Equivalence and conversion between fractions, decimals, and percentages are essential skills in mathematics, enabling students to navigate various problems with ease. This section focuses on practical techniques and examples to master these conversions.

Fractions to Decimals

To convert a fraction to a decimal:

• Divide the numerator by the denominator.

Example:

• Convert $\frac{3}{4}$ to a decimal.
• Answer: $\frac{3}{4} = 0.75$

Decimals to Fractions

Converting a decimal to a fraction involves:

1. Writing the decimal over 1.

2. Multiplying numerator and denominator to remove the decimal point.

3. Simplifying the resulting fraction.

Example:

• Convert 0.75 to a fraction.
• Answer: $0.75 = \frac{75}{100} = \frac{3}{4}$

More Examples:

Image courtesy of Online Math Learning

Simplifying Fractions

Simplification requires:

1. Finding the Greatest Common Divisor (GCD) of numerator and denominator.
2. Dividing both by the GCD.

Example:

Simplify $\frac{60}{84}$.

Solution:

• GCD of 60 and 84 is 12.
• Answer: $\frac{60}{84} = \frac{5}{7}$

Fractions to Percentages

Convert a fraction to a percentage by:

1. Dividing the numerator by the denominator.
2. Multiplying the result by 100.

Example:

• Convert $\frac{3}{5}$ to a percentage.
• Answer: $\frac{3}{5} = 0.6; \, 0.6 \times 100 = 60\%$

More Examples:

Image courtesy of Online Math Learning

Percentages to Fractions

To convert a percentage to a fraction:

• Write the percentage as a fraction of 100.
• Simplify if possible.

Example:

• Convert 28% to a fraction.
• Answer: $28\% = \frac{28}{100} = \frac{7}{25}$

Image courtesy of Kate Math Lessons

Decimals to Percentages

Multiply the decimal by 100.

Example:

• Convert 0.84 to a percentage.
• Answer: $0.84 \times 100 = 84\%$

Percentages to Decimals

Divide the percentage by 100.

Example:

• Convert 28% to a decimal.
• Answer: $28\% = 0.28$

Mixed Numbers to Improper Fractions

Convert by:

1. Multiplying the whole number part by the denominator.

2. Adding the result to the numerator.

3. Placing the result over the original denominator.

Example:

• Convert $2\frac{3}{4}$ to an improper fraction.
• Answer: $2 \times 4 + 3 = 11; \, \frac{11}{4}$

Improper Fractions to Mixed Numbers

Convert by:

1. Dividing the numerator by the denominator.

2. The quotient is the whole number part, and the remainder is the numerator of the fraction part.

Example:

• Convert $\frac{7}{3}$ to a mixed number.
• Answer: $\frac{7}{3} = 2\frac{1}{3}$

Practice Questions

1. Convert $\frac{5}{8}$ to a decimal and then to a percentage.

Solution:

$\frac{5}{8} = 0.625; \, 0.625 \times 100 = 62.5\%$

2. Simplify the fraction $\frac{150}{180}$ and convert it to a percentage.

Solution:

$\frac{150}{180} = \frac{5}{6}; \, \frac{5}{6} \approx 0.8333; \, 0.8333 \times 100 = 83.33\%$

3. Convert 40% to a fraction in its simplest form and then to a decimal.

Solution:

$40\% = \frac{40}{100} = \frac{2}{5}; \, \frac{2}{5} = 0.4$

4. Change 0.125 into a fraction and simplify it.

Solution:

$0.125 = \frac{125}{1000} = \frac{1}{8}$

5. Convert $3\frac{1}{2}$ into an improper fraction and then to a decimal.

Solution:

$3 \frac{1}{2} = \frac{7}{2}; \, \frac{7}{2} = 3.5$