PERCENTAGE

QUESTION 1 [5 marks] — Criterion B Medium

Investigate a mental-maths shortcut method for finding percentages, using 10%, 5%, and 1% as 'building blocks'.

a. Find 10% of \$240 and 1% of \$240 using simple mental division.

[2]

Show Solution

10% of 240 = 24. 1% of 240 = 2.4.

b. Using only your answers from (a) (adding or combining them, without recalculating from scratch), find 23% of \$240.

[3]

Show Solution

$23\% = 2\times10\% + 3\times1\% = 2(24)+3(2.4)=48+7.2=55.2$. So 23% of \$240 is \$55.20.

QUESTION 2 [5 marks] — Criterion C Medium

A student finds 45% of 80 using decimals ($0.45\times80$) and gets 36. A classmate finds it using fractions ($\frac{45}{100}\times80$) and also gets 36.

a. Show both methods in full, confirming both give 36.

[3]

Show Solution

Decimal method: $0.45\times80=36$. Fraction method: $\frac{45}{100}\times80=\frac{3600}{100}=36$. Both confirm 36.

b. Explain why these two methods always give the same result, using the definition of a decimal.

[2]

Show Solution

A decimal like 0.45 IS simply the fraction $\frac{45}{100}$ written in a different form — they represent exactly the same value, so multiplying by either form must always give an identical result.

QUESTION 3 [4 marks] — Criterion D Medium

A restaurant bill totals \$68.50 before tax and tip. Sales tax is 8%, and a customary tip of 18% is added to the pre-tax amount.

a. Calculate the sales tax amount.

[1]

Show Solution

$$8\% \times 68.50 = \$5.48$$

b. Calculate the tip amount (based on the pre-tax bill).

[1]

Show Solution

$$18\% \times 68.50 = \$12.33$$

c. Find the total amount the customer pays (bill + tax + tip).

[2]

Show Solution

$$68.50+5.48+12.33 = \$86.31$$

QUESTION 4 [6 marks] — Criterion A Medium

A property developer pays an annual property tax of 0.85% on a building valued at \$185,000.

a. Calculate the annual property tax.

[2]

Show Solution

$$0.0085\times185000=\$1572.50$$

b. The building's value is expected to increase by 12% over the next 2 years. Find the NEW building value, then the NEW annual tax at the SAME 0.85% rate, and the total INCREASE in tax paid per year as a result.

[4]

Show Solution

New value: $185000\times1.12=\$207200$. New tax: $0.0085\times207200=\$1761.20$. Increase: $1761.20-1572.50=\$188.70$ per year.

QUESTION 5 [7 marks] — Criterion D Hard

A hospital's medication dosage is calculated as 2.5% of a patient's body mass (in kg), given in milligrams (1kg body mass ? dosage in mg is $2.5\%$ of mass IN GRAMS, i.e. mass×1000×0.025).

a. For a patient with body mass 72kg, calculate the dosage in mg, showing your unit conversion clearly.

[3]

Show Solution

Mass in grams: $72\times1000=72000$g. Dosage: $2.5\%\times72000=1800$mg.

b. The maximum SAFE dosage for this medication is 2000mg. Find the maximum body mass (to the nearest kg) for which the standard 2.5% dosage formula remains safe, and discuss why medical dosage formulas like this often need a stated MAXIMUM cap regardless of body mass.

[4]

Show Solution

$2.5\%\times(m\times1000)\le2000 \Rightarrow 25m\le2000 \Rightarrow m\le80$kg. Beyond 80kg, the formula would exceed the safe maximum, so a cap is essential — very large patients shouldn't receive proportionally scaled-up doses indefinitely, since medication safety often depends on absolute concentration limits in the bloodstream, not just body-mass-proportional scaling; a hard maximum protects against overdose regardless of how large the patient is.

QUESTION 6 [5 marks] — Criterion B Medium

Investigate a mental-maths method for finding 'awkward' percentages like 17.5% (a common tax rate in some countries), by breaking it into 10% + 5% + 2.5%.

a. For a bill of \$340, find 10%, 5%, and 2.5% separately.

[3]

Show Solution

10%: \$34. 5%: \$17. 2.5%: \$8.50.

b. Add these together to find 17.5% of \$340, then verify by calculating $0.175\times340$ directly.

[2]

Show Solution

$34+17+8.50=\$59.50$. Direct: $0.175\times340=\$59.50$ ?.

QUESTION 7 [7 marks] — Criterion D Hard

A solar farm generates enough electricity to power 3,400 homes at full capacity, but currently operates at only 68% of a target output due to a technical issue.

a. If the 3,400 homes figure represents the TARGET (100%) capacity, find the number of homes actually being powered at the current 68% output.

[2]

Show Solution

$$0.68\times3400=2312 \text{ homes}$$

b. Engineers estimate repairs will restore output at a rate of 4 percentage points per week. Determine how many WHOLE weeks are needed to reach at least 95% capacity, and calculate the number of ADDITIONAL homes that will be powered once this target is reached (compared to the current 2,312).

[5]

Show Solution

Percentage points needed: $95-68=27$. Weeks needed: $27\div4=6.75\to7$ whole weeks (rounding up, since partial weeks don't complete the repair). Homes at 95%: $0.95\times3400=3230$. Additional homes powered: $3230-2312=918$.

Finding a percentage of a quantity