The average age of three boys is 16 years. If their ages are in the ratio 4:5:7, then the age of youngest boy is

Options

A. (a) 10 years
B. (b) 12 years
C. (c) 14 years
D. (d) 15 years

Answer

To find the age of the youngest boy given that the average age of three boys is 16 years and their ages are in the ratio 4:5:7, we can follow these steps:

### Step 1: Understand the average age
The average age of the three boys is given as 16 years. The formula for average is:
\[ \text{Average} = \frac{\text{Sum of ages}}{\text{Number of boys}} \]
Since there are three boys, we can express this as:
\[ 16 = \frac{\text{Sum of ages}}{3} \]

### Step 2: Calculate the sum of ages
To find the sum of their ages, we can rearrange the formula:
\[ \text{Sum of ages} = \text{Average} \times \text{Number of boys} = 16 \times 3 = 48 \text{ years} \]

### Step 3: Set up the ratio
The ages of the boys are in the ratio 4:5:7. Let the ages of the boys be represented as:
- Age of the first boy = \(4x\)
- Age of the second boy = \(5x\)
- Age of the third boy = \(7x\)

### Step 4: Write an equation for the sum of ages
Now, we can express the sum of their ages in terms of \(x\):
\[ 4x + 5x + 7x = 48 \]

### Step 5: Simplify the equation
Combine the terms on the left side:
\[ 16x = 48 \]

### Step 6: Solve for \(x\)
Now, divide both sides by 16 to find \(x\):
\[ x = \frac{48}{16} = 3 \]

### Step 7: Find the ages of the boys
Now that we have \(x\), we can find the ages of each boy:
- Age of the first boy = \(4x = 4 \times 3 = 12\) years
- Age of the second boy = \(5x = 5 \times 3 = 15\) years
- Age of the third boy = \(7x = 7 \times 3 = 21\) years

### Step 8: Identify the youngest boy's age
The youngest boy's age is:
\[ \text{Youngest boy's age} = 12 \text{ years} \]

### Final Answer
The age of the youngest boy is **12 years**.

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