A, B and C are batsmen. The ratio of the runs scored by them in a certain match are given below : `A: B = 5 : 3` and `B : C = 4 : 5`. In all they scored 564 runs. The number of runs scored by B is:

Options

A. 124
B. 104
C. 114
D. 144

Answer

To solve the problem step by step, we will first express the runs scored by A, B, and C in terms of a common variable based on the given ratios.

### Step 1: Understand the Ratios
The ratios provided are:
- A : B = 5 : 3
- B : C = 4 : 5

### Step 2: Express Runs in Terms of a Common Variable
Let's denote the runs scored by A, B, and C as follows:
- Let the runs scored by B be \( 3x \) (from the ratio A:B = 5:3).
- Then, the runs scored by A will be \( 5x \).
- From the second ratio B:C = 4:5, we can express C in terms of B:
- If B = \( 4y \), then C = \( 5y \).

Since B is common in both ratios, we need to equate the two expressions for B:
- \( 3x = 4y \)

### Step 3: Solve for x and y
From the equation \( 3x = 4y \), we can express y in terms of x:
- \( y = \frac{3x}{4} \)

Now substitute \( y \) back into the expression for C:
- C = \( 5y = 5 \left( \frac{3x}{4} \right) = \frac{15x}{4} \)

### Step 4: Total Runs Scored
Now we can express the total runs scored by A, B, and C:
- Total runs = A + B + C
- Total runs = \( 5x + 3x + \frac{15x}{4} \)

To add these, we need a common denominator:
- Convert \( 5x \) and \( 3x \) to have a denominator of 4:
- \( 5x = \frac{20x}{4} \)
- \( 3x = \frac{12x}{4} \)

Now we can add:
- Total runs = \( \frac{20x}{4} + \frac{12x}{4} + \frac{15x}{4} = \frac{47x}{4} \)

### Step 5: Set Up the Equation
According to the problem, the total runs scored by A, B, and C is 564:
- \( \frac{47x}{4} = 564 \)

### Step 6: Solve for x
Multiply both sides by 4 to eliminate the fraction:
- \( 47x = 564 \times 4 \)
- \( 47x = 2256 \)

Now divide by 47:
- \( x = \frac{2256}{47} = 48 \)

### Step 7: Calculate Runs Scored by B
Now that we have the value of \( x \), we can find the runs scored by B:
- B = \( 3x = 3 \times 48 = 144 \)

### Final Answer
The number of runs scored by B is **144**.