Let's break down the problem step by step.
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Equation for Judy's total money after \(d\) days: Judy starts with $100 and saves $5 per day. Therefore, her total money after \(d\) days can be represented by the equation: \[ J(d) = 100 + 5d \]
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Equation for Elenore's total money after \(d\) days: Elenore starts with $75 and saves $7 per day. Therefore, her total money after \(d\) days can be represented by the equation: \[ E(d) = 75 + 7d \]
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Equation to find the number of days until Judy and Elenore have the same amount of money: To find out when Judy and Elenore will have the same amount of money saved, we set their equations equal to each other: \[ 100 + 5d = 75 + 7d \]
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Solving for \(d\): To solve the equation, first, we will rearrange it: \[ 100 - 75 = 7d - 5d \] \[ 25 = 2d \] \[ d = \frac{25}{2} = 12.5 \] This means it will take 12.5 days for Judy and Elenore to have the same amount of money saved.
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Calculating how much money each has after 12.5 days: We will substitute \(d = 12.5\) back into both equations to find their totals.
For Judy: \[ J(12.5) = 100 + 5(12.5) = 100 + 62.5 = 162.5 \]
For Elenore: \[ E(12.5) = 75 + 7(12.5) = 75 + 87.5 = 162.5 \]
So, after 12.5 days, both Judy and Elenore will have:
- Judy: $162.50
- Elenore: $162.50