To solve this, let's break it down step by step.
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Equations for total money after saving for d days:
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Judy starts with $100 and saves $5 per day. The equation representing how much money Judy will have in total after saving for \(d\) days is: \[ J(d) = 100 + 5d \]
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Elenore starts with $75 and saves $7 per day. The equation representing how much money Elenore will have in total after saving for \(d\) days is: \[ E(d) = 75 + 7d \]
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Equation to find how many days it will take for Judy and Elenore to 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 \(100 + 5d = 75 + 7d\):
- Rearranging gives: \[ 100 - 75 = 7d - 5d \] \[ 25 = 2d \] Dividing both sides by 2 gives: \[ d = 12.5 \]
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Calculating how much money they have each after \(d = 12.5\) days:
We can substitute \(d = 12.5\) back into the equations for Judy and Elenore to find how much money they have:
- For Judy: \[ J(12.5) = 100 + 5(12.5) = 100 + 62.5 = 162.5 \]
- For Elenore: \[ E(12.5) = 75 + 7(12.5) = 75 + 87.5 = 162.5 \]
So, after 12.5 days, both Judy and Elenore will each have $162.50.
Final answers:
- Judy's equation: \(J(d) = 100 + 5d\)
- Elenore's equation: \(E(d) = 75 + 7d\)
- Equality equation: \(100 + 5d = 75 + 7d\)
- Amount each has after 12.5 days: $162.50