066i - Percentages of Quantities

Lesson 66 — Calculating Percentages of Quantities

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

Learning Intentions

• To calculate a percentage of a quantity using mental and written strategies.
• To apply percentage calculations to discounts and increases.

Success Criteria

I can:

1. Find , , and of a quantity mentally.
2. Build harder percentages from these benchmarks.
3. Calculate a percentage of a quantity by converting to a fraction or decimal.
4. Find a price after a percentage discount or increase.

Warmup

(6 minutes — the benchmark ladder, mini whiteboards)

Start with 240$ and find each in turn:

Answers: ; ; ; ; .

The moves to name: is halving; divides by ; is half of ; divides by ; is half of half.

Activities

Activity 1 — Building Percentages from Benchmarks (14 min)

The strategy: decompose any percentage into benchmark pieces.

I do. Find of 80$.

I do. Find of 400$.

Careful — :

The all-purpose written method — via fraction or decimal:

Which to use? Benchmarks for mental work with friendly numbers; the decimal method for anything awkward or with a calculator.

We do: Find mentally, showing the decomposition:

1. of 65$
2. of 120$
3. of 48$
4. of 350$

(Answers: ; ; ; .)

You do:

1. of
2. of 60$
3. of 250$
4. of 70$
5. of 800$
6. of 40$

(Answers: ; ; ; ; ; .)

Discussion of Q4 and Q6. For , it is quicker to find and subtract: . And exceeds the original — percentages over mean growth.

Activity 2 — Discounts and Increases (14 min)

I do — discount, two methods. A 6025%$.

Method A — find the discount, subtract:

Method B — pay the remaining percentage:

Method B is one step and scales better — it becomes the standard method in later years. Teach both; let students choose.

I do — increase. A 1415%$.

We do:

1. 9030%$ off.
2. 4520%$ off.
3. 8.5010%$.
4. 26035%$.

(Answers: 63$36$9.35$351$.)

You do — mixed:

1. 12040%$ off.
2. 6512%$ booking fee.
3. A 28,00015%$ of its value in a year. Find its new value.
4. A 18\tfrac1330%$ off?

(Answers: 72$72.80$23,800\tfrac13= $1230%= $12.60$ — the fraction discount is better for the buyer.)

Activity 3 — Inquiry: Percentage of What? (7 min)

Pairs. The classic asymmetry.

A 10010%10%$ on Tuesday.

1. Predict: is the final price 100$, more, or less?
2. Calculate the price after each day.
3. Explain the result.
4. Does the order matter — fall first, then rise?

Socratic scaffolding:

Prompt	Purpose
Take the vote first.	Most predict 100$ — record the prediction.
Monday’s rise.	of 110$.
Tuesday’s fall — of what?	Of 110$, the current price — not the original.
Calculate.	of 99$.
So the answer is…?	99$ — a net loss.
Why?	The fall acted on a larger base than the rise.
Reverse the order.	then of 991.1 \times 0.9 = 0.9 \times 1.1 = 0.99$.
Looking back	A percentage is always of something, and the something can change mid-problem.

Answers: 2. 110$99$. 3–4. As scaffolded.

Checks for Understanding

(5 minutes — exit ticket)

1. Find mentally: (a) of 855%$14025%$32$.
2. Calculate of 150$.
3. A 7520%$. Find the sale price.
4. A 508%$. Find the new price.
5. Reasoning. Explain why a rise followed by a fall does not return a price to its start.

Answers: 1. (a) 8.50$7$8$5480%75 = $601.08 \times 50 = $541.1 \times 0.9 = 0.991%$ net loss.

Common Misconceptions

Misconception	How to pre-empt it
Finding the discount and reporting it as the sale price.	Re-read the question: was the discount or the price asked for? Both methods end with a price.
Believing rise then equal fall restores the original.	The inquiry addresses this directly.
Adding percentage points to dollars: 60 - 25% = $59.75$.	A percentage is an operation on the quantity, not a number of dollars. Convert first.
Believing percentages cannot exceed .	and the booking-fee example normalise it.
Finding as garbled into .	Write the decomposition line before calculating: .
Rounding money to whole dollars mid-calculation.	Keep cents throughout; round only the final answer if asked.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). of a number is . What is the number?

Answer

is , so is .

E2 (AMC Junior style). A store offers ” off, then a further off the reduced price”. What single discount is this equivalent to?

Answer

Paying means a discount — not .

E3 (Challenge). After a discount, a jacket costs 102$. What was the original price?

Answer

10285%$ of the original:

12015%102$117.3015%$ was of the original, not the sale price.)*

E4 (Challenge). In a school of students, are boys. How many girls are there?

Answer

Girls are : girls.

E5 (Challenge). Which is larger: of , or of ? Explain the general result.

Answer

Both equal . In general of equals of , because both are . A handy mental trick: of is hard; of is instant.

Homework

1. Find mentally, showing your decomposition: (a) of 4515%$8035%$12090%$6011%$300$.
2. Calculate: (a) of (b) of 923%$1500130%$40$.
3. Find the sale price: (a) 8520%$14035%$59.9010%$.
4. Find the new price: (a) 2650%$48012.5%$18.405%$.
5. A restaurant bill of 9610%$ service charge. Find the total.
6. A 125030%10%40%$ off?
7. A population of grows by in a year. Find the new population.
8. of a number is . Find the number.
9. Reasoning. Explain why ” off, then off again” is not free.
10. Challenge. After a price rise, a ticket costs 54$. What did it cost before?

Answers: Q1 — (a) 9$12$42$54$3384$59.80$45$52$68$91$53.91$39$540$19.32$105.601250 \times 0.7 \times 0.9 = $787.5040%$750901015050%0.5 \times 0.5 = 0.251.2p = 54p = $45$.