Average of the runs of 133 players of a team is 38. If the average of the runs of the male players is 43 and the average of the runs of the female players is 24, then what will be the ratio of the total runs of male players and the total runs of female players respectively?

Options

A. `301:60`
B. `7:3`
C. `39:11`
D. `60:207`

Answer

To solve the problem step by step, we will follow the logical flow of the information provided:

### Step 1: Calculate the total runs scored by all players
Given that the average runs of 133 players is 38, we can find the total runs scored by all players using the formula:

\[ \text{Total Runs} = \text{Average} \times \text{Number of Players} \]

Substituting the values:

\[ \text{Total Runs} = 38 \times 133 = 5054 \]

### Step 2: Set up equations for male and female players
Let \( X \) be the number of male players and \( Y \) be the number of female players. Since the total number of players is 133, we have:

\[ X + Y = 133 \quad \text{(Equation 1)} \]

The total runs scored by male players can be expressed as \( 43X \) (since the average runs of male players is 43) and the total runs scored by female players can be expressed as \( 24Y \) (since the average runs of female players is 24). Therefore, the total runs scored by all players can also be expressed as:

\[ 43X + 24Y = 5054 \quad \text{(Equation 2)} \]

### Step 3: Substitute \( Y \) in terms of \( X \)
From Equation 1, we can express \( Y \) in terms of \( X \):

\[ Y = 133 - X \]

### Step 4: Substitute \( Y \) in Equation 2
Now substitute \( Y \) in Equation 2:

\[ 43X + 24(133 - X) = 5054 \]

Expanding this gives:

\[ 43X + 3192 - 24X = 5054 \]

### Step 5: Simplify the equation
Combining like terms:

\[ (43X - 24X) + 3192 = 5054 \]
\[ 19X + 3192 = 5054 \]

### Step 6: Solve for \( X \)
Now, isolate \( X \):

\[ 19X = 5054 - 3192 \]
\[ 19X = 1862 \]
\[ X = \frac{1862}{19} = 98 \]

### Step 7: Calculate \( Y \)
Now substitute \( X \) back into Equation 1 to find \( Y \):

\[ Y = 133 - 98 = 35 \]

### Step 8: Calculate total runs for male and female players
Now we can calculate the total runs for male and female players:

- Total runs for male players:
\[ \text{Total Runs for Males} = 43X = 43 \times 98 = 4214 \]

- Total runs for female players:
\[ \text{Total Runs for Females} = 24Y = 24 \times 35 = 840 \]

### Step 9: Find the ratio of total runs of male players to female players
Now we can find the ratio of the total runs of male players to female players:

\[ \text{Ratio} = \frac{\text{Total Runs for Males}}{\text{Total Runs for Females}} = \frac{4214}{840} \]

### Step 10: Simplify the ratio
To simplify the ratio, we can divide both numbers by their greatest common divisor (GCD):

\[ \text{Ratio} = \frac{4214 \div 28}{840 \div 28} = \frac{301}{60} \]

### Final Answer
Thus, the ratio of the total runs of male players to the total runs of female players is:

\[ \frac{301}{60} \]

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