2.12 A Fully Combined Problem (Markup + Successive Discount + False Weight)

Exam-level (Tier-II / NTPC higher-difficulty) questions occasionally stack two or even three of the above ideas into a single problem. The method never changes — work out the actual cost incurred and the actual revenue received, in that order, and only then compute the profit or loss percentage. Do not try to shortcut-combine the individual percentages by addition; as shown below, only careful sequential multiplication gives the right answer.

Worked Example 26: A dealer marks his goods 30% above the cost price. He then allows successive discounts of 10% and 5% to the customer. In addition, he uses a weight of 950 g in place of the standard 1 kg while weighing out the goods. Find his overall profit percentage.

Solution: Assume the true (correct) weight is 1000 g, with cost price Re 1 per gram, so CP of 1000 g = Rs 1000.

Step 1 — Actual cost incurred by the dealer for the goods he really hands over (950 g): Actual CP = 950 × 1 = Rs 950.

Step 2 — Marked price, based on the claimed 1000 g, marked up 30%: MP = 1000 × 1.30 = Rs 1300.

Step 3 — Apply the two successive discounts to the MP: After 10% discount: 1300 × 0.90 = 1170. After 5% discount: 1170 × 0.95 = Rs 1111.50 — this is the actual revenue (SP) received by the dealer.

Step 4 — Profit and profit%, computed on the actual cost (Step 1), not the claimed 1000 g: Profit = 1111.50 − 950 = Rs 161.50. Profit% = (161.50/950) × 100 = 17%.

Three separate effects — a 30% markup, a combined ≈14.5% successive discount, and a 5% false-weight shortfall — have combined to produce a clean 17% profit. This is the standard template for any question that layers markup, discount, and false weight together: actual cost first, actual revenue second, profit% last.

Worked Example 26A (markup + single discount + false weight, different numbers): A trader marks his goods 50% above cost price and allows a discount of 20% to attract customers. He also uses a weight of 900 g in place of 1 kg while weighing out the goods. Find his overall profit percentage.

Solution: True weight = 1000 g, CP (Re 1/g) = Rs 1000. Actual cost of goods physically handed over (900 g) = Rs 900. MP (based on claimed 1000 g) = 1000 × 1.50 = Rs 1500. SP after 20% discount = 1500 × 0.80 = Rs 1200 — this is the actual revenue received (based on the claimed 1000 g, discounted). Profit = 1200 − 900 = Rs 300. Profit% (on actual CP of 900) = (300/900) × 100 = 33.33% (100/3%).

Worked Example 26B (markup + THREE successive discounts + false weight — hardest variant): A trader marks his goods 25% above cost price. He allows three successive discounts of 10%, 5%, and 4% to a customer, and also uses a weight of 950 g in place of 1 kg while weighing out the goods. Find his overall profit percentage.

Solution: True weight = 1000 g, CP = Rs 1000. Actual cost of goods physically handed over (950 g) = Rs 950. MP = 1000 × 1.25 = Rs 1250. Apply the three successive discounts to the MP: 1250 × 0.90 = 1125; 1125 × 0.95 = 1068.75; 1068.75 × 0.96 = Rs 1026 — this is the actual revenue received. (Quicker: combine the three discount factors first — 0.90 × 0.95 × 0.96 = 0.8208 — then 1250 × 0.8208 = 1026.) Profit = 1026 − 950 = Rs 76. Profit% (on actual CP of 950) = (76/950) × 100 = 8%. This is the most heavily stacked variant of the false-weight family — a markup, three chained discounts, and a false weight all in one problem — but the resolution method is identical to Worked Example 26: compute actual cost, compute actual revenue (using the claimed weight throughout the markup/discount chain), and only then find profit%.