068i - Problem Solving with Rational Numbers

Lesson 68 — Problem Solving: Mixed Operations with Rational Numbers

Strand: Number | Descriptor: AC9M7N06 | Duration: 45 minutes

Learning Intentions

• To solve multi-step problems combining fractions, decimals and percentages.
• To plan a solution path before calculating.

Success Criteria

I can:

1. Identify which operations a worded problem requires, and in what order.
2. Move between forms mid-problem when it helps.
3. Track a changing quantity through several stages.
4. Check answers against an estimate and the context.

Warmup

(6 minutes — plan, don’t solve; pairs)

For each problem, write the solution plan only — the operations in order, no arithmetic.

1. “A 14025%$ off. You pay half now. How much is the first payment?”
2. “A tank is full. L are drained, leaving it full. What is its capacity?”
3. “Three friends split a 67.80$2.50$ tip. How much does each pay?”

Sample plans: 1. Find of , then halve; 2. Find the fraction drained (), set it equal to L, scale up; 3. Divide by , then add .

The habit: plan first. Most errors in multi-step problems are wrong order, not wrong arithmetic.

Activities

Activity 1 — Staged Problems (16 min)

Pairs. Full working, estimate first, sentence answer.

Problem 1 — The sale rack. A shirt marked 5625%\tfrac{1}{7}$ off what remains.

(a) Find the price after each stage.

(b) What single fraction of the original price is finally paid?

Problem 2 — The tank. A rainwater tank is full and holds L at that level.

(a) Find the tank’s capacity.

(b) of the full capacity is then used. What fraction of the capacity remains, and how many litres is that?

Problem 3 — The pay rise. Maya earns 22.50166%$ pay rise.

(a) Find her new hourly rate.

(b) Find her new weekly pay.

(c) How much extra per week is the rise worth?

Problem 4 — The recipe scale-up. A recipe for people uses cups of flour, cups of milk and g of butter. Scale it for people.

Socratic scaffolding for Problem 2:

Prompt	Purpose
Understand: what does ” full holds L” tell you?	is three-quarters of the capacity.
Plan the first step.	One quarter is , so capacity L.
Now ” of full capacity used”. Of or of ?	Of — the problem says full capacity. Read precisely.
Calculate the usage.	L.
What remains?	L.
As a fraction of capacity?	(or ).
Check	✓ — the fraction route agrees with the litre route.
Looking back	Mixing forms was forced on us: the problem gave a fraction and a decimal. Converting to one form early keeps control.

Answers:

1. (a) 4242 \times \tfrac67 = $36\tfrac34 \times \tfrac67 = \tfrac{9}{14}56 \times \tfrac{9}{14} = 36$ ✓)*
2. (a) L (b) of capacity L.
3. (a) 23.8523.85 \times 16 = $381.60381.60 - 360 = $21.606%360$)*.
4. Scale factor : flour cups; milk cups; butter g.

Activity 2 — The Estimation Gate (10 min)

Every answer must pass through an estimate before being accepted.

Task. For each, write an estimate, then calculate, then compare.

1. of 47.84$
3. of
4. 89.95 \div 4$

(Estimates and answers: 1. ; exact 41.86\approx 2.5 \times 4 = 109.75\approx \tfrac23 \times 312 \approx 208202.8\approx 90 \div 4 = 22.5$22.49$22.4875 \to $22.49\approx 0.83 + 0.72 \approx 1.55\tfrac56 + \tfrac{18}{25} = \tfrac{125+108}{150} = \tfrac{233}{150} = 1\tfrac{83}{150} \approx 1.553$.)

Discussion of Q5. The exact answer is far uglier than the decimal approximation. When is the approximation acceptable? (When the context is a measurement; not when exactness is demanded. The problem’s purpose decides.)

Activity 3 — Inquiry: the Shrinking Chocolate bar (8 min)

Pairs. A “shrinkflation” investigation.

A chocolate bar shrinks from g to g, but its price stays at 4.00$.

1. By what percentage did the mass fall?
2. Find the price per g before and after.
3. By what percentage did the price per gram rise?
4. Why are the answers to Q1 and Q3 different?

Socratic scaffolding:

Prompt	Purpose
Q1: the fall as a fraction of the original.	.
Q2: unit prices.	Before: 1.604.00 \div 2 = $2.00$/100 g.
Q3: the rise as a fraction of the original unit price.	.
Q4: why down but up?	The two percentages have different bases — the fall is measured against g, the rise against 1.60$.
Looking back	The same asymmetry as Lesson 66’s rise-then-fall inquiry. A percentage is always of its own base.

Answers: 1. ; 2. 1.60$2.0010025%$; 4. Different bases.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. A 9030%\tfrac19$ off the sale price. Find the final price.
2. A bottle is full and contains L at that level. Find its capacity.
3. Calculate, estimating first: .
4. A pay rate of 18.405%$. Find the new rate.
5. Reasoning. A bar shrinks in size at fixed price. Explain why the price per gram rises by more than .

Answers: 1. ; 561.5 \div 5 \times 8 = 2.4\approx 2 \times 2.4 = 4.8\tfrac74 \times \tfrac{12}{5} = \tfrac{21}{5} = 4.2$19.3290%\tfrac{1}{0.9} \approx 1.11111.1%$ rise.

Common Misconceptions

Misconception	How to pre-empt it
Applying a later percentage to the original rather than the current amount.	Problem 1 and the inquiry both hinge on this. Ask “percentage of what?” at every stage.
Calculating before planning.	The warmup makes the plan a deliverable in itself.
Refusing to change forms mid-problem.	Problem 2 forces a fraction–decimal mix. Convert early, to whichever form the numbers favour.
Treating full L as capacity L.	Ask “is this the whole or a part?” before any scaling step.
Reporting a decimal answer when the question demanded an exact fraction (or vice versa).	Match the answer’s form to the question’s purpose — discussed in Activity 2.
Skipping the estimate gate under time pressure.	No estimate, no marks — keep the routine.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A bar of soap loses of its mass each week. What fraction of the original remains after two weeks?

Answer

E2 (AMC Junior style). Ana spends of her savings, then of what remains, and has 48$ left. How much did she start with?

Answer

After stage 1: remains. After stage 2: remains.

E3 (Challenge). A price rises , then falls . What single percentage change is this?

Answer

A decrease overall.

E4 (Challenge). A container is full. Adding L makes it full. Find its capacity.

Answer

Not a whole number — either accept L or note that tidy problems are a curated luxury. (Teacher variant: adding L gives exactly L.)

E5 (Challenge). In a bag, of the marbles are red, are blue, and the remaining are green. How many marbles in the bag?

Answer

Homework

1. A 6425%10%$ off the sale price. Find the final price and the single equivalent discount.
2. A tank is full, holding L at that level. (a) Find its capacity. (b) If of the capacity is used, how many litres remain?
3. Tom’s hourly rate of 21.607.5%12$ hours a week. Find his new weekly pay.
4. A recipe for serves uses cups of flour and L of milk. Scale it for serves.
5. A bill of 86.40412.5%$ of their share as a tip. How much does each pay in total?
6. A puppy’s mass increases from kg to kg. Find the percentage increase.
7. Ana spends of her money, then of what remains, and has 36$ left. How much did she start with?
8. A juice box shrinks from mL to mL at the same price of 1.50100$ mL, and the percentage rise in unit price.
9. Reasoning. Explain why ” off then off” is a bigger total discount than ” off then off” — or is it? Justify with the multipliers.
10. Challenge. A number is increased by , and the result is decreased by . Show that the final value equals the original.

Answers: Q1 — 43.2032.5%540450 - 162 = 28821.60 \times 1.075 = $23.22$278.64\tfrac4330.8$21.60$2.70$24.30\tfrac{0.8}{3.2} = 25%\tfrac23 \times \tfrac34 = \tfrac12$72$0.50$0.60100= \tfrac{0.10}{0.50} = 20%0.75 \times 0.9 = 0.9 \times 0.75 = 0.675\tfrac65 \times \tfrac56 = 1$.