The average earning of each member of the Ambani family is 20% less than the average earning of each member of the Sahara family and the total earning of Ambani's family is 20% more than the total earning of Saharas's family. The no. of family members in the Sahara is what per cent of the no. of family members of Ambani :

Options

A. 25%
B. 20%
C. 66.66%
D. none of these

Answer

To solve the problem step by step, we will use the information given about the average earnings and total earnings of the Ambani and Sahara families.

### Step 1: Define Variables
Let:
- \( A \) = Average earning of each member of Ambani family
- \( S \) = Average earning of each member of Sahara family
- \( N_A \) = Number of members in Ambani family
- \( N_S \) = Number of members in Sahara family
- \( T_A \) = Total earning of Ambani family
- \( T_S \) = Total earning of Sahara family

### Step 2: Set Up the Equations
From the problem, we know:
1. The average earning of each member of Ambani family is 20% less than that of Sahara family:
\[ A = S - 0.2S = 0.8S \]

2. The total earning of Ambani family is 20% more than that of Sahara family:
\[ T_A = T_S + 0.2T_S = 1.2T_S \]

### Step 3: Express Total Earnings in Terms of Averages
The total earnings can also be expressed in terms of average earnings and the number of members:
\[ T_A = A \times N_A \]
\[ T_S = S \times N_S \]

### Step 4: Substitute the Expressions
Substituting the expressions for total earnings into the second equation:
\[ A \times N_A = 1.2(S \times N_S) \]

### Step 5: Substitute the Value of A
Now substitute \( A = 0.8S \) into the equation:
\[ (0.8S) \times N_A = 1.2(S \times N_S) \]

### Step 6: Simplify the Equation
We can cancel \( S \) from both sides (assuming \( S \neq 0 \)):
\[ 0.8N_A = 1.2N_S \]

### Step 7: Rearrange to Find the Ratio
Rearranging gives:
\[ \frac{N_S}{N_A} = \frac{0.8}{1.2} = \frac{8}{12} = \frac{2}{3} \]

### Step 8: Convert to Percentage
To find the percentage of the number of family members in Sahara compared to Ambani:
\[ \text{Percentage} = \left(\frac{N_S}{N_A}\right) \times 100 = \left(\frac{2}{3}\right) \times 100 = 66.67\% \]

### Final Answer
The number of family members in the Sahara is 66.67% of the number of family members of Ambani.

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