A man sold 12 candles for $10, incurring a loss of b%. He then sold the same…

Concept: When the same quantity of goods is sold once at a loss of x% and once at a profit of x%, the Cost Price (CP) can be recovered from the Selling Price (SP) using CP = SP x 100/(100-loss%) for a loss and CP = SP x 100/(100+profit%) for a profit. Since the CP is identical in both sales (same batch of candles), equating these two expressions for CP solves for the unknown percentage.

Application:

1. Let CP be the cost price of the 12 candles, the same in both sales.

2. First sale: SP1 = $10 at a loss of b%, so CP = 10 x 100/(100-b).

3. Second sale: SP2 = $12 at a profit of b%, so CP = 12 x 100/(100+b).

4. Equate the two expressions for CP: 10 x 100/(100-b) = 12 x 100/(100+b).

5. Cross-multiply: 1000(100+b) = 1200(100-b).

6. Expand both sides: 100000 + 1000b = 120000 - 1200b.

7. Collect the b terms: 1000b + 1200b = 120000 - 100000, so 2200b = 20000.

8. Solve for b: b = 20000/2200 = 100/11, approximately 9.09.

Cross-check: Substitute b = 100/11 back into either expression to verify: 100-b = 1000/11, so CP = 1000 / (1000/11) = 11. Also 100+b = 1200/11, so CP = 1200 / (1200/11) = 11. Both give the same Cost Price ($11), confirming b is approximately 9.09.