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Finding CP or SP

Introduction

In profit and loss questions, you are often given the Cost Price (CP), Selling Price (SP), or the percentage of profit/loss, and asked to find the missing value.

Mastering these relations is crucial for solving problems quickly and accurately.

Pattern: Finding CP or SP

Pattern

The key idea is to use the relation between CP, SP, and Profit/Loss %.

Profit %: SP = CP × (1 + Profit%/100)
Loss %: SP = CP × (1 - Loss%/100)
Finding CP from SP: CP = SP ÷ (1 ± Profit/Loss%/100)

Step-by-Step Example

Question

A shopkeeper sells an item for ₹600 at a profit of 20%. Find its cost price.

Options:

  • A. ₹480
  • B. ₹520
  • C. ₹500
  • D. ₹550

Solution

  1. Step 1: Write the relation between SP and CP.

    Formula → SP = CP × (1 + Profit%/100).

  2. Step 2: Substitute given values and simplify.

    Substitute values → 600 = CP × 1.2.

  3. Step 3: Solve for CP.

    CP = 600 ÷ 1.2 = ₹500.

  4. Final Answer:

    ₹500 → Option C

  5. Quick Check:

    20% of 500 = 100 → SP = 500 + 100 = 600 ✅

Quick Variations

1. If CP and profit % are given → Find SP.

2. If SP and profit % are given → Find CP.

3. If SP and loss % are given → Find CP.

4. If CP and loss % are given → Find SP.

Trick to Always Use

  • Step 1: Remember SP = CP × (1 ± %/100).
  • Step 2: Rearrange when CP needs to be found.
  • Step 3: Always apply + for profit and - for loss.
  • Step 4: Quick verify with mental calculation.

Summary

Summary

  • Use SP = CP × (1 + profit%/100) to get SP when profit% is given.
  • Use SP = CP × (1 - loss%/100) to get SP when loss% is given.
  • To find CP from SP, divide SP by (1 ± %/100) depending on profit/loss.
  • Always perform a quick check: compute % of CP and add/subtract to verify SP.

Example to remember:
If SP = ₹600 at 20% profit → CP = ₹600 ÷ 1.2 = ₹500

Practice

(1/5)
1. An article is sold for ₹720 at a profit of 20%. Find the cost price.
easy
A. ₹600
B. ₹650
C. ₹700
D. ₹750

Solution

  1. Step 1: Write the SP-CP relation for profit.

    Formula → SP = CP × (1 + Profit%/100).

  2. Step 2: Substitute values.

    720 = CP × 1.2.

  3. Step 3: Solve for CP.

    CP = 720 ÷ 1.2 = ₹600.

  4. Final Answer:

    Cost Price = ₹600 → Option A

  5. Quick Check:

    20% of 600 = 120 → 600 + 120 = 720 ✅
Hint: CP = SP ÷ (1 + Profit%/100).
Common Mistakes: Subtracting profit directly from SP instead of reversing the formula.
2. An item was sold at ₹480 at a loss of 20%. Find the cost price.
easy
A. ₹500
B. ₹550
C. ₹600
D. ₹650

Solution

  1. Step 1: Write the SP-CP relation for loss.

    Formula → SP = CP × (1 - Loss%/100).

  2. Step 2: Substitute values.

    480 = CP × 0.8.

  3. Step 3: Solve for CP.

    CP = 480 ÷ 0.8 = ₹600.

  4. Final Answer:

    Cost Price = ₹600 → Option C

  5. Quick Check:

    20% of 600 = 120 → 600 - 120 = 480 ✅
Hint: CP = SP ÷ (1 - Loss%/100).
Common Mistakes: Adding loss instead of subtracting when back-calculating CP.
3. The cost price of an article is ₹1,200. It is sold at a loss of 15%. Find the selling price.
medium
A. ₹960
B. ₹1,000
C. ₹1,020
D. ₹1,050

Solution

  1. Step 1: Use the SP formula for loss.

    Formula → SP = CP × (1 - Loss%/100).

  2. Step 2: Substitute values.

    SP = 1200 × (1 - 0.15).

  3. Step 3: Calculate the final SP.

    SP = 1200 × 0.85 = ₹1,020.

  4. Final Answer:

    Selling Price = ₹1,020 → Option C

  5. Quick Check:

    15% of 1200 = 180 → 1200 - 180 = 1020 ✅
Hint: SP = CP × (100 - Loss%)/100.
Common Mistakes: Subtracting 15 directly from 1200 instead of percentage calculation.
4. An article is bought at ₹1,500 and sold at a profit of 25%. Find the selling price.
medium
A. ₹1,750
B. ₹1,800
C. ₹1,850
D. ₹1,875

Solution

  1. Step 1: Use the SP formula for profit.

    Formula → SP = CP × (1 + Profit%/100).

  2. Step 2: Substitute values.

    SP = 1500 × (1 + 0.25).

  3. Step 3: Calculate the final SP.

    SP = 1500 × 1.25 = ₹1,875.

  4. Final Answer:

    Selling Price = ₹1,875 → Option D

  5. Quick Check:

    25% of 1500 = 375 → 1500 + 375 = 1875 ✅
Hint: SP = CP × (100 + Profit%)/100.
Common Mistakes: Adding 25 directly to CP instead of 25% of CP.
5. An item is sold for ₹960 at a profit of 20%. Find the cost price.
medium
A. ₹750
B. ₹780
C. ₹800
D. ₹820

Solution

  1. Step 1: Write the SP-CP relation for profit.

    Formula → SP = CP × (1 + Profit%/100).

  2. Step 2: Substitute values.

    960 = CP × 1.2.

  3. Step 3: Solve for CP.

    CP = 960 ÷ 1.2 = ₹800.

  4. Final Answer:

    Cost Price = ₹800 → Option C

  5. Quick Check:

    20% of 800 = 160 → 800 + 160 = 960 ✅
Hint: Always divide SP by 1.2 when profit is 20%.
Common Mistakes: Subtracting 20% of SP instead of CP.

Mock Test

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