A factory requires 42 machines to produce a given number of articles in 56 days. How many machines would be required to produce the same number of articles in 48 days?

Answer

To solve the problem step by step, we will use the concept of inverse proportions. Here’s how we can approach the solution:

### Step 1: Understand the Relationship
We know that the number of machines (let's denote it as \( M \)) and the number of days (let's denote it as \( D \)) are inversely proportional. This means that if the number of days decreases, the number of machines required increases to maintain the same output.

### Step 2: Set Up the Proportion
We have the following information:
- 42 machines are required for 56 days.
- We need to find out how many machines (let's call this \( M \)) are required for 48 days.

Using the inverse proportion relationship, we can set up the equation:
\[ M_1 \times D_1 = M_2 \times D_2 \]
Where:
- \( M_1 = 42 \) (machines for 56 days)
- \( D_1 = 56 \) (days)
- \( M_2 = ? \) (machines for 48 days)
- \( D_2 = 48 \) (days)

### Step 3: Substitute the Known Values
Substituting the known values into the equation gives us:
\[ 42 \times 56 = M \times 48 \]

### Step 4: Solve for \( M \)
Now, we need to isolate \( M \):
\[ M = \frac{42 \times 56}{48} \]

### Step 5: Calculate the Right Side
First, calculate \( 42 \times 56 \):
\[ 42 \times 56 = 2352 \]
Now, divide by 48:
\[ M = \frac{2352}{48} \]

### Step 6: Perform the Division
Now, we perform the division:
\[ M = 49 \]

### Conclusion
Therefore, the number of machines required to produce the same number of articles in 48 days is **49 machines**.

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