Question

Grace and Andy shared a sum of money in the ratio 5:8 at first. Grace received $108 from her mother and Andy received $39 from his father. They now have the same amount of money. How muxh did each of the have at first?

Answer 1

Let's denote the initial amounts of money that Grace and Andy received as \( G \) and \( A \), respectively. According to the problem, they shared their money in the ratio \( 5:8 \). This can be expressed using a common variable \( k \):

\[
G = 5k \quad \text{and} \quad A = 8k
\]

Next, Grace receives $108 from her mother and Andy receives $39 from his father. After these additions, their total amounts become:

\[
G + 108 \quad \text{and} \quad A + 39
\]

According to the problem, they now have the same amount of money:

\[
G + 108 = A + 39
\]

Substituting the expressions for \( G \) and \( A \):

\[
5k + 108 = 8k + 39
\]

Now, we can solve for \( k \):

\[
5k + 108 - 39 = 8k
\]
\[
5k + 69 = 8k
\]
\[
69 = 8k - 5k
\]
\[
69 = 3k
\]
\[
k = \frac{69}{3} = 23
\]

Now, we can find the initial amounts \( G \) and \( A \):

\[
G = 5k = 5 \times 23 = 115
\]
\[
A = 8k = 8 \times 23 = 184
\]

Thus, Grace initially had $115 and Andy initially had $184.

In summary:

- Grace initially had \( \boxed{115} \)
- Andy initially had \( \boxed{184} \)