You earn $9.75 per hour at a part-time job. Your friend earns $9.35 per hour at a part-time job. The only other income you and your friend earn is a weekly allowance.
Determine whether it is possible for you and your friend to work the same number of hours and earn the same total amount in a given week for the following situations. Explain your reasoning.
Your allowance is $20 per week and your friend’s allowance is $10 per week. Write and solve an equation for each of the other three situations. Compare your solutions to your answers. What do you notice?
Situation 1: Your friend works 10 hours in a week
- Your earnings: $9.35 * 10 = $93.50
- Your friend's earnings: $9.35 * 10 = $93.50
- Your total income: $93.50 + $20 = $113.50
- Your friend's total income: $93.50 + $10 = $103.50
In this situation, you would earn more than your friend.
Situation 2: Your friend works 15 hours in a week
- Your earnings: $9.75 * 15 = $146.25
- Your friend's earnings: $9.35 * 15 = $140.25
- Your total income: $146.25 + $20 = $166.25
- Your friend's total income: $140.25 + $10 = $150.25
In this situation, you would earn more than your friend.
Situation 3: Your friend works 20 hours in a week
- Your earnings: $9.75 * 20 = $195
- Your friend's earnings: $9.35 * 20 = $187
- Your total income: $195 + $20 = $215
- Your friend's total income: $187 + $10 = $197
In this situation, you would earn more than your friend.
Situation 4: Your friend works x hours in a week
- Your earnings: $9.75 * x = 9.75x
- Your friend's earnings: $9.35 * x = 9.35x
- Your total income: 9.75x + 20
- Your friend's total income: 9.35x + 10
To determine if it is possible for you and your friend to earn the same total amount, we need to solve the equation: 9.75x + 20 = 9.35x + 10
9.75x - 9.35x = 10 - 20
0.40x = -10
x = -10 / 0.40
x = -25
The solution to this equation is x = -25. This means your friend would have to work -25 hours in a week to earn the same total amount as you. Since it is not possible to work negative hours, it is not possible for you and your friend to work the same number of hours and earn the same total amount in a given week in this situation.
Comparing the solutions, we notice that in situations 1, 2, and 3, you would always earn more than your friend regardless of the number of hours worked. However, in situation 4, there is no possible number of hours that would allow your friend to earn the same total amount as you.