Frank weighs 160 pounds and is on a diet to gain two pounds a week so that he can make the football team. John weighs 208 pounds and is on a diet to lose three pounds a week so that he can be on the wrestling team in a lower weight class. Homework Help ✎
a. If Frank and John can meet these goals with their diets, when
will they weigh the same, and how much will they weigh at that time?

To solve this problem, we need to set up equations that model the weight changes for both Frank and John over time and then find the point at which they will weigh the same.

### Step 1: Define Variables
- Let \( t \) represent the number of weeks.
- Frank's initial weight is 160 pounds, and he gains 2 pounds per week.
- John's initial weight is 208 pounds, and he loses 3 pounds per week.

### Step 2: Set Up the Equations
- Frank's weight after \( t \) weeks: \( \text{Frank's weight} = 160 + 2t \)
- John's weight after \( t \) weeks: \( \text{John's weight} = 208 - 3t \)

### Step 3: Set the Equations Equal to Each Other
We want to find the point at which their weights are the same, so we set the two equations equal:
\[
160 + 2t = 208 - 3t
\]

### Step 4: Solve for \( t \)
Add \( 3t \) to both sides:
\[
160 + 5t = 208
\]
Subtract 160 from both sides:
\[
5t = 48
\]
Divide both sides by 5:
\[
t = \frac{48}{5} = 9.6 \text{ weeks}
\]

### Step 5: Calculate Their Weight at That Time
Substitute \( t = 9.6 \) back into either of the original equations. Let's use Frank's equation:
\[
\text{Frank's weight} = 160 + 2(9.6) = 160 + 19.2 = 179.2 \text{ pounds}
\]

So, they will weigh the same in 9.6 weeks, and their weight at that time will be 179.2 pounds.

### Final Answer:
Frank and John will weigh the same after 9.6 weeks, and they will both weigh 179.2 pounds at that time.

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Would you like more details on any part of this solution? Here are some related questions to think about:

1. What if Frank's weight gain rate changes to 3 pounds per week?
2. How would the result change if John loses 4 pounds per week instead?
3. What if both Frank and John start at the same weight but with different weight change rates?
4. How would the situation change if Frank started at a different weight?
5. What would happen if Frank’s weight gain decreases after a certain number of weeks?

**Tip:** When solving equations with multiple variables, always clearly define your variables first, as this makes it easier to set up and solve the equations correctly.

Explore the solution to the problem where Frank and John are on different weight change diets. Calculate when they will weigh the same and their weights at that time. Suitable for Grades 7-9.

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