During its first week of business a market sold a total of 108 apples and oranges. The second week, five times the number of apples and three times the number of oranges were sold. A total of 452 apples and oranges were sold during the second week. How many more apples than oranges were sold in the first week?

Answer

To solve the problem step by step, we will define variables, set up equations based on the information given, and then solve those equations.

### Step 1: Define Variables
Let:
- \( x \) = number of apples sold in the first week
- \( y \) = number of oranges sold in the first week

### Step 2: Set Up the First Equation
According to the problem, the total number of apples and oranges sold in the first week is 108. Therefore, we can write the equation:
\[ x + y = 108 \]
This is our **Equation 1**.

### Step 3: Set Up the Second Equation
In the second week, the market sold five times the number of apples and three times the number of oranges. The total number of apples and oranges sold in the second week is 452. Therefore, we can write the equation:
\[ 5x + 3y = 452 \]
This is our **Equation 2**.

### Step 4: Solve the Equations
Now we have two equations:
1. \( x + y = 108 \) (Equation 1)
2. \( 5x + 3y = 452 \) (Equation 2)

To eliminate \( y \), we can multiply Equation 1 by 3:
\[ 3(x + y) = 3(108) \]
This simplifies to:
\[ 3x + 3y = 324 \]
Now we have a new equation:
3. \( 3x + 3y = 324 \) (Equation 3)

### Step 5: Subtract Equation 2 from Equation 3
Now we can subtract Equation 2 from Equation 3:
\[ (3x + 3y) - (5x + 3y) = 324 - 452 \]
This simplifies to:
\[ 3x + 3y - 5x - 3y = -128 \]
Which further simplifies to:
\[ -2x = -128 \]
Dividing both sides by -2 gives:
\[ x = 64 \]

### Step 6: Find the Number of Oranges
Now that we have \( x \), we can find \( y \) using Equation 1:
\[ x + y = 108 \]
Substituting \( x = 64 \):
\[ 64 + y = 108 \]
Subtracting 64 from both sides gives:
\[ y = 108 - 64 = 44 \]

### Step 7: Calculate the Difference
Now we need to find how many more apples than oranges were sold in the first week:
\[ x - y = 64 - 44 = 20 \]

### Final Answer
Thus, **20 more apples than oranges were sold in the first week**.

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