If the ratio of mean and median of a certain data is 5 : 7, then find the ratio of its mode and mean.

Options

A. 2 : 5
B. 11 : 5
C. 6 : 5
D. 2 : 3

Explanation

To solve the problem step by step, we will use the relationship between mean, median, and mode in statistics.

### Step-by-Step Solution:

1. **Understanding the Given Ratio**:
We are given that the ratio of mean (M) to median (Me) is 5:7. This can be expressed mathematically as:
\[ \frac{M}{Me} = \frac{5}{7} \]

2. **Expressing Median in Terms of Mean**:
From the ratio, we can express the median in terms of the mean:
\[ Me = \frac{7}{5} M \]

3. **Using the Mode Formula**:
The relationship between mode (Mo), median (Me), and mean (M) can be expressed using the formula:
\[ Mo = 3 \times Me - 2 \times M \]

4. **Substituting Median into the Mode Formula**:
Substitute the expression for median (Me) from step 2 into the mode formula:
\[ Mo = 3 \left(\frac{7}{5} M\right) - 2M \]

5. **Simplifying the Mode Expression**:
Now, simplify the right-hand side:
\[ Mo = \frac{21}{5} M - 2M \]
To combine the terms, convert \(2M\) into a fraction with a denominator of 5:
\[ Mo = \frac{21}{5} M - \frac{10}{5} M = \frac{11}{5} M \]

6. **Finding the Ratio of Mode to Mean**:
Now, we can find the ratio of mode to mean:
\[ \frac{Mo}{M} = \frac{\frac{11}{5} M}{M} = \frac{11}{5} \]

### Final Answer:
The ratio of mode to mean is:
\[ \frac{Mo}{M} = \frac{11}{5} \]