Question

Judy has $100 and Elenore has $75. Judy is saving $5 per day and Elenore is saving $7 per day.

After the number of days it took both to have the same amount of money saved, how much money do they have each?

Answer 1

Let's define a variable \( d \) to represent the number of days that pass.

Initially:
- Judy has $100.
- Elenore has $75.

After \( d \) days:
- Judy's total amount will be: \( 100 + 5d \)
- Elenore's total amount will be: \( 75 + 7d \)

We need to find the number of days \( d \) when both have the same amount of money. Therefore, we set up the equation:

\[
100 + 5d = 75 + 7d
\]

Now, let's solve for \( d \).

1. Subtract \( 5d \) from both sides:
\[
100 = 75 + 2d
\]

2. Subtract 75 from both sides:
\[
25 = 2d
\]

3. Divide by 2:
\[
d = 12.5
\]

Since \( d \) represents days, we need \( d \) to be a whole number. It looks like we can't achieve exact equality at a whole number of days. However, both will have the same amount after 12 full days or after rounding to the next integer, 13 days.

Calculating for 12 days:
- Judy's total amount after 12 days:
\[
100 + 5 \times 12 = 100 + 60 = 160
\]
- Elenore's total amount after 12 days:
\[
75 + 7 \times 12 = 75 + 84 = 159
\]

Calculating for 13 days:
- Judy's total amount after 13 days:
\[
100 + 5 \times 13 = 100 + 65 = 165
\]
- Elenore's total amount after 13 days:
\[
75 + 7 \times 13 = 75 + 91 = 166
\]

At 12 days, they do not have equal amounts, but we see that after 12 days, Judy will have $160 and Elenore will have $159, just $1 apart. However, after 13 days, they reach amounts of $165 (Judy) and $166 (Elenore).

Thus, the values they reach closest to equality are:
- After 12 days, Judy has **$160** and Elenore has **$159**.
- After 13 days, the amounts become **$165** and **$166**, respectively, so neither achieves exactly the same amount.

Final conclusion:
After **12 days**, Judy has **$160** and Elenore has **$159**.