If the sum of the mode and mean of a certain frequency distribution is 129 and the median of the observations is 63, mode and mean are respectively

Options

A. 69 and 60
B. 65 and 64
C. 68 and 61
D. none of these

Answer

To solve the problem step by step, we will use the information provided in the question about the mode, mean, and median of a frequency distribution.

### Step 1: Set Up the Equations
We are given:
1. The sum of the mode (Mo) and mean (M) is 129.
\[ Mo + M = 129 \quad \text{(Equation 1)} \]
2. The median (Me) is given as 63.
\[ Me = 63 \]

### Step 2: Use the Relationship Between Mean, Median, and Mode
We can use the relationship between the mean, median, and mode:
\[ 3 \times Me = 2 \times M + Mo \]
Substituting the value of the median:
\[ 3 \times 63 = 2M + Mo \]
Calculating the left side:
\[ 189 = 2M + Mo \quad \text{(Equation 2)} \]

### Step 3: Solve the Equations Simultaneously
Now we have two equations:
1. \( Mo + M = 129 \) (Equation 1)
2. \( 189 = 2M + Mo \) (Equation 2)

We can express \( Mo \) from Equation 1:
\[ Mo = 129 - M \]
Now, substitute this expression for \( Mo \) into Equation 2:
\[ 189 = 2M + (129 - M) \]
Simplifying this gives:
\[ 189 = 2M + 129 - M \]
\[ 189 = M + 129 \]
Now, isolate \( M \):
\[ M = 189 - 129 \]
\[ M = 60 \]

### Step 4: Find the Mode
Now that we have the mean, we can find the mode using Equation 1:
\[ Mo + 60 = 129 \]
\[ Mo = 129 - 60 \]
\[ Mo = 69 \]

### Final Answer
Thus, the mode and mean are:
- Mode (Mo) = 69
- Mean (M) = 60

### Summary
The mode and mean are respectively 69 and 60.

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