How to Solve: Profit and Loss

How to Solve: Profit and Loss

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"Imagine you’re running a lemonade stand. If you sell each cup for $2 but it costs you $1.50 to make, how much profit do you make? And if you raise the price to $2.50, what’s your new profit percentage? Mastering profit and loss lets you answer these questions in seconds—and crush your exam!

What You Need To Know First

Before diving into profit and loss, you must understand: 1. Percentage calculations – How to find 10% of 50 or increase 80 by 25%. 2. Basic algebra – Solving for an unknown (e.g., if Cost Price + Profit = Selling Price, find Cost Price). 3. Simple word problems – Translating sentences like "A shopkeeper gains 20% by selling a book for $120" into equations.

If any of these feel shaky, review them first!

Key Vocabulary

Formulas To Know

(MEMORISE THESE—most exams don’t provide them!)

1. Profit
   Formula: Profit = Selling Price (SP) – Cost Price (CP)
   Variables:
2. SP = Price the item is sold for.

3. CP = Price the item was bought for.

4. Loss
   Formula: Loss = Cost Price (CP) – Selling Price (SP)

5. Profit Percentage
   Formula: Profit % = (Profit / CP) × 100
   Alternative (if SP is given): Profit % = [(SP – CP) / CP] × 100

6. Loss Percentage
   Formula: Loss % = (Loss / CP) × 100
   Alternative (if SP is given): Loss % = [(CP – SP) / CP] × 100

7. Selling Price (when Profit % is given)
   Formula: SP = CP × (1 + Profit%/100)
   
   Example: If CP = $100 and Profit% = 20%, then SP = 100 × (1 + 0.20) = $120.

8. Selling Price (when Loss % is given)
   Formula: SP = CP × (1 – Loss%/100)
   
   Example: If CP = $100 and Loss% = 15%, then SP = 100 × (1 – 0.15) = $85.

9. Cost Price (when Profit % and SP are given)
   Formula: CP = SP / (1 + Profit%/100)
   
   Example: If SP = $120 and Profit% = 20%, then CP = 120 / (1 + 0.20) = $100.

10. Cost Price (when Loss % and SP are given)
    Formula: CP = SP / (1 – Loss%/100)
    
    Example: If SP = $85 and Loss% = 15%, then CP = 85 / (1 – 0.15) = $100.

Step-by-Step Method

Follow these 5 steps for every profit and loss problem:

1. Identify what’s given and what’s asked.
2. Underline key numbers and terms (CP, SP, Profit%, Loss%).

3. Circle the unknown (what you need to find).

4. Write down the relevant formula.

5. Pick from the list above based on what’s given/asked.

6. Plug in the known values.

7. Substitute numbers into the formula carefully.

8. Solve for the unknown.

9. Do the math step by step (show all working!).

10. Check your answer.

11. Does it make sense? (e.g., Profit% can’t be negative; SP should be > CP if there’s a profit.)

Worked Example Using the Steps

Question: A shopkeeper buys a shirt for $80 and sells it for $100. Find the profit percentage.

Step 1: Identify given and asked - Given: CP = $80, SP = $100 - Asked: Profit %

Step 2: Write the formula - Profit % = (Profit / CP) × 100 - But first, find Profit = SP – CP

Step 3: Plug in values - Profit = $100 – $80 = $20 - Profit % = ($20 / $80) × 100

Step 4: Solve - Profit % = 0.25 × 100 = 25%

Step 5: Check - 25% of $80 = $20 (matches Profit). ✔️

Answer: 25%

Worked Examples

Example 1 – Basic

Question: A toy costs $40 to make and is sold for $50. Find the profit percentage.

Step 1: CP = $40, SP = $50. Find Profit %. Step 2: Profit = SP – CP = $50 – $40 = $10 Step 3: Profit % = (Profit / CP) × 100 = (10 / 40) × 100 Step 4: Profit % = 0.25 × 100 = 25% Answer: 25%

What we did and why: - We found the profit first ($10), then expressed it as a percentage of the cost price ($40). This follows the formula directly.

Example 2 – Medium

Question: A book is sold for $180 at a 20% profit. Find the cost price.

Step 1: SP = $180, Profit% = 20%. Find CP. Step 2: Use CP = SP / (1 + Profit%/100) Step 3: CP = 180 / (1 + 0.20) = 180 / 1.20 Step 4: CP = $150 Answer: $150

What we did and why: - Since we had SP and Profit%, we rearranged the formula to solve for CP. Dividing by 1.20 is the same as increasing CP by 20% to get SP.

Example 3 – Exam Style

Question: A merchant sells two chairs for $240 each. On one, he gains 20%, and on the other, he loses 20%. Find his overall profit or loss percentage.

Step 1: For each chair, SP = $240. Find CP for both. - Chair 1 (Profit 20%): CP = SP / (1 + 0.20) = 240 / 1.20 = $200 - Chair 2 (Loss 20%): CP = SP / (1 – 0.20) = 240 / 0.80 = $300

Step 2: Total CP = $200 + $300 = $500 Step 3: Total SP = $240 × 2 = $480 Step 4: Loss = CP – SP = $500 – $480 = $20 Step 5: Loss % = (Loss / Total CP) × 100 = (20 / 500) × 100 = 4% Answer: 4% loss

What we did and why: - We treated each chair separately to find their CPs, then combined them. The key was realizing the profit and loss percentages apply to different CPs, not the same one.

Common Mistakes

Exam Traps

1-Minute Recap

"Okay, let’s lock this in—here’s what you need to remember for profit and loss: 1. Profit = SP – CP. If SP > CP, it’s a profit. If CP > SP, it’s a loss. 2. Profit% = (Profit / CP) × 100. Always divide by CP, not SP! 3. SP = CP × (1 + Profit%/100) for profit, or SP = CP × (1 – Loss%/100) for loss. 4. If you have SP and %, rearrange to find CP: CP = SP / (1 ± %/100). 5. Watch for traps: Profit% is on CP, not SP. And 20% profit + 20% loss ≠ 0% overall!

Now go practice 3 problems tonight. Write down every step—no shortcuts! You’ve got this!