Percentage Calculation
Complete guide to percentage calculations with formulas, examples, and real-world applications
Introduction to Percentages
Percentage calculation is a fundamental skill in mathematics that you encounter daily in real life. From discounts and price increases to interest rates and statistics, percentages help us understand proportional relationships.
A percentage literally means "per hundred" (from Latin "per centum"). It expresses a number as a fraction of 100.
- \(\displaystyle G\) = Basic value (the whole, the original amount, 100%)
- \(\displaystyle P\) = Percentage (the percentage rate, expressed in %)
- \(\displaystyle W\) = Percentage value (the part, the absolute amount)
Fundamental Percentage Formula
All percentage calculations are based on one fundamental relationship. From this single formula, we can derive all other calculations.
\(\displaystyle \frac{W}{G} = \frac{P}{100}\)
or rearranged: \(\displaystyle W = \frac{G \cdot P}{100}\) and \(\displaystyle G = \frac{W \cdot 100}{P}\) and \(\displaystyle P = \frac{W \cdot 100}{G}\)
The fundamental equation states that the ratio of the percentage value to the basic value equals the ratio of the percentage to 100. This relationship allows us to solve for any unknown variable.
Case 1: Calculate the Percentage Value (W)
This is the most common percentage calculation. We know the basic value and the percentage, and we want to find the percentage value.
Given: Basic value \(G\) and Percentage \(P\)
Find: Percentage value \(W\)
\(\displaystyle W = \frac{G \cdot P}{100}\)
Example 1: Wage Increase
A salary of $1,000 is increased by 3%. Calculate the amount of the increase.
Solution
\(\displaystyle G = 1000\) (basic value = original salary)
\(\displaystyle P = 3\) (percentage = 3%)
\(\displaystyle W = \frac{G \cdot P}{100} = \frac{1000 \cdot 3}{100} = \frac{3000}{100} = 30\)
New salary = \(\displaystyle 1000 + 30 = 1030\)
Example 2: Discount Calculation
A product originally costs $200. It is on sale at 25% off. Calculate the discount amount and the final price.
Solution
\(\displaystyle G = 200\) (original price)
\(\displaystyle P = 25\) (discount percentage)
\(\displaystyle W = \frac{200 \cdot 25}{100} = 50\)
Sale price = \(\displaystyle 200 - 50 = 150\)
Case 2: Calculate the Basic Value (G)
In this case, we know the percentage value and the percentage, and we need to find the original amount (the basic value).
Given: Percentage value \(W\) and Percentage \(P\)
Find: Basic value \(G\)
\(\displaystyle G = \frac{W \cdot 100}{P}\)
Example 3: Finding the Original Price
You bought a product on sale for $120. The discount was 40%. What was the original price?
Solution
So the discount is 40%, but the percentage value is 60% of the original price.
\(\displaystyle W = 120\) (sale price = 60% of original)
\(\displaystyle P = 60\) (you pay 60% when reduced by 40%)
\(\displaystyle G = \frac{W \cdot 100}{P} = \frac{120 \cdot 100}{60} = \frac{12000}{60}\)\(\displaystyle = 200\)
Verify: \(\displaystyle 200 \cdot 0.60 = 120\) ✓
Case 3: Calculate the Percentage (P)
In this case, we know the basic value and the percentage value, and we want to determine what percentage the value represents.
Given: Basic value \(G\) and Percentage value \(W\)
Find: Percentage \(P\)
\(\displaystyle P = \frac{W \cdot 100}{G}\)
Example 4: Percentage Increase Needed
A stock falls from $1,000 to $850. By what percentage must it rise to reach the original value again?
Solution
\(\displaystyle G = 850\) (current stock price = base for increase)
\(\displaystyle W = 1000\) (target value = the increase needed)
\(\displaystyle P = \frac{W \cdot 100}{G} = \frac{1000 \cdot 100}{850} = \frac{100000}{850}\)\(\displaystyle \approx 117.65\)
Required increase = \(\displaystyle 117.65 - 100 = 17.65\%\)
Real-World Application: Gains and Losses
A common real-world example is calculating gains and losses with percentage changes. This demonstrates an important principle: equal percentage losses and gains do not cancel out.
The Stock Portfolio Example
Complete Scenario
Step 1: Initial Investment
Step 2: Stock Price Falls by 15%
New value: \(\displaystyle 1000 - 150 = 850\)
Step 3: Stock Price Rises by 15%
\(\displaystyle W_2 = \frac{850 \cdot 15}{100} = 127.50\)
New value: \(\displaystyle 850 + 127.50 = 977.50\)
Result:
Overall loss: \(\displaystyle 1000 - 977.50 = 22.50\)
A 15% loss followed by a 15% gain does NOT result in breaking even. The 15% gain is applied to a smaller base (after the loss), so you end up with less than you started with.
Summary: The Three Percentage Cases
| Case | Given | Find | Formula |
|---|---|---|---|
| Case 1 Find Percentage Value |
\(\displaystyle G, P\) | \(\displaystyle W\) | \(\displaystyle W = \frac{G \cdot P}{100}\) |
| Case 2 Find Basic Value |
\(\displaystyle W, P\) | \(\displaystyle G\) | \(\displaystyle G = \frac{W \cdot 100}{P}\) |
| Case 3 Find Percentage |
\(\displaystyle G, W\) | \(\displaystyle P\) | \(\displaystyle P = \frac{W \cdot 100}{G}\) |
Key Terminology and Distinctions
- "Reduced BY 40%": You pay 60% of the original price (original - 40% = 60%)
- "Reduced TO 40%": You pay 40% of the original price
- "Increased BY 20%": New value = original + 20% (120% total)
Example: Correct Terminology
Item costs $100:
- "Reduced by 30%" → Pay: \(\displaystyle 100 \times 0.70 = 70\)
- "Reduced to 30%" → Pay: \(\displaystyle 100 \times 0.30 = 30\)
- "Increased by 20%" → Pay: \(\displaystyle 100 \times 1.20 = 120\)
Percentage Points vs. Percentages
Percentage points and percentages are different concepts:
- Percentage: Expressed relative to a base value (e.g., "20% of 100 = 20")
- Percentage Points: Absolute difference (e.g., "from 30% to 50% is a change of 20 percentage points")
Example
If unemployment rises from 5% to 6%:
- The change is 1 percentage point
- But it's a 20% increase (from 5 to 6 is 20% of 5)
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