Susan earns $10 per hour tutoring elementary students in math. She also earns $15 per hour cleaning in the library. Her goal each week is to earn at least $240 while working no more than 20 hours.

Question

Part B

Which combinations allow Susan to reach her weekly goal? Select the three correct combinations.

1. 9 hours tutoring and 10 hours cleaning the library
2. 10 hours tutoring and 10 hours cleaning the library
3. 11 hours tutoring and 8 hours cleaning the library
4. 12 hours tutoring and 8 hours cleaning the library
5. 13 hours tutoring and 7 hours cleaning the library
6. 14 hours tutoring and 7 hours cleaning the library

Answer 1

First, solve the system of inequality:

t: hours tutoring Math

c: hours cleaning library

t + c <= 20

10t + 15c >= 240

Replace t <= 20 - c in second inequality:

10 (20 - c) + 15c >= 240

200 - 10c + 15c >= 240

200 + 5c >= 240

5c >= 240 - 200

5c >= 40

c >= 40 / 5

c >= 8

t <= 20 - c

t <=20 - 8

t <= 12

Then, analyze which combinations are correct, and satisfy:

t <= 12

c >= 8

10t + 15c >= 240

first combination:

t = 9 <= 12

c = 10 >= 8

10(9) + 15(10)

90 + 150 = 240

240 >= 240

CORRECT COMBINATION

second combination:

t = 10 <= 12

c = 10 >= 8

10(10) = 15(10)

100 + 150

250 >= 240

CORRECT COMBINATION

third combination:

t = 11 <= 12

c = 8 >= 8

10(11) + 15(8)

110 + 120

230 < 240

INCORRECT. Doesn't satisfy.

forth combination:

t = 12 <= 12

c = 8 >= 8

10(12) + 15(8)

120 + 120

240 >= 240

CORRECT COMBINATION

fifth combination

t = 13 > 12

INCORRECT. Doesn't satisfy.

sixth combination:

t = 14 > 12

INCORRECT. Doesn't satisfy.

So, the correct combinations are First, Second and Fourth.

Answer 2

1. 9 hours tutoring and 10 hours cleaning the library: 9(10)+10(15) = 240 this satisfies both
2. 10 hours tutoring and 10 hours cleaning the library: 10(10)+10(15) = 250 this satisfies both
3. 11 hours tutoring and 8 hours cleaning the library: 11(10)+8(15) = 230 no, not enough
4. 12 hours tutoring and 8 hours cleaning the library: 12(10)+8(15) = 240 this satisfies both
5. 13 hours tutoring and 7 hours cleaning the library: 13(10)+7(15) = 230 no, not enough
6. 14 hours tutoring and 7 hours cleaning the library - This one totals 21 hours, so this is not correct