NCEA Level 1Percentages and fractionsAS 91946

Percentages and fractions: 7 exam questions step by step

What this lesson covers

  • A percentage is always a percentage of something, so the base must be identified first.
  • Percentage change compares the change with the original amount, not with the new one.
Key idea

7 questions, one first question

  • 1. A percentage of what? Find the base.
  • 2. Multiply for a part or a new price.
  • 3. Divide to undo a change.
  • 4. Answer in a sentence, to the cent.
Your turn

Question 1 · Achieved, in context

  • A jacket costs $84 before GST.
  • GST is 15%.
  • Find the price including GST.
Worked example

Q1: add 15% GST to $84

  • GST is 15% of the base.
  • Or multiply by 1.15.
  1. 15% of 84 = 0.15 × 84base: $84
  2. = $12.60
  3. $84 + $12.60 = $96.60add the GST on
  4. 84 × 1.15 = 96.6115% in one step
  5. Price with GST: $96.60
Your turn

Question 2 · Achieved, in context

  • A marae kitchen has 36 kg of potatoes.
  • 5/12 are used on Saturday,
  • and 1/3 on Sunday.
  • How many kg are left? What fraction?
Worked example

Q2: fractions of 36 kg

  • Divide by the bottom,
  • multiply by the top.
  1. 5/12 of 36 = 36 ÷ 12 × 5
  2. = 3 × 5 = 15 kg
  3. 1/3 of 36 = 36 ÷ 3 = 12 kg
  4. used: 15 + 12 = 27 kg
  5. left: 36 − 27 = 9 kg
  6. 9/36 = 1/4÷ 9 top and bottom
  7. 1 − 5/12 − 4/12 = 3/12 = 1/4
  8. 9 kg left, a quarter of them
Your turn

Question 3 · Merit, in context

  • A phone costs $517.50 including GST.
  • Find the price before GST.
  • Careful: do not just take 15% off.
Worked example

Q3: take GST off $517.50

  • With GST = 115% of base.
  • Undo it: divide by 1.15.
  1. base × 1.15 = $517.50100% + 15%
  2. base = 517.50 ÷ 1.15undo × with ÷
  3. = $450.00
  4. 450 × 1.15 = 517.50
  5. 517.50 × 0.85 = 439.87515% of the wrong base
  6. GST = 517.50 − 450 = $67.50
  7. Before GST: $450.00
Your turn

Question 4 · Merit, in context

  • Petrol went from $2.48 a litre
  • to $2.79 a litre.
  • Find the percentage increase.
Worked example

Q4: $2.48 to $2.79

  • Change ÷ original × 100.
  • Original = the old price.
  1. change = 2.79 − 2.48 = $0.31
  2. % change = 0.31 ÷ 2.48 × 100
  3. = 12.5%
  4. 0.31 ÷ 2.79 × 100 ≈ 11.1%divided by the new price
  5. 2.48 × 1.125 = 2.79
  6. Petrol rose by 12.5%.
Your turn

Question 5 · Merit, in context

  • A surfboard is $640 in two shops.
  • Shop A: 25% off, then 10% off that.
  • Shop B: 35% off.
  • Which is cheaper, and by how much?
Worked example

Q5: two discounts or one?

  • Each % is of a new base.
  • 25% then 10% is not 35%.
  1. A: 640 × 0.75 = $48025% off: pay 75%
  2. 480 × 0.9 = $43210% off the $480
  3. B: 640 × 0.65 = $41635% off: pay 65%
  4. 432 − 416 = $16
  5. 0.75 × 0.9 = 0.675A: you pay 67.5%
  6. A takes off 32.5%, not 35%
  7. Shop B is $16 cheaper.
Your turn

Question 6 · Excellence, in context

  • Mere earns $62,000 a year.
  • Her KiwiSaver rate goes 3% to 4%.
  • A site: her contributions "rose 1%".
  • Is that right? Explain.
Worked example

Q6: percent or percentage points?

  • Points: a difference.
  • Percent: ÷ the original.
  1. 3% of $62,000 = $1860per year
  2. 4% of $62,000 = $2480per year
  3. increase = 2480 − 1860 = $620
  4. 620 ÷ 1860 × 100 ≈ 33.3%change ÷ original
  5. 4% − 3% = 1 percentage point
  6. 1 ÷ 3 × 100 ≈ 33.3%from the rates
  7. Rate: +1 point. Dollars: +33.3%
Your turn

Question 7 · Excellence, in context

  • Sam puts $4000 in a term deposit
  • at 4.5% a year, compounded yearly.
  • Find the interest after 3 years.
  • Then compare with simple interest.
Worked example

Q7: compound interest

  • Each year: × 1.045.
  • Round only at the end.
  1. Year 1: 4000 × 1.045 = 4180
  2. Year 2: 4180 × 1.045 = 4368.1
  3. Year 3: 4368.1 × 1.045
  4. = 4564.6645
  5. 4000 × 1.045³ = 4564.6645
  6. balance = $4564.66to the cent, at the end
  7. interest = 4564.66 − 4000
  8. = $564.66
Worked example

Q7: compare with simple interest

  • Simple: on $4000 only.
  • Compound: on interest too.
  1. interest = $564.66
  2. 4000 × 0.045 = $180simple, per year
  3. 3 × $180 = $540three years
  4. 564.66 − 540 = $24.66
  5. Compound: $564.66 interest,
  6. $24.66 more than simple.
Summary

What solved all seven

  • Find the base, then multiply or divide.
StepHow
Find basea percentage of what?
Add GSTmultiply by 1.15
Undo GSTdivide by 1.15, never × 0.85
% changechange ÷ original × 100
Discountseach one off the new price
% pointsa difference of two rates
Roundingto the cent, only at the end

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