Total runs scored by three batsmen x, y and z are 1584. Ratio of runs scored by x and y is 4 : 3 and y and z is 5 : 3. How many runs are scored by x?

Options

A. 742
B. 614
C. 516
D. 720

Answer

To solve the problem step by step, we will follow the given ratios and the total runs scored by the batsmen.

### Step 1: Understand the Ratios
We are given:
- The total runs scored by batsmen x, y, and z is 1584.
- The ratio of runs scored by x and y is 4:3.
- The ratio of runs scored by y and z is 5:3.

### Step 2: Set Up the Ratios
Let’s denote the runs scored by x, y, and z as follows:
- Runs scored by x = 4k (from the ratio of x and y)
- Runs scored by y = 3k (from the ratio of x and y)

Now, we need to express z in terms of y. Since the ratio of y and z is 5:3, we can express z as:
- Runs scored by z = (3/5) * Runs scored by y = (3/5) * 3k = (9/5)k

### Step 3: Find a Common Variable
To combine the ratios, we need to express y in a way that allows us to equate both ratios. We can express y in terms of a new variable p:
- Let y = 15p (from the second ratio where y is 5 parts and z is 3 parts)

Now, we can express x and z in terms of p:
- From the ratio x:y = 4:3, we have:
- x = (4/3) * y = (4/3) * 15p = 20p
- From the ratio y:z = 5:3, we have:
- z = (3/5) * y = (3/5) * 15p = 9p

### Step 4: Combine the Runs
Now we can express the total runs scored by x, y, and z:
- Total runs = x + y + z = 20p + 15p + 9p = 44p

### Step 5: Set Up the Equation
We know that the total runs scored is 1584:
\[ 44p = 1584 \]

### Step 6: Solve for p
To find p, divide both sides by 44:
\[ p = \frac{1584}{44} \]
Calculating this gives:
\[ p = 36 \]

### Step 7: Calculate Runs Scored by x
Now, we can find the runs scored by x:
\[ x = 20p = 20 \times 36 = 720 \]

### Conclusion
The runs scored by batsman x is **720**.

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