PERCENTAGE

QUESTION 1 [2 marks] — Criterion A Medium

In a survey, 65% of respondents said they prefer tea over coffee, and this represented 130 people.

a. Using the unitary method, find how many people represent 1% of respondents.

[1]

Show Solution

$$130 \div 65 = 2 \text{ people per 1\%}$$

b. Hence find the total number of people surveyed (100%).

[1]

Show Solution

$$2 \times 100 = 200 \text{ people}$$

QUESTION 2 [3 marks] — Criterion A Medium

A shop states that 8 items cost \$36, and this represents 40% of a customer's total weekly grocery budget.

a. Find the price of 1 item.

[1]

Show Solution

$$36 \div 8 = \$4.50 \text{ per item}$$

b. Using the unitary method, find the customer's total weekly grocery budget (100%).

[2]

Show Solution

1% of budget $=\$36\div40=\$0.90$. Total budget $=\$0.90\times100=\$90$.

QUESTION 3 [6 marks] — Criterion B Medium

Investigate the unitary method as a general two-step process.

a. If 35% of a quantity is 105, use the unitary method (find 1%, then find 100%) to find the whole quantity.

[2]

Show Solution

1% $=105\div35=3$. Whole quantity (100%) $=3\times100=300$.

b. Now find the whole quantity directly using division: $105 \div 0.35$. Compare this to your unitary method answer.

[2]

Show Solution

$105\div0.35=300$ — identical to the unitary method answer, confirming both approaches are mathematically equivalent, just organised differently.

c. Explain, in general terms using $p\%$ and value $v$, why 'finding 1% then multiplying by 100' always gives the same result as 'dividing by $p$ then multiplying by 100', i.e. dividing by $\frac{p}{100}$.

[2]

Show Solution

Finding 1% means dividing by $p$ (since $p\%$ corresponds to value $v$, 1% corresponds to $v/p$). Multiplying by 100 to reach 100% gives $\frac{v}{p}\times100 = \frac{100v}{p} = v \div \frac{p}{100}$ — showing both methods are algebraically the same operation.

QUESTION 4 [4 marks] — Criterion C Medium

A classmate finds the whole amount when 'given that 20% of a number is 50' by calculating $50\times20=1000$.

a. Explain the classmate's error, and demonstrate the correct unitary method approach.

[3]

Show Solution

The classmate multiplied by 20 instead of first finding 1% and then scaling to 100% (or dividing by the decimal 0.20). Correct approach: 1% $=50\div20=2.5$. Whole (100%) $=2.5\times100=250$.

b. Verify the correct answer by checking: does 20% of 250 actually equal 50?

[1]

Show Solution

$20\%\times250=0.20\times250=50$ ? — confirms 250 is correct.

QUESTION 5 [5 marks] — Criterion D Medium

A charity fun-run raised \$3,750, which was reported as being 75% of their fundraising target.

a. Use the unitary method to find the charity's full fundraising target (100%).

[3]

Show Solution

1% $=3750\div75=50$. Target (100%) $=50\times100=\$5000$.

b. The charity needs to decide whether to extend the campaign by one more week. Based on your answer, how much more money do they need to reach their target?

[2]

Show Solution

$$5000-3750=\$1250 \text{ more needed}$$

QUESTION 6 [3 marks] — Criterion A Medium

A crowdfunding campaign for a community water well raised \$4,200, which its organizers announced represents exactly 70% of the total goal.

a. Using the unitary method, find 1% of the target, then find the full target amount (100%).

[3]

Show Solution

1% $=4200\div70=60$. Target $=60\times100=\$6000$.

QUESTION 7 [5 marks] — Criterion B Medium

Investigate the unitary method as applied to finding a percentage that ISN'T a whole number, using 'if 8.5% of a quantity is 68, find the whole quantity'.

a. Using the unitary method (find 1% first), solve for the whole quantity.

[3]

Show Solution

1% $=68\div8.5=8$. Whole (100%) $=8\times100=800$.

b. Verify your answer using DIRECT division ($68\div0.085$) instead, and explain why both methods must give the same result.

[2]

Show Solution

$68\div0.085=800$ ? — matches. Both methods are mathematically equivalent: 'find 1% then scale to 100%' is really just dividing by $8.5$ then multiplying by $100$, which is the exact same overall operation as dividing directly by $\frac{8.5}{100}=0.085$.

QUESTION 8 [5 marks] — Criterion C Medium

A classmate uses the unitary method to solve 'if 45% of a number is 90, find the number' by writing: '1% = 90 ÷ 45 = 2, so the number is... 45 × 2 = 90? That's just the same number I started with, something's wrong.'

a. Identify the classmate's error, and clearly explain the CORRECT final step of the unitary method.

[3]

Show Solution

The classmate correctly found 1% = 2, but then multiplied by 45 (the ORIGINAL percentage) instead of by 100 (to reach the WHOLE, 100%). The unitary method's final step ALWAYS scales up to 100%, regardless of what percentage was originally given: correct answer $=2\times100=200$.

b. Verify the correct answer (200) by checking that 45% of 200 does indeed equal 90.

[2]

Show Solution

$45\%\times200=0.45\times200=90$ ?, confirming 200 is correct.

QUESTION 9 [6 marks] — Criterion C Hard

A national park's annual visitor revenue was \$84,000 last year, which reportedly represented a 14% increase from the previous year's revenue (i.e. last year's revenue is 114% of the previous year's).

a. Using the unitary method (treating last year's \$84,000 as 114%), find 1%, and hence find the PREVIOUS year's revenue (100%).

[3]

Show Solution

1% $=84000\div114\approx736.84$. Previous year's revenue $\approx736.84\times100\approx\$73{,}684$.

b. Communicate your full solution as a clear, well-organized written explanation (in full sentences, not just calculations), suitable for inclusion in a park management report, explicitly stating what '114%' represents and why this specific percentage (rather than simply 14%) was used as the basis for the unitary method calculation.

[3]

Show Solution

Since revenue increased by 14%, this year's revenue represents the original 100% PLUS an additional 14%, totalling $100\%+14\%=114\%$ of the previous year's figure. Using the unitary method with 114% (not 14%) as the reference percentage is essential, because \$84,000 corresponds to the ENTIRE new revenue figure — including the original base amount — not just the increase portion alone. Dividing \$84,000 by 114 gives the value of 1% of the previous year's revenue, which when multiplied by 100 recovers the previous year's full (100%) revenue of approximately \$73,684.

The unitary method in percentage