A, B and C play cricket. A's runs are to B's runs and B's runs are to C's as `3 : 2`. They get altogether 342 runs. How many runs did A make?

Options

A. 162
B. 108
C. 72
D. None of these

Answer

To solve the problem step by step, we need to find out how many runs A made given the ratios of runs scored by A, B, and C, and the total runs scored by all three players.

### Step 1: Understand the Ratios
We are given that:
- A's runs to B's runs = 3 : 2
- B's runs to C's runs = 3 : 2

### Step 2: Express Runs in Terms of a Common Variable
Let’s denote:
- A's runs = 3x
- B's runs = 2x

From the second ratio (B's runs to C's runs), we can express C's runs in terms of B's runs:
- B's runs = 3y
- C's runs = 2y

Since B's runs are common in both ratios, we can set them equal:
- 2x = 3y

### Step 3: Solve for x and y
From the equation 2x = 3y, we can express y in terms of x:
- y = (2/3)x

### Step 4: Substitute y back to find C's runs
Now, substitute y back into the expression for C's runs:
- C's runs = 2y = 2 * (2/3)x = (4/3)x

### Step 5: Write the Total Runs Equation
Now we can express the total runs scored by A, B, and C:
- Total runs = A's runs + B's runs + C's runs
- Total runs = 3x + 2x + (4/3)x

To combine these, we need a common denominator:
- Total runs = (9/3)x + (6/3)x + (4/3)x = (19/3)x

### Step 6: Set the Total Runs Equal to 342
We know the total runs scored by A, B, and C is 342:
- (19/3)x = 342

### Step 7: Solve for x
To find x, multiply both sides by 3:
- 19x = 342 * 3
- 19x = 1026

Now, divide by 19:
- x = 1026 / 19
- x = 54

### Step 8: Calculate A's Runs
Now that we have x, we can find A's runs:
- A's runs = 3x = 3 * 54 = 162

### Final Answer
A made 162 runs.

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