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False Weights / Cheat Problems

Introduction

False-weights (cheat) problems appear frequently in aptitude tests. A seller (or buyer) gives less (or takes more) quantity than claimed while charging the price for the claimed quantity. The trick is to convert the weight cheat into an effective price-per-unit change and then compute profit or loss.

Two common setups:

  • Seller gives less quantity than claimed but charges full price → implicit gain.
  • Buyer takes more quantity than paid for → implicit loss to the seller.

Pattern: False Weights / Cheat Problems

Pattern

Key idea: Treat the declared quantity (e.g. 1 kg) as the unit of sale. Find the actual quantity given for that unit. Convert that to an effective price per true unit, then compare with cost price per true unit.

Formula (declared = 1 unit):

  • Actual quantity given = r unit (where r < 1 if cheating).
  • Effective SP per true unit = Price charged ÷ actual quantity given.
  • Profit% = ((Effective SP per unit - CP per unit) ÷ CP per unit) × 100.
  • If selling at cost price (SP = CP) but giving r < 1, then Gain% = (1/r - 1) × 100.

Step-by-Step Example

Question

A shopkeeper sells sugar at the price of ₹40 per kg but uses a weight such that a buyer receives only 900 g when paying for 1 kg. If his cost price is ₹30 per kg, find his profit percentage.

Options:

  • A. 48.15%
  • B. 45.00%
  • C. 40.00%
  • D. 50.00%

Solution

  1. Step 1: Convert units to true kg

    900 g = 900 ÷ 1000 = 0.9 kg.

  2. Step 2: Compute effective selling price per true kg

    Customer pays ₹40 for 0.9 kg. Effective SP per kg = 40 ÷ 0.9 = 44.44.

  3. Step 3: Calculate profit per kg

    CP per kg = ₹30. Profit per kg = 44.44 - 30 = 14.44.

  4. Step 4: Compute profit percentage

    Profit% = (14.44 ÷ 30) × 100 ≈ 48.15%.

  5. Final Answer:

    48.15% → Option A

  6. Quick Check:

    CP for 0.9 kg = 30 × 0.9 = ₹27. SP for 0.9 kg = ₹40. Profit = 40 - 27 = ₹13. Profit% = (13 ÷ 27) × 100 = 48.15% ✅

Question

A merchant claims to sell 1 kg of rice at ₹50. He actually gives 950 g for each "1 kg". If his cost price is ₹40 per kg, find his profit percentage.

Options:

  • A. 30.00%
  • B. 31.58%
  • C. 32.00%
  • D. 28.00%

Solution

  1. Step 1: Convert actual quantity to kg

    950 g = 950 ÷ 1000 = 0.95 kg.

  2. Step 2: Compute effective selling price per true kg

    Customer pays ₹50 for 0.95 kg. Effective SP per kg = 50 ÷ 0.95 = 52.63.

  3. Step 3: Compute profit per kg

    CP per kg = ₹40. Profit per kg = 52.63 - 40 = 12.63.

  4. Step 4: Compute profit percentage

    Profit% = (12.63 ÷ 40) × 100 ≈ 31.58%.

  5. Final Answer:

    31.58% → Option B

  6. Quick Check:

    CP for 0.95 kg = 40 × 0.95 = ₹38. SP for 0.95 kg = ₹50. Profit = 12. Profit% = (12 ÷ 38) × 100 = 31.58% ✅

Question

A shopkeeper sells at the listed price but gives only 95% of the quantity. He also marks up price by 10% on CP. If CP = ₹200 per unit, find overall profit%.

Options:

  • A. 15.00%
  • B. 15.79%
  • C. 16.00%
  • D. 14.50%

Solution

  1. Step 1: Compute marked price after markup

    CP = ₹200. Marked price = 200 × (1 + 10/100) = 220.

  2. Step 2: Account for false weight

    Actual given = 95% = 0.95 unit.

  3. Step 3: Compute effective SP per true unit

    SP per true unit = 220 ÷ 0.95 = 231.58.

  4. Step 4: Compute profit and percentage

    Profit per unit = 231.58 - 200 = 31.58. Profit% = (31.58 ÷ 200) × 100 = 15.79%.

  5. Final Answer:

    15.79% → Option B

  6. Quick Check:

    CP for 0.95 unit = 200 × 0.95 = ₹190. SP charged = 220. Profit = 30. Profit% = (30 ÷ 190) × 100 = 15.79% ✅

Quick Variations

  • 1. If seller gives x% less, Gain% = x / (100 - x) × 100.
  • 2. If buyer cheats (takes more), Loss% = (Extra taken ÷ actual given) × 100.
  • 3. If both markup and false weight exist, compute effective SP/unit first, then compare with CP.

Trick to Always Use

  • Convert everything to per true unit.
  • Effective SP/unit = Price charged ÷ Actual quantity given.
  • Compare SP vs CP per unit → find profit or loss percentage.

Summary

Summary

  • Focus on declared vs actual quantity.
  • Effective SP per unit = SP ÷ actual_given.
  • Profit% = ((SP_effective - CP) ÷ CP) × 100.
  • Shortcut (selling at CP): Gain% = (1/r - 1) × 100.

Practice

(1/5)
1. A shopkeeper sells 1 kg sugar for ₹50 but actually gives 950 g only. What is his gain percentage if CP = ₹40 per kg?
easy
A. 20%
B. 22.22%
C. 31.58%
D. 25%

Solution

  1. Step 1: Convert actual weight to kg

    Actual quantity = 950 g = 0.95 kg.

  2. Step 2: Compute effective selling price per true kg

    Effective SP per kg = 50 ÷ 0.95 = 52.63157895.

  3. Step 3: Compute profit per kg

    Profit per kg = 52.63157895 - 40 = 12.63157895.

  4. Step 4: Compute profit percentage

    Profit% = (12.63157895 ÷ 40) × 100 = 31.58%.

  5. Final Answer:

    31.58% → Option C

  6. Quick Check:

    CP for 0.95 kg = 40 × 0.95 = 38; SP = 50; profit = 12; profit% = 12 ÷ 38 × 100 = 31.58% ✅
Hint: Find effective SP per actual unit: Price ÷ (actual fraction), then compare with CP/unit.
Common Mistakes: Forgetting to convert grams to kg or treating SP as per actual quantity instead of per declared unit.
2. A trader sells goods at cost price but uses a false weight of 800 g instead of 1 kg. Find his gain percentage.
easy
A. 20%
B. 25%
C. 30%
D. 40%

Solution

  1. Step 1: Convert actual quantity

    Declared 1 kg, actual given = 800 g = 0.8 kg.

  2. Step 2: Use cheat formula

    When selling at CP per declared kg, gain% = (1/r - 1) × 100, where r = 0.8.

  3. Step 3: Compute gain%

    Gain% = (1/0.8 - 1) × 100 = (1.25 - 1) × 100 = 25%.

  4. Final Answer:

    25% → Option B

  5. Quick Check:

    CP for 0.8 kg = 0.8 units; SP charged = 1 unit; profit = 0.2; profit% = 0.2 ÷ 0.8 × 100 = 25% ✅
Hint: If selling at CP, use Gain% = (1/r - 1) × 100 where r is actual fraction of declared weight.
Common Mistakes: Using (1 - r) × 100 instead of (1/r - 1) × 100.
3. A merchant claims to sell 1 litre of milk at ₹60 but actually gives only 900 ml. If CP is ₹45 per litre, find his gain percentage.
medium
A. 33.33%
B. 28.57%
C. 36%
D. 48.15%

Solution

  1. Step 1: Convert ml to litres

    Actual given = 900 ml = 0.9 L.

  2. Step 2: Compute effective selling price per litre

    Effective SP per litre = 60 ÷ 0.9 = 66.6666667.

  3. Step 3: Compute profit per litre

    Profit per litre = 66.6666667 - 45 = 21.6666667.

  4. Step 4: Compute profit percentage

    Profit% = (21.6666667 ÷ 45) × 100 ≈ 48.15%.

  5. Final Answer:

    48.15% → Option D

  6. Quick Check:

    CP for 0.9 L = 45 × 0.9 = 40.5; SP = 60; profit = 19.5; profit% = 19.5 ÷ 40.5 × 100 = 48.15% ✅
Hint: Compute SP per true unit (price ÷ actual fraction) then compare with CP/unit.
Common Mistakes: Mixing litres and ml or computing profit% relative to wrong base.
4. A trader sells goods at ₹80 per kg but cheats by giving 950 g. If CP is ₹60 per kg, find profit percentage.
medium
A. 25%
B. 30%
C. 33.33%
D. 40.35%

Solution

  1. Step 1: Convert grams to kg

    Actual given = 950 g = 0.95 kg.

  2. Step 2: Compute effective SP per kg

    Effective SP per kg = 80 ÷ 0.95 = 84.2105263.

  3. Step 3: Compute profit per kg

    Profit per kg = 84.2105263 - 60 = 24.2105263.

  4. Step 4: Compute profit percentage

    Profit% = (24.2105263 ÷ 60) × 100 ≈ 40.35%.

  5. Final Answer:

    40.35% → Option D

  6. Quick Check:

    CP for 0.95 kg = 60 × 0.95 = 57; SP = 80; profit = 23; profit% = 23 ÷ 57 × 100 ≈ 40.35% ✅
Hint: Compare 'price ÷ actual weight' with CP per unit to get profit% directly.
Common Mistakes: Using profit per unit but dividing by wrong CP (use same unit basis).
5. A merchant uses a false weight of 850 g instead of 1 kg and sells at CP. Find his profit percentage.
hard
A. 17.65%
B. 15%
C. 20%
D. 25%

Solution

  1. Step 1: Convert false weight

    Declared 1 kg, actual = 850 g = 0.85 kg.

  2. Step 2: Apply cheat formula

    Gain% = (1/r - 1) × 100 where r = 0.85.

  3. Step 3: Compute gain%

    Gain% = (1/0.85 - 1) × 100 = 0.1764706 × 100 = 17.65%.

  4. Final Answer:

    17.65% → Option A

  5. Quick Check:

    CP for 0.85 kg = 0.85 units; SP charged = 1 unit; profit = 0.15; profit% = 0.15 ÷ 0.85 × 100 = 17.65% ✅
Hint: When selling at CP with a false weight r, profit% = (1/r - 1) × 100.
Common Mistakes: Rounding too early or using the wrong decimal for r.

Mock Test

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