Percentages as change (% can be more than 100%)

A t-shirt usually sells for $24. The shop has a 25% off sale.

• How much do you pay for the t-shirt?

Signs of fluency and understanding

• Recognises that 25% equals one quarter. Calculates one quarter of 24 as $6 and subtracts this from the whole ($24) to get $18. May calculate three quarters of $24 by dividing $24 by four to get $6, then multiplying by three to get $18.

What to notice if they don’t solve the problem fluently

• Recognises that 25% is equivalent to one quarter but does not apply this to accurately calculate 24 ÷ 4 = $6 and/or subtract 24 – 6 = $18. This indicates that the student needs to become more proficient at whole number division and subtraction.
• Does not recognise 25% as one quarter. May try finding 10% then 5% and subtracting repeatedly. The memory load required for this strategy may become prohhibitive. This indicates that the student needs more practice working with common fraction to percentage conversions and with finding a fraction of a quantity.

Supporting activity

• Simple discounts

A hoodie usually sells for $55. The shop has a 40% off sale.

• How much do you pay for the hoodie?

Signs of fluency and understanding

• Calculates 10% of $55 as $5.50. Uses 10% to find 40% of $55, which equals $22.00, and subtracts this to find $33. May use 10% to find 60% (discounted cost) using 6 x 5.5 = $33.
• Knows that 40% equals two fifths, then calculates 2/5 x 55 = $22 and subtracts, or calculates 3/5 x 55 = $33.

What to notice if they don’t solve the problem fluently

• Does not recognise 40% as a fraction (i.e., 4/10, made up of four amounts of 10%), so is unable to proceed. This indicates that the student needs to work on recognising percentages as common fractions.
• Recognises that 40% is made up of four amounts of 10%. Attempts to divide 55 by 10 and may get $5.50. Finds the memory load associated with multiplying 5.5 by four then subtracting $22 from $55 becomes burdensome. Access to paper may support their work. This indicates a need for the student to build up their calculation strategies with whole numbers and simple decimals.

Supporting activity

• More complex discounts

A hat usually sells for $35. The shop increases the price to $42.

• What is the percentage increase compared to the earlier price?

Signs of fluency and understanding

• Calculates the increase in price as $7. Expresses the increase as a fraction of 35 and uses equivalence to find 7/35 = 1/5. Knows that 1/5 = 20%. Concludes that the price has increased by 20%.
• Recognises that seven is a common factor in both $35 and $42. Uses equivalent fractions to express 42/35 as 6/5 = 120%. Concludes that the price has increased by 20%.

What to notice if they don’t solve the problem fluently

• Identifies that the hat has increased in price by $7 but is unsure how to calculate the percentage increase. This indicates the student will benefit from learning diagrams to represent percentage increase.
• Expresses the increase of $7 out of $42 (one sixth) and is not sure how to find one-sixth as a percentage. This indicates that the student has not understood that the original price of $35 is the whole (one unit) in this problem.

Supporting activity

• Percentage increase

Before you could buy a super scarf for $48. Now a super scarf costs $108.

• By what percentage has the price increased compared to the earlier price?

Use a calculator, or pencil and paper, to work this problem out.

Signs of fluency and understanding

• Calculates 108 ÷ 48 = 2.25 using a calculator. Recognises that 2.25 = 225% so the increase equals 125%, allowing for the original 100%.
• Finds the increase in cost using subtraction: 108 – 48 = $60. Expresses the increase as a fraction, 60/48. Uses common factors to simplify 60/48 =10/8 = 5/4, or uses division to find 100 ÷ 48 = 1.25. Concludes from their calculations that the increase equals 125%.

What to notice if they don’t solve the problem fluently

• Recognises that the increase equals $60 but is unsure which fraction, 48/60 or 60/48, represents the increase. This indicates that the student needs support in identifying the unit of comparison and will benefit from instruction on creating supporting diagrams. A preference for 48/60 may indicate that the student believes that percentages cannot be greater than 100%.
• Experiments with calculating using the numbers available when provided with a calculator. For example, the student may try 48 ÷ 108 = 44.4444% and conclude that 44% is a reasonable answer for the increase. Trial calculation indicates that the student needs support to structure the problem and identify the unit of comparison.

Supporting activity

• More complex percentage increase

The jeans cost $96, and the shirt costs $60.

• What percentage of the cost of the jeans is the cost of the shirt?
• What percentage of the cost of the shirt is the cost of the jeans?

Signs of fluency and understanding

• Calculates 96 ÷ 60 = 1.6 and concludes that the jeans are 160% of the price of the shirt. They may use equivalent fractions to get the same result: 96/60 = 32/20 = 16/10 = 160%.
• Calculates 60 ÷ 96 = 0.625 and concludes that the shirt is 62.5% of the price of the jeans. They may use equivalent fractions to get the same result, 60/96 = 20/32 = 10/16 = 5/8 = 62.5%.

What to notice if they don’t solve the problem fluently

• Attempts various calculations but gets confused as to what each percentage refers to. They might calculate 96 ÷ 60 = 1.6 but be unsure what percentage that represents. This indicates the student needs more support with converting fractions to percentages, including percentages and fractions greater than one.
• Correctly calculates the decimal to percentage conversions, 96 ÷ 60 = 1.6 = 160% and 60 ÷ 96 = 0.625 = 62.5, but is unsure which comparison each percentage refers to. This indicates that the student needs to identify the unit of comparison and recognise situations where the percentage is greater than 100%.

Supporting activity

• Comparing in both directions