Reverse Percentage Problems: Working Backwards
Most percentage problems ask you to find the result when you know the original value. Reverse percentage problems flip this: you know the result and need to find the original value. These problems appear frequently in shopping, finance, and data analysis, making them essential to master.
What Are Reverse Percentage Problems?
Reverse percentage problems start with the end result and work backwards to find the original value. Common scenarios include:
- Finding the original price when you know the sale price and discount percentage
- Determining the total amount when you know what percentage a portion represents
- Calculating pre-tax prices when you know the total with tax
- Finding original values after percentage increases or decreases
The Basic Reverse Percentage Formula
When you know that "X is Y% of what number?", use:
Original Value = X / (Y / 100)
Or more simply:
Original Value = X / (Y%)
Where X is the known result and Y% is the percentage.
Example: 30 is 20% of what number?
- Original value: 30 / 0.20 = 150
Finding Original Prices After Discounts
Sale Price to Original Price
If an item costs $60 after a 25% discount, what was the original price?
- Sale price = Original price × (1 - discount%)
- $60 = Original price × 0.75
- Original price = $60 / 0.75 = $80
Formula: Original Price = Sale Price / (1 - Discount Percentage / 100)
Verification: $80 × 0.25 = $20 discount, $80 - $20 = $60 ✓
Multiple Discounts
When multiple discounts are applied, work backwards through each:
- Final price: $51
- Last discount: 15% off
- Price before last discount: $51 / 0.85 = $60
- First discount: 20% off
- Original price: $60 / 0.80 = $75
Finding Original Amounts After Increases
Increased Value to Original
If a salary is $66,000 after a 10% raise, what was the original salary?
- New salary = Original × (1 + increase%)
- $66,000 = Original × 1.10
- Original = $66,000 / 1.10 = $60,000
Formula: Original Value = New Value / (1 + Increase Percentage / 100)
Verification: $60,000 × 0.10 = $6,000 increase, $60,000 + $6,000 = $66,000 ✓
Working with Tax-Inclusive Prices
Finding Pre-Tax Price
If the total with 8% tax is $108, what's the pre-tax price?
- Total = Pre-tax × (1 + tax%)
- $108 = Pre-tax × 1.08
- Pre-tax = $108 / 1.08 = $100
Verification: $100 × 0.08 = $8 tax, $100 + $8 = $108 ✓
Tip-Inclusive Totals
If a restaurant bill totals $132 including an 18% tip, what was the pre-tip amount?
- Total = Pre-tip × (1 + tip%)
- $132 = Pre-tip × 1.18
- Pre-tip = $132 / 1.18 = $111.86
Finding Totals from Percentages
Part to Whole
If 45 students represent 30% of a class, how many students are in the class?
- Part = Whole × percentage
- 45 = Whole × 0.30
- Whole = 45 / 0.30 = 150 students
Verification: 150 × 0.30 = 45 ✓
Multiple Parts
If you know several parts and their percentages:
- Product A sales: $30,000 (25% of total)
- Product B sales: $45,000 (37.5% of total)
- Total from A: $30,000 / 0.25 = $120,000
- Total from B: $45,000 / 0.375 = $120,000
Both calculations confirm the total is $120,000.
Salary and Compensation Calculations
Gross Salary from Net
If your take-home pay is $3,600 after 20% deductions, what's your gross salary?
- Net = Gross × (1 - deduction%)
- $3,600 = Gross × 0.80
- Gross = $3,600 / 0.80 = $4,500
Verification: $4,500 × 0.20 = $900 deductions, $4,500 - $900 = $3,600 ✓
Commission Calculations
If a salesperson earns $8,000 commission at 5% rate, what were total sales?
- Commission = Sales × commission%
- $8,000 = Sales × 0.05
- Sales = $8,000 / 0.05 = $160,000
Grade and Score Calculations
Total Points from Percentage
If you scored 85% on a test and got 85 points, how many total points were possible?
- Score = Total × percentage
- 85 = Total × 0.85
- Total = 85 / 0.85 = 100 points
Verification: 85 / 100 = 85% ✓
Finding Missing Scores
If you know your average and most scores:
- Test 1: 90% (worth 30%)
- Test 2: 85% (worth 30%)
- Test 3: ? (worth 40%)
- Overall average: 88%
To find Test 3 score:
- Weighted contribution from Tests 1 & 2: 0.30 × 90 + 0.30 × 85 = 52.5
- Needed from Test 3: (88 - 52.5) / 0.40 = 88.75%
Investment and Interest Calculations
Principal from Interest
If you earned $500 interest at 5% annual rate, what was your principal?
- Interest = Principal × rate × time
- $500 = Principal × 0.05 × 1
- Principal = $500 / 0.05 = $10,000
Future Value to Present Value
If an investment grows to $11,000 at 10% annual return, what was the original investment?
- Future value = Present × (1 + rate)^years
- $11,000 = Present × (1.10)¹
- Present = $11,000 / 1.10 = $10,000
Population and Statistics
Total Population from Sample
If a survey of 400 people represents 2% of a population, what's the total population?
- Sample = Total × percentage
- 400 = Total × 0.02
- Total = 400 / 0.02 = 20,000
Market Share Calculations
If your company has $2.5 million in sales representing 12.5% market share, what's the total market size?
- Market share = Company sales / Total market
- 0.125 = $2,500,000 / Total market
- Total market = $2,500,000 / 0.125 = $20,000,000
Common Problem Types
Type 1: "X is Y% of what number?"
Direct application of the formula:
- 75 is 15% of what number?
- Answer: 75 / 0.15 = 500
Type 2: "After a Y% discount, the price is X. What was the original?"
- After 20% discount, price is $80
- Answer: $80 / 0.80 = $100
Type 3: "After a Y% increase, the value is X. What was the original?"
- After 25% increase, value is $125
- Answer: $125 / 1.25 = $100
Type 4: "X represents Y% of the total. What is the total?"
- $450 represents 30% of total
- Answer: $450 / 0.30 = $1,500
Step-by-Step Problem Solving
Problem: A store marks up items by 40%. If an item sells for $140, what did the store pay for it?
Step 1: Identify what you know
- Final price: $140
- Markup: 40%
Step 2: Determine the relationship
- Final price = Cost × (1 + markup%)
- $140 = Cost × 1.40
Step 3: Solve for the unknown
- Cost = $140 / 1.40 = $100
Step 4: Verify
- $100 × 0.40 = $40 markup
- $100 + $40 = $140 ✓
Common Mistakes
1. Using the Wrong Formula
- Wrong: Original = Result × percentage
- Correct: Original = Result / percentage
2. Forgetting to Convert Percentage
Always convert percentages to decimals:
- 25% = 0.25, not 25
3. Confusing Increase and Decrease
For increases: divide by (1 + %) For decreases: divide by (1 - %)
4. Wrong Base for Calculations
When working backwards, ensure you're using the correct percentage relationship.
Real-World Applications
Retail Pricing
Finding wholesale costs from retail prices:
- Retail price: $120
- Markup: 50%
- Wholesale cost: $120 / 1.50 = $80
Budget Planning
Determining total budget from allocated portion:
- Allocated: $15,000
- This is 25% of total budget
- Total budget: $15,000 / 0.25 = $60,000
Loan Calculations
Finding principal from interest payments:
- Monthly interest: $150
- Interest rate: 6% annually (0.5% monthly)
- Principal: $150 / 0.005 = $30,000
Advanced Scenarios
Multiple Percentage Changes
If a value increased 20% then decreased 15% to reach $102, what was the original?
- Final = Original × 1.20 × 0.85
- $102 = Original × 1.02
- Original = $102 / 1.02 = $100
Percentage of a Percentage
If 60% of students passed, and 40% of those passed with honors, and 24 students passed with honors, how many students total?
- Honors = Total × 0.60 × 0.40
- 24 = Total × 0.24
- Total = 24 / 0.24 = 100 students
Verification Strategies
Always verify reverse percentage calculations:
- Forward Check: Apply the percentage to your answer and confirm you get the known result
- Alternative Method: Solve using a different approach if possible
- Reasonableness: Check if your answer makes sense in context
Conclusion
Reverse percentage problems are essential for real-world calculations, from shopping to finance to data analysis. Mastering these techniques helps you work backwards from known results to find original values, making you better equipped to solve complex problems. Practice with various scenarios and use our Percentage Calculator to verify your work and build confidence.
FAQs
Q: How do I know whether to divide or multiply when working backwards?
A: If you're finding the original from a result, divide by the percentage (as a decimal). If you're finding a result from an original, multiply by the percentage.
Q: What if I have multiple percentage changes?
A: Multiply all the percentage factors together, then divide the final value by that product. For example, 20% increase then 10% decrease: divide by (1.20 × 0.90) = 1.08.
Q: How do I handle reverse percentages with taxes or tips?
A: Divide the total by (1 + tax% or tip%) to find the pre-tax/pre-tip amount. For example, $110 with 10% tax: $110 / 1.10 = $100.
Q: Can I use reverse percentages for compound growth?
A: Yes. For compound growth over multiple periods, divide by (1 + rate)^periods. For example, $121 after 2 years at 10%: $121 / (1.10)² = $100.
Sources
- Khan Academy – Reverse percentage problems and working backwards
- Math is Fun – Percentage calculations and reverse problems
- Purplemath – Percentage word problems and solution strategies