Question

Alma was given some money as a gift and deposited it into a savings account. She makes a withdrawal of the same amount at the end of each month. At the end of the 5th month, her balance was $442. At the end of the 13th month, her balance was $226. How much money was Alma given as a gift?

$

Answer 1

Let \( P \) be the initial amount of money Alma was given (the principal amount), and let \( W \) be the amount she withdraws each month.

From the problem, we have two pieces of information:

1. At the end of the 5th month, her balance is given by:
\[
P - 5W = 442
\]
(Equation 1)

2. At the end of the 13th month, her balance is given by:
\[
P - 13W = 226
\]
(Equation 2)

Now we can set up the equations based on the information we have:

From Equation 1, we rearrange it to express \( P \):
\[
P = 442 + 5W
\]

Substituting this expression for \( P \) into Equation 2:
\[
442 + 5W - 13W = 226
\]

This simplifies to:
\[
442 - 8W = 226
\]
Subtracting 442 from both sides:
\[
-8W = 226 - 442
\]
\[
-8W = -216
\]

Dividing both sides by -8 gives:
\[
W = \frac{-216}{-8} = 27
\]

Now that we have \( W = 27 \), we can substitute back to find \( P \) using Equation 1:
\[
P = 442 + 5 \times 27
\]
Calculating \( 5 \times 27 \):
\[
5 \times 27 = 135
\]

Now substituting this back into the equation for \( P \):
\[
P = 442 + 135 = 577
\]

Thus, the amount Alma was given as a gift is:
\[
\boxed{577}
\]