Using Percentages

Percentage Increases and Decreases

Understanding how to increase or decrease a value by a percentage is essential in algebra and real-world applications such as finance, sales, and data analysis. The process involves either adding or subtracting a percentage of the original value to itself.

• Percentage Increase: Add a specified percentage of the original value to itself. For example, increasing a value by 10% means adding 10% of the original value to the original value.

• Percentage Decrease: Subtract a specified percentage of the original value from itself. For example, decreasing a value by 20% means subtracting 20% of the original value from the original value.

• Multiplier: A decimal value representing the total percentage change. For an increase of p%, the multiplier is ; for a decrease, it is .

Steps for Calculating Percentage Change:

1. Find the multiplier.

2. Multiply the original value by the multiplier to get the new value.

3. Round the final answer as required.

• Key Points:

  • Increase multiplier:

  • Decrease multiplier:

  • A multiplier less than 1 always represents a decrease.

  • For multiple changes, multiply the multipliers together.

  • Only round the final answer, not intermediate steps.

Examples

• Example 1: Increase £340 by 15% Multiplier: New value = £391

• Example 2: Decrease £520 by 8% Multiplier: New value = £478.40

• Example 3: Two-step change: increase 10%, then decrease 5% New value = £209

• Example 4: Additive method, increase 72 by 25% New value = 90

Repeated Percentage Change

When a value undergoes several consecutive percentage changes, multiply the corresponding multipliers together.

• Formula:

Example: An apartment bought for decreases by 4% in year 1, 6.5% in year 2, and increases by x% in year 3 to .

• Multipliers: , ,

• Equation:

• Solve for : Third-year increase = 3.75%

Reverse Percentage Problems

To find the original value before a percentage change, use the multiplier method in reverse.

• Step 1: Determine the multiplier for the percentage change.

• Step 2: Set up the equation:

• Step 3: Rearrange:

Example: Sale price of a watch is €198 after a 28% reduction. Find the normal price.

• Multiplier:

• Equation:

• Normal price = €275

Finding the Original Value After a Percentage Increase

When given the increased value and the percentage increase, find the original value using the multiplier method or algebra.

• Example: Ava drove 943 km in April, which is 15% more than in March. Find March's distance.

• Multiplier:

• March distance: km

• March distance = 820 km

Percentage Change Formula

Percentage change quantifies how much a value has increased or decreased compared to its original value.

• Percentage change:

• Percentage increase:

• Percentage decrease:

• Always divide by the original value.

Example: Otis sells 75 ice creams on Friday and 87 on Saturday. Find the percentage increase.

• Percentage increase:

• Multiplier method: (which is 116% of the original; 16% increase)

Percentage Profit and Loss

Profit and loss percentages are applications of percentage change, commonly used in business and commerce.

• Profit: Selling price − Cost price

• Percentage profit:

• Loss: Cost price − Selling price

• Percentage loss:

Example: Anna makes 320 cups at 6 francs each, sells 80 boxes (4 cups per box) for 2160 francs. Find percentage profit.

• Cost: francs

• Profit: francs

• Percentage profit:

Complex Profit Example: Multiple Costs

When multiple costs are involved, sum all costs before calculating profit and percentage profit.

• Example: Joshua covers a 45 m2 floor. Each 1.5 m2 requires one box of tiles (£64/box), plus 5 bags of adhesive (£12/bag). He charges £3000. Find percentage profit.

• Number of boxes:

• Cost of tiles:

• Cost of adhesive:

• Total cost:

• Profit:

• Percentage profit:

Profit from Ratio and Percentage

When items are sold in a ratio and each has a different profit percentage, calculate the profit for each type and sum them.

• Example: Norberto sells 200 loaves (white:brown = 3:2). White: £1.50 each, 40% profit; Brown: £1.75 each, 60% profit.

• Ratio share:

• White loaves: ; Brown loaves:

• Income: (white), (brown)

• Profit: (white), (brown)

• Total profit:

Summary Table: Key Percentage Formulas

Additional info: These concepts are foundational for Intermediate Algebra and are widely applicable in finance, business, and everyday problem-solving. Mastery of percentage calculations is essential for success in algebraic word problems and quantitative reasoning.