A man buys two chairs for a total cost of Rs 900. By selling one for `(4)/(5)` of its cost and the other for `(5)/(4)` of its cost, he makes a profit of Rs 90 on the whole transtransaction. The cost of the lower priced chair is.

Options

A. Rs.360
B. Rs.400
C. Rs.420
D. Rs.300

Answer

To solve the problem step by step, we will denote the cost price (CP) of the two chairs as \( x \) and \( y \). We know the following:

1. The total cost of the two chairs is Rs 900.
2. The selling price (SP) of the first chair is \( \frac{4}{5} \) of its cost price.
3. The selling price (SP) of the second chair is \( \frac{5}{4} \) of its cost price.
4. The total profit made from selling both chairs is Rs 90.

### Step 1: Set up the equations
From the information given, we can write the following equations:

\[ x + y = 900 \quad \text{(1)} \]

The selling prices can be expressed as:
- For the first chair: \( SP_1 = \frac{4}{5}x \)
- For the second chair: \( SP_2 = \frac{5}{4}y \)

### Step 2: Calculate the total selling price
The total selling price of both chairs is:

\[ SP_1 + SP_2 = \frac{4}{5}x + \frac{5}{4}y \quad \text{(2)} \]

### Step 3: Calculate the total profit
The total profit is given by:

\[ (SP_1 + SP_2) - (x + y) = 90 \]

Substituting equation (1) into this gives:

\[ \left(\frac{4}{5}x + \frac{5}{4}y\right) - 900 = 90 \]

### Step 4: Simplify the profit equation
Rearranging the equation gives:

\[ \frac{4}{5}x + \frac{5}{4}y = 990 \quad \text{(3)} \]

### Step 5: Solve equations (1) and (3)
Now we have two equations:
1. \( x + y = 900 \)
2. \( \frac{4}{5}x + \frac{5}{4}y = 990 \)

From equation (1), we can express \( y \) in terms of \( x \):

\[ y = 900 - x \quad \text{(4)} \]

Substituting equation (4) into equation (3):

\[ \frac{4}{5}x + \frac{5}{4}(900 - x) = 990 \]

### Step 6: Clear the fractions
To eliminate the fractions, multiply the entire equation by 20 (the least common multiple of 5 and 4):

\[ 16x + 25(900 - x) = 19800 \]

### Step 7: Expand and simplify
Expanding gives:

\[ 16x + 22500 - 25x = 19800 \]

Combining like terms:

\[ -9x + 22500 = 19800 \]

### Step 8: Solve for \( x \)
Rearranging gives:

\[ -9x = 19800 - 22500 \]
\[ -9x = -2700 \]
\[ x = 300 \]

### Step 9: Find \( y \)
Using equation (4):

\[ y = 900 - x = 900 - 300 = 600 \]

### Conclusion
The cost of the lower priced chair is Rs 300.