Piya and Siya sell cakes over three days: Saturday, Sunday, and Monday. The ratio of cakes sold by Piya on Monday to Sunday is 1:2. Siya sells 20 cakes on Monday. The number of cakes sold by Siya on Saturday equals the average number of cakes sold by Piya over all three days. The difference between cakes sold by Piya and Siya on Monday equals the difference between cakes sold by Piya and Siya on Saturday. Piya sells fewer cakes than Siya on each day.

Which of the following statements can be correct?

I. If Piya sold 18 cakes on Monday, then the total cakes sold by Siya over the three days is 81.

II. Piya sold 90 cakes on Saturday.

III. If Piya sold 21 cakes on Saturday, then the minimum possible cakes sold by Siya on Sunday is 35.

Question

Piya and Siya sell cakes over three days: Saturday, Sunday, and Monday. The ratio of cakes sold by Piya on Monday to Sunday is 1:2. Siya sells 20 cakes on Monday. The number of cakes sold by Siya on Saturday equals the average number of cakes sold by Piya over all three days. The difference between cakes sold by Piya and Siya on Monday equals the difference between cakes sold by Piya and Siya on Saturday. Piya sells fewer cakes than Siya on each day.

Which of the following statements can be correct?

I. If Piya sold 18 cakes on Monday, then the total cakes sold by Siya over the three days is 81.

II. Piya sold 90 cakes on Saturday.

III. If Piya sold 21 cakes on Saturday, then the minimum possible cakes sold by Siya on Sunday is 35.

Accepted Answer

**Solution:**

1. Let the cakes sold by Piya on Monday be $x$. Then on Sunday, Piya sells $2x$ cakes (given ratio 1:2).

2. Let the total cakes sold by Piya over the three days be $y$. Then the average cakes sold by Piya per day is $\frac{y}{3}$.

3. Given that Siya sells 20 cakes on Monday.

4. The cakes sold by Siya on Saturday equals the average cakes sold by Piya, so Siya's Saturday sales = $\frac{y}{3}$.

5. The difference between cakes sold by Piya and Siya on Monday equals the difference on Saturday:
   $$|x - 20| = \left| 	ext{Piya's Saturday} - \frac{y}{3} ight|$$

6. Piya sells fewer cakes than Siya on each day, so:
   $$x < 20, \quad 2x < 	ext{Siya's Sunday}, \quad 	ext{Piya's Saturday} < \frac{y}{3}$$

7. Check statement I: If $x=18$, then Piya's Sunday sales = $2 	imes 18 = 36$. Total Piya sales $y = x + 2x + 	ext{Saturday sales} = 18 + 36 + 	ext{Saturday sales}$.

Since Siya's Saturday sales = average of Piya's sales = $\frac{y}{3}$, and the difference condition must hold, solving leads to contradictions or negative values, so statement I is incorrect.

8. Check statement II: If Piya sold 90 cakes on Saturday, then total Piya sales $y = 18 + 36 + 90 = 144$. Siya's Saturday sales = $\frac{144}{3} = 48$. The difference condition and Piya selling less than Siya on each day can be satisfied. Hence, statement II can be correct.

9. Check statement III: If Piya sold 21 cakes on Saturday, then total Piya sales $y = 18 + 36 + 21 = 75$. Siya's Saturday sales = $\frac{75}{3} = 25$. Using the difference condition and Piya selling less than Siya, the minimum Siya's Sunday sales comes out to be less than 35, so statement III is incorrect.

**Answer:** Only II

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