AI Percentage Solver

Tutorial on Percentages

1. Understand the Term ‘Percentage’

Definition:
– Percentage: A percentage is a way of expressing a number as a fraction of 100. It is denoted using the percent sign (%). For example, 25% means 25 out of 100, or 25/100.

Examples:
– 50% means 50 out of 100.
– 75% means 75 out of 100.

2. Convert Decimals to Percentages and Vice Versa

Convert Decimal to Percentage:
– Multiply the decimal by 100 and add the percent sign (%).

Example: Convert 0.45 to a percentage.
\[ 0.45 \times 100 = 45 \]
\[ \text{So, } 0.45 = 45\% \]

Convert Percentage to Decimal:
– Divide the percentage by 100 and remove the percent sign (%).

Example: Convert 85% to a decimal.
\[ \frac{85}{100} = 0.85 \]
\[ \text{So, } 85\% = 0.85 \]

3. Calculate the Percentage of a Quantity

Formula:
\[ \text{Percentage of a Quantity} = \left( \frac{\text{Percentage}}{100} \right) \times \text{Quantity} \]

Example: Calculate 20% of 150.
\[ 20\% \text{ of } 150 = \left( \frac{20}{100} \right) \times 150 \]
\[ = 0.2 \times 150 \]
\[ = 30 \]

4. Express One Quantity as a Percentage of Another Quantity

Formula:
\[ \text{Percentage} = \left( \frac{\text{Part}}{\text{Whole}} \right) \times 100 \]

Example: Express 25 as a percentage of 200.
\[ \text{Percentage} = \left( \frac{25}{200} \right) \times 100 \]
\[ = 0.125 \times 100 \]
\[ = 12.5\% \]

5. Calculate Percentage Error and Percentage Change

Percentage Error:
– Formula:
\[ \text{Percentage Error} = \left( \frac{|\text{Approximate Value} – \text{Exact Value}|}{\text{Exact Value}} \right) \times 100 \]

Example: If the approximate value is 47 and the exact value is 50, calculate the percentage error.
\[ \text{Percentage Error} = \left( \frac{|47 – 50|}{50} \right) \times 100 \]
\[ = \left( \frac{3}{50} \right) \times 100 \]
\[ = 6\% \]

Percentage Change:
– Formula:
\[ \text{Percentage Change} = \left( \frac{\text{New Value} – \text{Original Value}}{\text{Original Value}} \right) \times 100 \]

Example: If the original value is 60 and the new value is 75, calculate the percentage change.
\[ \text{Percentage Change} = \left( \frac{75 – 60}{60} \right) \times 100 \]
\[ = \left( \frac{15}{60} \right) \times 100 \]
\[ = 25\% \]

Key Concepts:

1. Understanding Percentage:
– Percentage is a way of expressing a number as a fraction of 100.

2. Conversion:
– To convert a decimal to a percentage, multiply by 100 and add %.
– To convert a percentage to a decimal, divide by 100 and remove %.

3. Calculations:
– Percentage of a quantity is found by multiplying the quantity by the percentage (expressed as a decimal).
– To express one quantity as a percentage of another, divide the part by the whole and multiply by 100.

4. Errors and Changes:
– Percentage error measures the accuracy of an approximate value compared to the exact value.
– Percentage change measures the change between an original value and a new value, expressed as a percentage of the original value.