Two friends, Sam and Alex, are saving money. Sam starts with $20 and saves $5 per week. Alex starts with $40 and saves $3 per week. After how many weeks will they have the same amount of money?

Answer 1

To find out after how many weeks Sam and Alex will have the same amount of money, we can write equations for how much money each will have over time.

Let \( w \) represent the number of weeks.

For Sam:

• Initial amount: $20
• Savings per week: $5

The amount of money Sam has after \( w \) weeks is given by: \[ \text{Sam's amount} = 20 + 5w \]

For Alex:

• Initial amount: $40
• Savings per week: $3

The amount of money Alex has after \( w \) weeks is given by: \[ \text{Alex's amount} = 40 + 3w \]

We want to find the value of \( w \) for which Sam's amount equals Alex's amount: \[ 20 + 5w = 40 + 3w \]

Now, we can solve this equation for \( w \).

1. Subtract \( 3w \) from both sides: \[ 20 + 5w - 3w = 40 \] \[ 20 + 2w = 40 \]

2. Subtract 20 from both sides: \[ 2w = 40 - 20 \] \[ 2w = 20 \]

3. Divide both sides by 2: \[ w = \frac{20}{2} = 10 \]

Thus, Sam and Alex will have the same amount of money after \( \boxed{10} \) weeks.

To verify, we can calculate how much each person will have after 10 weeks:

• Sam's amount after 10 weeks: \[ 20 + 5 \times 10 = 20 + 50 = 70 \]

• Alex's amount after 10 weeks: \[ 40 + 3 \times 10 = 40 + 30 = 70 \]

Both Sam and Alex will have $70 after 10 weeks, confirming that our solution is correct.