2.9 Using CP = 100 as a Variable (Speed Technique)

Whenever a question gives you only ratios or percentages and no actual rupee value, assume CP = 100 (or, for multi-article problems, assume CP per article = 100 or CP of one unit = 1). This avoids fractions and lets you work entirely in whole numbers.

Worked Example 21: By selling an article at three-fourths (3/4) of its marked price, a shopkeeper makes a loss of 10%. What profit or loss percentage would he make if he sold it at the full marked price?

Solution: Let CP = 100. SP at (3/4)MP gives a 10% loss ⇒ SP = 90. So (3/4) × MP = 90 ⇒ MP = 90 × (4/3) = 120. If sold at full MP = 120, and CP = 100, Profit = 20 ⇒ Profit% = 20%.

Worked Example 22: If the cost price of 15 articles is equal to the selling price of 12 articles, find the profit percentage.

Solution: Let CP of 1 article = Re 1. Then CP of 15 articles = Rs 15 = SP of 12 articles. ⇒ SP of 1 article = 15/12 = 1.25. Profit per article = 1.25 − 1 = 0.25 ⇒ Profit% = 25%.

Worked Example 23: A shopkeeper marks his goods 40% above cost price and then allows a discount of 10%. Find his profit percentage.

Solution: Let CP = 100. MP = 140. SP = 140 × 0.9 = 126. Profit = 126 − 100 = 26 ⇒ Profit% = 26%.

Worked Example 23A (fraction-of-MP variant with a different fraction — edge case): By selling an article at five-sixths (5/6) of its marked price, a trader makes a loss of 5%. What profit percentage would he have made if he had sold it at the full marked price?

Solution: Let CP = 100. SP at (5/6)MP gives 5% loss ⇒ SP = 95. So (5/6) × MP = 95 ⇒ MP = 95 × (6/5) = 114. If sold at full MP = 114 with CP = 100, Profit = 14 ⇒ Profit% = 14%. Compare with Worked Example 21: a smaller loss (5% vs 10%) at a larger fraction of MP (5/6 vs 3/4) still needs to be checked by calculation each time — there is no shortcut that lets you guess the answer just from “loss is smaller.”

Worked Example 23B (mirror of Example 22 — the loss-producing direction, edge case): If the selling price of 18 articles is equal to the cost price of 15 articles, find the trader’s loss percentage.

Solution: Let CP of 1 article = Re 1. SP of 18 articles = CP of 15 articles = Rs 15. ⇒ SP of 1 article = 15/18 = 5/6. Loss per article = 1 − 5/6 = 1/6 ⇒ Loss% = (1/6) × 100 = 50/3 % ≈ 16.67%. Contrast this carefully with Worked Example 22: there, CP of more articles equalled SP of fewer articles, producing a profit; here, SP of more articles equals CP of fewer articles, producing a loss. Always check which count is attached to CP and which to SP before assuming the sign of the result.