4. $$ \frac{1}{3} $$ of Julie's money was equal to $$ \frac{3}{5} $$ of Nancy's money. After Julie gave Nancy $$ \$ 42 $$, both of them would have the same amount of money. How much money did Nancy have at first?

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UpStudy AI · Accepted Answer

Let's solve the problem step by step.

### Let:
- $$ J $$ = Julie's initial amount of money
- $$ N $$ = Nancy's initial amount of money

### Given:
1. $$ \frac{1}{3} $$ of Julie's money is equal to $$ \frac{3}{5} $$ of Nancy's money:
   $$
   \frac{1}{3}J = \frac{3}{5}N
   $$
   
2. After Julie gives Nancy \$42, both have the same amount:
   $$
   J - 42 = N + 42
   $$

### Step 1: Express $$ J $$ in terms of $$ N $$
From the second equation:
$$
J = N + 84
$$

### Step 2: Substitute $$ J $$ into the first equation
$$
\frac{1}{3}(N + 84) = \frac{3}{5}N
$$

### Step 3: Eliminate the fractions by multiplying both sides by 15 (the least common multiple of 3 and 5):
$$
15 \times \frac{1}{3}(N + 84) = 15 \times \frac{3}{5}N
$$
$$
5(N + 84) = 9N
$$

### Step 4: Solve for $$ N $$
$$
5N + 420 = 9N
$$
$$
420 = 4N
$$
$$
N = 105
$$

**Nancy originally had \$105.**
Nancy originally had \$105.

of Julie’s money was equal to of Nancy’s money. After Julie gave Nancy
, both of them would have the same amount of money. How much money
did Nancy have at first?

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of Julie’s money was equal to of Nancy’s money. After Julie gave Nancy , both of them would have the same amount of money. How much money did Nancy have at first?