The price of rice is increased by `20%`. Due to this, a family uses rice `40kg` less. So that the expenditure increases by `5%` only . Find original consumption?

Options

A. `326` kg
B. `320` kg
C. `350` kg
D. `300` kg

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the problem
The price of rice has increased by 20%, and as a result, the family uses 40 kg less rice. Despite this decrease in quantity, their total expenditure on rice only increases by 5%. We need to find the original consumption of rice before the price increase.

### Step 2: Define variables
Let the original price of rice per kg be \( P \).
Then, the new price after a 20% increase is:
\[ \text{New Price} = P + 0.2P = 1.2P \]

Let the original consumption of rice be \( C \) kg.

### Step 3: Calculate the original expenditure
The original expenditure on rice can be calculated as:
\[ \text{Original Expenditure} = P \times C \]

### Step 4: Calculate the new expenditure
After the price increase, the family consumes \( C - 40 \) kg of rice. The new expenditure will be:
\[ \text{New Expenditure} = 1.2P \times (C - 40) \]

### Step 5: Set up the equation for expenditure increase
According to the problem, the new expenditure is 5% more than the original expenditure. Therefore, we can set up the equation:
\[ 1.2P \times (C - 40) = 1.05 \times (P \times C) \]

### Step 6: Simplify the equation
Expanding both sides gives:
\[ 1.2P(C - 40) = 1.05PC \]
\[ 1.2PC - 48P = 1.05PC \]

### Step 7: Rearranging the equation
Now, we can rearrange the equation to isolate \( C \):
\[ 1.2PC - 1.05PC = 48P \]
\[ (1.2 - 1.05)PC = 48P \]
\[ 0.15PC = 48P \]

### Step 8: Solve for \( C \)
Dividing both sides by \( 0.15P \) (assuming \( P \neq 0 \)):
\[ C = \frac{48P}{0.15P} \]
\[ C = \frac{48}{0.15} = 320 \text{ kg} \]

Thus, the original consumption of rice was **320 kg**.