In a family of 8 adults and some minors, the average consumption of rice per head per month is 10.8 kg, while the average consumption for adults is 15 kg per head and for minors it is 6 kg per head. The number of minors in the family is :

Options

A. 8
B. 6
C. 7
D. 9

Answer

To solve the problem step by step, we will use the concept of averages and the method of allegation.

### Step 1: Understand the given information
We have:
- Number of adults = 8
- Average consumption of rice per head for the family = 10.8 kg
- Average consumption of rice per head for adults = 15 kg
- Average consumption of rice per head for minors = 6 kg

### Step 2: Set up the equation using the average formula
Let the number of minors be \( x \).

The total number of people in the family = Number of adults + Number of minors = \( 8 + x \).

The total consumption of rice for adults = Number of adults × Average consumption for adults = \( 8 \times 15 = 120 \) kg.

The total consumption of rice for minors = Number of minors × Average consumption for minors = \( x \times 6 \) kg.

The total consumption of rice for the family = Total consumption for adults + Total consumption for minors = \( 120 + 6x \) kg.

### Step 3: Write the equation for the average consumption
The average consumption for the whole family can be expressed as:
\[ \text{Average consumption} = \frac{\text{Total consumption}}{\text{Total number of people}} \]
Substituting the values we have:
\[ 10.8 = \frac{120 + 6x}{8 + x} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 10.8(8 + x) = 120 + 6x \]

### Step 5: Expand and simplify the equation
Expanding the left side:
\[ 86.4 + 10.8x = 120 + 6x \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ 10.8x - 6x = 120 - 86.4 \]
\[ 4.8x = 33.6 \]

### Step 7: Solve for \( x \)
Dividing both sides by 4.8:
\[ x = \frac{33.6}{4.8} = 7 \]

### Conclusion
The number of minors in the family is \( 7 \).

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