2.3 Relation Between CP and SP Given Profit% or Loss%

This is the single most-used formula block in the entire chapter — internalise it completely.

When profit% = p is given: > SP = CP × (100 + p)/100 > CP = SP × 100/(100 + p)

When loss% = l is given: > SP = CP × (100 − l)/100 > CP = SP × 100/(100 − l)

Think of (100+p)/100 or (100−l)/100 as a multiplying factor — exactly like you used in the Percentage chapter for successive percentage change. This is the fastest way to move between CP and SP in a single step, without writing “let CP = x” every time.

Worked Example 4: The cost price of an article is Rs 250. If it is sold at a profit of 20%, find the selling price.

Solution: SP = 250 × (120/100) = Rs 300.

Worked Example 5: An item is sold for Rs 690 at a profit of 15%. Find its cost price.

Solution: CP = 690 × 100/115 = Rs 600.

Worked Example 6: A television is sold for Rs 920, thereby gaining 15%. Had it been sold for Rs 800, what would have been the loss or gain percentage?

Solution: First find CP: CP = 920 × 100/115 = 800. So if sold for Rs 800, SP = CP exactly ⇒ no profit, no loss (0%).

Worked Example 6A: A trader wants to earn a profit of 24% on an article whose cost price is Rs 950. Find the required selling price, and also find the price he must quote if a customer bargains him down by Rs 38 from this SP — what would his resulting profit% become?

Solution: Required SP = 950 × 124/100 = 1178. If the price is reduced by Rs 38, new SP = 1178 − 38 = 1140. New profit = 1140 − 950 = 190. New profit% = (190/950) × 100 = 20%. This kind of “what if he had sold for Rs ___ less/more” question is extremely common in Tier-I and CHSL — always compute the original SP first using the multiplying factor, then simply add/subtract the stated rupee change before recomputing profit%.

Worked Example 6B (reverse: two target profits give CP): A trader sold an article for Rs 1955, incurring a loss of 15%. If, instead, he wants to earn a profit of 8% on the same article, what selling price must he now quote?

Solution: First recover CP from the loss transaction: CP = 1955 × 100/85 = Rs 2300. Required new SP = 2300 × 108/100 = Rs 2484. This is the mirror image of Worked Example 6A — here the original transaction was a loss, and we are asked to move to a profit target, so both the “100 − l” and “100 + p” multiplying factors get used in the same problem.

Worked Example 6C (fractional/decimal percentages — edge case): An article is bought for Rs 2400. It is sold at a loss of 7.5%. Find the selling price. Also find what selling price would instead yield a profit of 12.5% on the same article.

Solution: SP (at 7.5% loss) = 2400 × (100 − 7.5)/100 = 2400 × 0.925 = Rs 2220. SP (at 12.5% profit) = 2400 × (100 + 12.5)/100 = 2400 × 1.125 = Rs 2700. Decimal/fractional percentages like 7.5% and 12.5% appear regularly in CGL Tier-II and NTPC CBT-2; the multiplying-factor method handles them exactly as smoothly as whole-number percentages — there is no need for a separate technique.