How to solve percentages?

Understand the Problem

The question is asking for guidance on how to solve percentage-related problems. This involves understanding the concept of percentages and knowing how to calculate them in various contexts, such as finding a percentage of a number, calculating percentage increase or decrease, or determining the original value before a percentage change.

Answer

To solve percentage problems: understand percentages, convert them to decimals or fractions, and use the correct formulas for finding a percentage of a number, calculating percentage increase/decrease: $\frac{\text{New} - \text{Original}}{\text{Original}} \times 100$, or finding the original value after a percentage change: $\frac{\text{Final}}{1 \pm \text{Percentage Change}}$.

Steps to Solve

Understanding Percentages

A percentage is a way of expressing a number as a fraction of 100. The word "percent" means "per hundred." So, 50% is equivalent to $\frac{50}{100}$ or 0.5.

Finding a Percentage of a Number

To find a percentage of a number, convert the percentage to a decimal or fraction and then multiply it by the number.

For example, to find 25% of 80:

Convert 25% to a decimal: $25% = \frac{25}{100} = 0.25$

Multiply: $0.25 \times 80 = 20$

Calculating Percentage Increase

To calculate the percentage increase, use the formula:

$$ \text{Percentage Increase} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100 $$

For example, if a price increases from $20 to $25:

$$ \text{Percentage Increase} = \frac{25 - 20}{20} \times 100 = \frac{5}{20} \times 100 = 25% $$

Calculating Percentage Decrease

To calculate the percentage decrease, use the formula:

$$ \text{Percentage Decrease} = \frac{\text{Original Value} - \text{New Value}}{\text{Original Value}} \times 100 $$

For example, if a price decreases from $20 to $15:

$$ \text{Percentage Decrease} = \frac{20 - 15}{20} \times 100 = \frac{5}{20} \times 100 = 25% $$

Finding the Original Value Before a Percentage Change

If you know the final value after a percentage change and want to find the original value, you can rearrange the percentage increase/decrease formulas.

After a Percentage Increase:

If $ \text{Final Value} = \text{Original Value} + (\text{Percentage Increase} \times \text{Original Value}) $, then $ \text{Final Value} = \text{Original Value} \times (1 + \text{Percentage Increase})$.

So, $ \text{Original Value} = \frac{\text{Final Value}}{1 + \text{Percentage Increase}}$.

For example: A price increased by 10% to become $110. What was the original price? $ \text{Original Value} = \frac{110}{1 + 0.10} = \frac{110}{1.10} = $100 $
After a Percentage Decrease:

If $ \text{Final Value} = \text{Original Value} - (\text{Percentage Decrease} \times \text{Original Value}) $, then $ \text{Final Value} = \text{Original Value} \times (1 - \text{Percentage Decrease})$.

So, $ \text{Original Value} = \frac{\text{Final Value}}{1 - \text{Percentage Decrease}}$.

For example: A price decreased by 20% to become $80. What was the original price? $ \text{Original Value} = \frac{80}{1 - 0.20} = \frac{80}{0.80} = $100 $

The steps to solve percentage problems involve understanding what percentages represent, converting them to decimals or fractions when necessary, and applying the correct formulas for percentage increase, decrease, or finding original values.

More Information

Percentages are a ubiquitous tool used in many aspects of daily life from finance, to statistics, to everyday discounts while shopping!

Tips

A common mistake is confusing percentage increase and decrease calculations, especially when finding the original value before the change. Always ensure you are dividing by $(1 + \text{Percentage Increase})$ or $(1 - \text{Percentage Decrease})$ as appropriate, and not simply subtracting the percentage from the final value. Also, watch out for misinterpreting the base value on which the percentage is calculated.

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