0
0

False Weights / Cheat Problems

Introduction

False-weights (cheat) problems aptitude tests में काफी आते हैं। Seller (या buyer) दावा की गई quantity से कम (या ज्यादा) देता/लेता है लेकिन price उतनी ही quantity का लेता/देता है। Trick यह है कि weight cheat को effective price-per-unit में convert किया जाए और फिर profit या loss निकाला जाए।

दो common setups:

  • Seller claimed quantity से कम quantity देता है लेकिन पूरी कीमत लेता है → implicit gain.
  • Buyer paid quantity से ज्यादा quantity लेता है → seller को implicit loss.

Pattern: False Weights / Cheat Problems

Pattern

Key idea: Declared quantity (जैसे 1 kg) को unit मान लें। पता करें कि वास्तव में कितनी quantity दी गई। उसे true unit के हिसाब से price में convert करें और फिर CP से compare करके profit/loss पाएँ।

Formula (declared = 1 unit):

  • Actual quantity given = r unit (cheating हो तो r < 1).
  • Effective SP per true unit = Price charged ÷ actual quantity given.
  • Profit% = ((Effective SP per unit - CP per unit) ÷ CP per unit) × 100.
  • यदि SP = CP पर बेच रहा है लेकिन r < 1 दे रहा है, तो Gain% = (1/r - 1) × 100.

Step-by-Step Example

Question

एक shopkeeper ₹40 per kg के भाव से चीनी बेचता है, लेकिन weight ऐसा रखता है कि buyer को 1 kg की जगह सिर्फ 900 g मिलते हैं। अगर उसका CP ₹30 per kg है, तो उसका profit% निकालें।

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 0.9 kg के लिए ₹40 देता है। 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:

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

Question

एक merchant दावा करता है कि वह 1 kg rice ₹50 में बेचता है, लेकिन वास्तव में हर “1 kg” में सिर्फ 950 g देता है। यदि उसका CP ₹40 per kg है, तो profit percentage निकालें।

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 ₹50 देता है 0.95 kg के लिए। 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:

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

Question

एक shopkeeper listed price पर बेचता है लेकिन quantity सिर्फ 95% देता है। वह CP पर 10% का markup भी करता है। यदि CP = ₹200 per unit है, तो overall profit% निकालें।

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 × 1.1 = 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:

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

Quick Variations

  • 1. Seller x% कम देता है → Gain% = x / (100 - x) × 100.
  • 2. Buyer ज्यादा लेता है → Loss% = (Extra taken ÷ actual given) × 100.
  • 3. Markup + false weight दोनों हों तो effective SP/unit निकालकर CP से compare करें।

Trick to Always Use

  • सबकुछ true unit पर convert करें।
  • Effective SP/unit = Price charged ÷ Actual quantity given.
  • SP और CP per unit compare करके profit या loss percentage निकालें।

Summary

Summary

  • Declared vs actual quantity पर ध्यान दें।
  • Effective SP per unit = SP ÷ actual_given.
  • Profit% = ((SP_effective - CP) ÷ CP) × 100.
  • Shortcut (अगर SP = CP हो): 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

Ready for a challenge?

Take a 10-minute AI-powered test with 10 questions (Easy-Medium-Hard mix) and get instant SWOT analysis of your performance!

10 Questions
5 Minutes