More complex discounts

1.

Pose this problem:

• You go to a shop and buy an item that usually costs $64. They give you a 12.5% off discount.
• How much do you pay for the item? 

You might adapt this problem to include a more specific context that is more relevant to your students' interests, cultural backgrounds, and learning from other curriculum areas.

Draw this double number line model to represent the problem:

• What percentage is taken off? (12.5%)
• What percentage of the price do you pay? (87.5% since 100 – 12.5 = 87.5%)
• How can you work out 12.5% of $64? (Methods include finding 10% then 5% then 2.5% or finding 50% then 25% then 12.5%)
• What is 87.5% of $64? (12.5% of 64 = $8.00 so 87.5% of 64 = 64 – 8 = $76.00)

Develop the double number line as each question is answered, shown below:

2.

Demonstrate how calculator algorithms can be used to answer the same problem; begin with a semi-written method recorded as 87.5/100 x 64 and develop a set of steps, 64 ÷ 100 x 87.5 = $56.00.

• What does 64 ÷ 100 = 0.64 tell you? (1% of $64 = $0.64).
• Why do you multiply by 87.5? (The discounted price equals 87.5% of the normal price).

A simpler method is to key in 64 x 87.5% =. In this case, the calculator knows to enact the calculations above. Explore what happens if only 64 x 87.5 = is keyed in. The answer 5600 is one hundred times greater than the actual answer because no division by 100 has occurred.

3.

Pose similar problems and support students to create double number line models and use calculators to check the reasonableness of their answers. Discuss how sensible rounding of the numbers can be used to get a reasonable estimate for the answer; you might introduce relevant te reo Māori kupu such as whakahekenga ōrau (percentage discount) and rārangi tau (number line).

Examples might be:

• You go to a shop and buy an item that usually costs $96. They give you a 33% off discount.
• How much do you pay for the item?

Students should recognise that 33% equals one third so the discount is 1/3 x 96 = $32. One method is to subtract 96 – 32 = $64.00 to get a close estimate of the discounted price.

The calculation 96 ÷ 100 x 67 = $64.32 gives the exact discounted price. Similarly keying in 96 x 67% = gives the same answer.

• You go to a shop and buy an item that usually costs $160. They give you a 37.5% off discount.
• How much do you pay for the item?

Students might recognise that 37.5% equals three eighths so the discount is 3/8 x 160 = $60. One method is to subtract 160 – 60 = $100.00 to get the discounted price. Students should note that 100- 37.5 = 62.5% gives the percentage you will pay of the original price.
The calculation 160 ÷ 100 x 62.5 = $100 gives the exact discounted price. Similarly keying in 160 x 62.5% = gives the same answer.

• You go to a shop and buy an item that usually costs $78. They give you a 22% off discount.
• How much do you pay for the item?

22% is between 20% and 25% so either might be used to estimate. $78 is close to $80 which might be used to make the estimation easier.

• You go to a shop and buy an item that usually costs $125. They give you a 41% off discount.
• How much do you pay for the item?

41% is about 40% of two fifths. $125 divides easily into fifths to make the estimation easier.

1.

Introduce problems in which both the discount percentage and original price are untidy. See if students can generalise an algorithm that works for any discount problem, and use estimation strategies to check if an answer is reasonable. For example:

• Find the price of an item that usually costs $237, and is discounted by 17%.

17% is close to 20% or one fifth. $237 is close to $250. 1/5 x 250 = $50. Note that the estimate is likely to be high since both the percentage and original price were both rounded up. 1/5 x 200 = $40 might be a better estimate of the discount.

Calculating that the discounted price is 83% of the original price is important (100 – 17 = 83%). 237 x 83 % = gives $196.31 which is close to the estimate of 237- 40 = $197.

2.

Pose increasingly difficult problems that involve finding discount percentages. For example:

• An item usually costs $72 but you pay only $48. What percentage discount did you get?

The calculation 48 ÷ 72 = 0.6 represents the discounted price as a fraction of the original price. 0.66 means sixty-six hundredths so the discounted price is about 66% of the original price. 66.6% is actually two thirds so the discount was one third or 33.3%.

A double number line representation of the problem might be: