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 \( d \).
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Subtract \( 5d \) from both sides: \[ 100 = 75 + 2d \]
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Subtract 75 from both sides: \[ 25 = 2d \]
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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.