Let's break this problem down step by step.
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Equation for Judy's total amount after saving for \( d \) days:
Judy starts with $100 and saves $5 each day. Therefore, the equation representing how much money Judy will have after \( d \) days is: \[ J(d) = 100 + 5d \]
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Equation for Elenore's total amount after saving for \( d \) days:
Elenore starts with $75 and saves $7 each day. Thus, the equation representing how much money Elenore will have after \( d \) days is: \[ E(d) = 75 + 7d \]
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Equation to find out when Judy and Elenore will have the same amount of money saved:
To find out when Judy and Elenore have the same amount of money, we set their equations equal to each other: \[ 100 + 5d = 75 + 7d \]
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Solving the equation:
Let's solve the equation for \( d \): \[ 100 + 5d = 75 + 7d \] Rearranging gives: \[ 100 - 75 = 7d - 5d \] \[ 25 = 2d \] \[ d = \frac{25}{2} = 12.5 \]
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After \( d = 12.5 \) days, how much money do they each have?
We can substitute \( d = 12.5 \) back into the equations for Judy and Elenore to find out how much money they will have:
For Judy: \[ J(12.5) = 100 + 5(12.5) = 100 + 62.5 = 162.5 \]
For Elenore: \[ E(12.5) = 75 + 7(12.5) = 75 + 87.5 = 162.5 \]
Final answer:
- After 12.5 days, both Judy and Elenore will have $162.50 each.