 2 — Extended Problems (14 min)

Pairs. These need a plan before arithmetic.

Problem 1 — The paint order. Purple paint mixes red : blue . A hall needs L of purple.

(a) How much of each colour?

(b) Red comes in L tins and blue in L tins. How many tins of each must be bought?

(c) Red tins cost 32$27$. Find the total cost.

Problem 2 — The three-town map. On a map, town A is cm from town B, and B is cm from C.

(a) Find both real distances.

(b) A cyclist rides A→B→C at an average of km/h. How long does the trip take, to the nearest minute?

Problem 3 — The mixed bag. A bag holds red, green and blue tokens in ratio . There are more blue tokens than green.

(a) How many tokens in total?

(b) What fraction are red? What percentage (1 d.p.)?

Socratic scaffolding for Problem 1(b):

Prompt	Purpose
From (a), how much of each colour?	Red L, blue L.
Red tins are L. How many?	tins exactly.
Blue tins are L. How many?	— not exact.
Round which way?	Up — running short of blue ruins the mix. tins.
Does buying extra blue break the ratio?	The mix is still made ; the extra L stays in the tin.
Looking back	The rounding rule from Lesson 61 (buy up) applies inside ratio problems too.

Answers:

1. (a) L red, L blue (b) red tins, blue tins (c) 354$.
2. (a) km and km (b) total km; hours exactly.
3. (a) blue green parts , so one part ; total (b) red .

Activity 3 — Inquiry: Design a Question (6 min)

Pairs, then swap.

Write your own ratio problem that:

• uses a three-part ratio,
• gives the difference between two of the shares (not the total),
• has whole-number answers.

Solve it yourself first, then swap with another pair.

Socratic prompts for pairs designing:

Prompt	Purpose
Start from the answer. Pick a value for one part.	E.g. one part .
Choose a ratio.	E.g. .
What difference will you reveal?	E.g. between the largest and smallest: parts .
Write the story.	”Three friends share marbles in ratio ; the biggest share exceeds the smallest by …”
Why does building backwards guarantee whole numbers?	You chose the part value to be whole — the shares inherit it.

This closes the loop from Lesson 36’s “build and swap” — constructing problems is the surest test of understanding the structure.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Simplify .
2. Share 2102 : 5$.
3. On a map, a road is cm. Find its real length in km.
4. In a club, the junior : senior ratio is , and there are more juniors than seniors. Find the total membership.
5. Reasoning. A student shares 602:3$24$40$. Use both checks to show something is wrong, and fix it.

Answers: 1. ; 2. 60$150560,000= 5.6= 1615 \times 16 = 24024 + 40 = 64 \ne 60= 12$24$36$ — the student found one share correctly but slipped on the other.

Common Misconceptions

Misconception	How to pre-empt it
Choosing a method at random rather than reading what is given.	The warmup’s method-matching makes selection itself the skill.
Forgetting that “difference” problems divide by the difference of parts.	Problem 3 and CFU Q4 rehearse it; the Lesson 71 table stays displayed.
Rounding tin/bus/ticket quantities to the nearest.	Problem 1(b) re-applies Lesson 61’s round-up rule inside a ratio context.
Simplifying a three-part ratio only pairwise.	All parts divide by the same HCF at once.
Presenting a percentage without stating what it is a percentage of.	Sentence answers naming the base.
Believing a designed problem is correct without solving it.	The design task requires authors to solve before swapping.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). The ratio of three angles at a point is . Find the largest angle.

Answer

Angles at a point total ; one part ; largest .

E2 (AMC Junior style). A scale model of a tower is cm tall. How tall is the real tower?

Answer

cm m.

E3 (Challenge). Alloy A is copper : zinc ; alloy B is copper : zinc . Which alloy has the higher proportion of copper?

Answer

A: copper. B: copper. Alloy B.

E4 (Challenge). In a box, the ratio of pens to pencils is . Adding pens makes the ratio . How many pencils are there?

Answer

Pencils unchanged at ; pens go , so : pencils. (Check: pens ; ✓)

E5 (Challenge). Three numbers are in ratio . The sum of their squares is . Find the numbers.

Answer

Let them be : , so and . The numbers are , , .

Homework

1. Simplify: (a) (b) (c) (d) mL : L.
2. Share: (a) 965 : 7$2202 : 4 : 5$.
3. The ratio has first part . Find the second part and the total.
4. Two numbers in ratio differ by . Find both.
5. Scale : (a) map cm → real km? (b) real km → map cm?
6. Bronze mixes copper : tin . How much of each in kg of bronze?
7. A choir’s soprano : alto : tenor ratio is . There are more sopranos than tenors. Find the size of the choir.
8. Juice concentrate is mixed with water. (a) What fraction of the drink is concentrate? (b) How much concentrate in L of drink?
9. Reasoning. Explain the difference between dividing by the sum of parts and dividing by the difference of parts, and when each applies.
10. Challenge. A model car is built to scale . The real car is m long and its wheels are cm across. Find the model’s length and wheel size in centimetres.

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) 40, $56$40, $80, $1001236120117524192.250.252= 8= 4= 12 \times 4 = 48\tfrac18250480 \div 24 = 2060 \div 24 = 2.5$ cm.