If x runs are scored by A y runs by B and z runs by C, then `x:y = y:z=3:2`. If total number of runs scored by A, B and C is 342, the runs scored by each would be respectively

Options

A. 144, 96, 64
B. 162, 108, 72
C. 180, 120, 80
D. 189, 126, 84

Answer

To solve the problem step by step, we will first set up the ratios and then use the total runs scored to find the individual scores of A, B, and C.

### Step 1: Set up the ratios
We are given that:
- \( x:y = 3:2 \)
- \( y:z = 3:2 \)

From these ratios, we can express \( y \) and \( z \) in terms of \( x \).

### Step 2: Express y and z in terms of x
From the ratio \( x:y = 3:2 \), we can write:
\[ y = \frac{2}{3}x \]

From the ratio \( y:z = 3:2 \), we can express \( z \) in terms of \( y \):
\[ z = \frac{2}{3}y \]

Substituting the expression for \( y \) into the equation for \( z \):
\[ z = \frac{2}{3} \left(\frac{2}{3}x\right) = \frac{4}{9}x \]

### Step 3: Write the total runs scored
Now we have:
- \( x \) (runs scored by A)
- \( y = \frac{2}{3}x \) (runs scored by B)
- \( z = \frac{4}{9}x \) (runs scored by C)

The total runs scored by A, B, and C is given as 342:
\[ x + y + z = 342 \]

Substituting the expressions for \( y \) and \( z \):
\[ x + \frac{2}{3}x + \frac{4}{9}x = 342 \]

### Step 4: Find a common denominator and simplify
The common denominator for the fractions is 9. We can rewrite the equation:
\[ x + \frac{6}{9}x + \frac{4}{9}x = 342 \]
Combining the terms:
\[ \frac{9}{9}x + \frac{6}{9}x + \frac{4}{9}x = 342 \]
\[ \frac{19}{9}x = 342 \]

### Step 5: Solve for x
To find \( x \), multiply both sides by \( \frac{9}{19} \):
\[ x = 342 \times \frac{9}{19} \]
Calculating this gives:
\[ x = 342 \div 19 \times 9 = 18 \times 9 = 162 \]

### Step 6: Calculate y and z
Now we can find \( y \) and \( z \):
\[ y = \frac{2}{3}x = \frac{2}{3} \times 162 = 108 \]
\[ z = \frac{4}{9}x = \frac{4}{9} \times 162 = 72 \]

### Final Result
Thus, the runs scored by A, B, and C are:
- A (x) = 162
- B (y) = 108
- C (z) = 72

### Summary
The runs scored by A, B, and C are respectively 162, 108, and 72.

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