A person bought two clocks. The cost price of one of them exceeds the cost price of the other by 1/4th. He sold the dearer one at a gain of 10% and the other at a gain of 7.5% and thus got Rs. 98 in all as S.P. Find the cost price of the cheaper one.

Options

A. Rs. 40
B. Rs. 50
C. Rs. 30
D. Rs. 60

Answer

To solve the problem step by step, we will denote the cost price of the cheaper clock as \( x \).

### Step 1: Define the Cost Prices
Let the cost price of the cheaper clock be \( x \). According to the problem, the cost price of the dearer clock exceeds that of the cheaper clock by \( \frac{1}{4} \). Therefore, we can express the cost price of the dearer clock as:
\[ \text{Cost Price of Dearer Clock} = x + \frac{1}{4}x = \frac{5}{4}x \]

### Step 2: Calculate Selling Prices
The selling price (S.P.) of the dearer clock is calculated with a gain of 10%. The formula for selling price with profit is:
\[ \text{Selling Price} = \text{Cost Price} + \text{Profit} \]
Thus, the selling price of the dearer clock becomes:
\[ \text{S.P. of Dearer Clock} = \frac{5}{4}x + 10\% \text{ of } \frac{5}{4}x = \frac{5}{4}x + \frac{10}{100} \cdot \frac{5}{4}x = \frac{5}{4}x + \frac{1}{40} \cdot 5x = \frac{5}{4}x + \frac{1}{8}x \]
To combine these, we need a common denominator:
\[ \frac{5}{4}x = \frac{10}{8}x \quad \text{so,} \quad \text{S.P. of Dearer Clock} = \frac{10}{8}x + \frac{1}{8}x = \frac{11}{8}x \]

For the cheaper clock, which is sold at a gain of 7.5%, the selling price is:
\[ \text{S.P. of Cheaper Clock} = x + 7.5\% \text{ of } x = x + \frac{7.5}{100}x = x + \frac{3}{40}x \]
Again, combining these:
\[ \text{S.P. of Cheaper Clock} = x + \frac{3}{40}x = \frac{40}{40}x + \frac{3}{40}x = \frac{43}{40}x \]

### Step 3: Total Selling Price
According to the problem, the total selling price of both clocks is Rs. 98. Therefore, we can set up the equation:
\[ \text{S.P. of Dearer Clock} + \text{S.P. of Cheaper Clock} = 98 \]
Substituting the expressions we derived:
\[ \frac{11}{8}x + \frac{43}{40}x = 98 \]

### Step 4: Solve for \( x \)
To solve this equation, we need a common denominator. The least common multiple of 8 and 40 is 40. Thus, we rewrite the first term:
\[ \frac{11}{8}x = \frac{55}{40}x \]
Now, substituting back into the equation:
\[ \frac{55}{40}x + \frac{43}{40}x = 98 \]
Combine the fractions:
\[ \frac{55 + 43}{40}x = 98 \]
\[ \frac{98}{40}x = 98 \]
Now, multiply both sides by 40:
\[ 98x = 98 \times 40 \]
\[ 98x = 3920 \]
Now divide by 98:
\[ x = \frac{3920}{98} = 40 \]

### Step 5: Conclusion
Thus, the cost price of the cheaper clock is:
\[ \text{Cost Price of Cheaper Clock} = x = 40 \]