Pay Rates

Activity One

In this activity, students meet rates in a context of time worked and pay. Use problem-solving groups of not more than four to sort out the information needed for question 1 and then get the students to discuss and share strategies that might lead to a solution. Check that they have found the correct answers to question 1 because they are needed in the questions that follow.
If the students plan to use a trial-and-error strategy to solve question 2, get them to make a sensible estimate first; this will encourage the more thoughtful approach described as trial and improvement.
A simple estimate can be made by noting that the girls both earn around $5 an hour. This means that Mrs White must be paying out at the rate of about $10 an hour. If this is the case, both must work for about 20 hours (10 x 20 = $200).
If the students now take the estimate of 20 hours as a starting point, they find that $5 x 20 = $100, $4.50 x 20 = $90, and $100 + $90 = $190. This leaves $10, so the girls must have worked 1 more hour: 20 + 1 = 21 hours.
Alternatively, the students could combine the girls’ pay and see that Mrs White is paying out a total of $9.50 per hour. 200 ÷ 9.5 = 200 ÷ 9 = 400 ÷ 19. A strategy for dividing 400 by 19 might go like this: 190 ÷ 19 = 10, and another 190 ÷ 19 = 10. That means 380 ÷ 19 = 20, which leaves 20 ÷ 19 = 1 and a little bit. So the girls must both work 21 hours. It’s a short step from here to the answer for question 2b.

Activity Two

In question 1, the girls’ pay rates are adjusted. For Karen, the increase is 10%. This can be thought of as an addition: 10% of $5.00 is 50c; adding 50c to $5.00 gives $5.50 per hour. But encourage your students to think multiplicatively when making percentage increases or decreases and to express the mathematics in one-step equations. In Karen’s case, the equation is: $5 x 1.10 = $5.50.
In Kylie’s case, the equation is: $4.50 x 1.20 = 4.50 x 1 + 4.50 x 0.20 = 4.50 + 0.90 = $5.40. Students can return to the methods they used in Activity One to find the answers to question 2.
Question 3 extends students’ thinking and should appeal to their sense of justice. Students need to see that Karen’s pay should be $5.40, the same as Kylie’s. They should also see that as a percentage, this increase must be less than the 10% that Karen was awarded. The increase from $5.00 is 40 cents. Here is one strategy for working out what this increase of 40 cents is as a percentage:
“1% of $5.00 is 5c. 5c x 8 = 40c. So Karen’s increase should have been 8% (8 x 1%).”
Have the students share their different ways of solving the question. Restate their explanations to clarify them and then record them, using numbers, on a chart or whiteboard.

Answers to Activities

Activity One
1. Kylie gets more pay, but Karen gets paid more per hr. (Kylie earns 27 ÷ 6 = $4.50/hr; Karen earns 20 ÷ 4 = $5.00/hr.)
2. a. Kylie is paid $4.50/hr and Karen $5.00/hr, so Mrs White must be paying
4.50 + 5.00 = $9.50/hr. 200 ÷ $9.50 = 21.05 hrs, so Mrs White must
be paying the girls for 21 hrs each, at a total cost of 21 x 9.50 = $199.50.
b. Kylie is paid 21 x 4.50 = $94.50 and Karen is paid 21 x 5.00 = $105.00.
Activity Two
1. Karen gets $5.50. (5.00 + 0.50 = $5.50)
Kylie gets $5.40. (4.50 + 0.90 = $5.40)
2. 18 hrs.
(5.50 + 5.40 = $10.90. 200 ÷ 10.90 = 18.35 hrs, or 18 whole hrs.) Karen gets $99.00 (18 x 5.50 = $99.00). Kylie gets $97.20 (18 x 5.40 = $97.20).
3. 8%. (Karen needs to increase her pay from $5.00 to $5.40 to match Kylie’s new rate. This is an increase of 40c for every $5.00 or 8c for every 100c, which is 8%.)