A string is cut into 2 pieces that have lengths in the ratio of 2:9. If the difference between the lengths of the 2 pieces of string is 42 inches, what is the lengths in inches of the shorter pieces?

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the lengths of the two pieces
Let the lengths of the two pieces of the string be represented as \(2k\) and \(9k\), where \(k\) is a common multiplier. This representation maintains the ratio of the lengths as \(2:9\).

### Step 2: Set up the equation for the difference in lengths
According to the problem, the difference between the lengths of the two pieces is given as 42 inches. Therefore, we can set up the equation:
\[ 9k - 2k = 42 \]

### Step 3: Simplify the equation
Now, simplify the left side of the equation:
\[ 7k = 42 \]

### Step 4: Solve for \(k\)
To find the value of \(k\), divide both sides of the equation by 7:
\[ k = \frac{42}{7} = 6 \]

### Step 5: Calculate the length of the shorter piece
Now that we have the value of \(k\), we can find the length of the shorter piece, which is \(2k\):
\[ \text{Length of shorter piece} = 2k = 2 \times 6 = 12 \text{ inches} \]

### Final Answer
The length of the shorter piece is **12 inches**.

---