The ratio of cost price and marked price of an article is $A : B$. The ratio of selling price and marked price is $(B + 1) : (B + 2)$. If the discount offered on the article is Rs. 30, then its selling price becomes Rs. 120 ($A$ and $B$ are natural numbers and $A

Question

The ratio of cost price and marked price of an article is $A : B$. The ratio of selling price and marked price is $(B + 1) : (B + 2)$. If the discount offered on the article is Rs. 30, then its selling price becomes Rs. 120 ($A$ and $B$ are natural numbers and $A < B$). Which of the following is/are correct?

(i) $B$ is a prime number.

(ii) Maximum profit can be Rs. 70.

(iii) Cost price can be Rs. 150.

Accepted Answer

**Solution:**

1. Let the marked price be $M$.

2. Given ratio of cost price to marked price is $A : B$, so:
   $$\frac{C}{M} = \frac{A}{B} \implies C = \frac{A}{B} M$$

3. Ratio of selling price to marked price is $(B+1) : (B+2)$, so:
   $$\frac{S}{M} = \frac{B+1}{B+2} \implies S = \frac{B+1}{B+2} M$$

4. Discount $D = M - S = 30$

Substitute $S$:
   $$M - \frac{B+1}{B+2} M = 30$$

$$M \left(1 - \frac{B+1}{B+2}ight) = 30$$

$$M 	imes \frac{(B+2) - (B+1)}{B+2} = 30$$

$$M 	imes \frac{1}{B+2} = 30 \implies M = 30 (B+2)$$

5. Selling price $S$:
   $$S = \frac{B+1}{B+2} M = \frac{B+1}{B+2} 	imes 30 (B+2) = 30 (B+1)$$

6. Given selling price $S = 120$:
   $$30 (B+1) = 120 \implies B+1 = 4 \implies B = 3$$

7. Since $B=3$, which is a prime number, statement (i) is correct.

8. Marked price:
   $$M = 30 (B+2) = 30 (3+2) = 30 	imes 5 = 150$$

9. Cost price:
   $$C = \frac{A}{B} M = \frac{A}{3} 	imes 150 = 50 A$$

Since $A < B = 3$, possible values for $A$ are 1 or 2.

So, cost price can be:
   - If $A=1$, $C=50$
   - If $A=2$, $C=100$

Cost price cannot be Rs. 150, so statement (iii) is incorrect.

10. Profit:
    $$	ext{Profit} = S - C = 120 - 50A$$

For $A=1$, profit = $120 - 50 = 70$

For $A=2$, profit = $120 - 100 = 20$

Maximum profit possible is Rs. 70, so statement (ii) is correct.

**Answer:** (a) Only (i) and (ii)

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