The ratio of the length to the perimeter of a rectangle is 5:16. Find the ratio of the length to the breadth.

Answer

To solve the problem, we need to find the ratio of the length to the breadth of a rectangle given that the ratio of the length to the perimeter is 5:16.

Let's denote:
- Length of the rectangle as \( l \)
- Breadth (or width) of the rectangle as \( w \)
- Perimeter of the rectangle as \( P \)

### Step 1: Write the formula for the perimeter of a rectangle.
The formula for the perimeter \( P \) of a rectangle is given by:
\[ P = 2(l + w) \]

### Step 2: Set up the ratio given in the problem.
According to the problem, the ratio of the length to the perimeter is:
\[ \frac{l}{P} = \frac{5}{16} \]

### Step 3: Substitute the perimeter formula into the ratio.
We can substitute the formula for the perimeter into the ratio:
\[ \frac{l}{2(l + w)} = \frac{5}{16} \]

### Step 4: Cross-multiply to eliminate the fraction.
Cross-multiplying gives us:
\[ 16l = 5 \cdot 2(l + w) \]
This simplifies to:
\[ 16l = 10(l + w) \]

### Step 5: Distribute on the right side.
Distributing \( 10 \) on the right side, we have:
\[ 16l = 10l + 10w \]

### Step 6: Rearrange the equation to isolate terms involving \( l \) and \( w \).
Subtract \( 10l \) from both sides:
\[ 16l - 10l = 10w \]
This simplifies to:
\[ 6l = 10w \]

### Step 7: Divide both sides to find the ratio of length to breadth.
To find the ratio \( \frac{l}{w} \), we divide both sides by \( 10w \):
\[ \frac{l}{w} = \frac{10}{6} \]
This simplifies to:
\[ \frac{l}{w} = \frac{5}{3} \]

### Final Answer:
Thus, the ratio of the length to the breadth of the rectangle is:
\[ \text{Length: Breadth} = 5:3 \]