Mixed Operations with Fractions and Decimals

When a math problem includes both a fraction and a decimal, you cannot add, subtract, multiply, or divide them directly. The golden rule for mixed operations is simple: Convert both numbers to the same form.

You can either change the decimal into a fraction, or change the fraction into a decimal. Let's look at both strategies.

Strategy 1: Convert the Decimal to a Fraction

This method works for any problem, and it is the best choice when the fraction would turn into a repeating decimal (like $\frac{1}{3}$ or $\frac{5}{9}$).

Example: Compute $0.75 - \frac{1}{3}$

1. Convert: Change $0.75$ into a fraction. $0.75 = \frac{75}{100} = \frac{3}{4}$
2. Find a Common Denominator: The denominators are $4$ and $3$. The least common multiple is $12$. $\frac{3}{4} = \frac{9}{12}$ $\frac{1}{3} = \frac{4}{12}$
3. Subtract: $\frac{9}{12} - \frac{4}{12} = \frac{5}{12}$

Strategy 2: Convert the Fraction to a Decimal

If the fraction can be easily turned into a terminating decimal (like $\frac{1}{2} = 0.5$ or $\frac{1}{4} = 0.25$), converting to decimals is often much faster!

Example: Calculate $\frac{1}{4} + 0.3$

1. Convert: Change $\frac{1}{4}$ to a decimal. $\frac{1}{4} = 0.25$
2. Add: Line up the decimal points and add. $0.25 + 0.30 = 0.55$

Comparing Mixed Numbers

You can use these same conversion strategies to compare expressions.

Example: Which is greater: $\frac{2}{3} + 0.5$ or $1.5$?

Since $\frac{2}{3}$ is a repeating decimal ($0.666...$), let's convert everything to fractions.

1. Convert $0.5$ to a fraction: $0.5 = \frac{1}{2}$
2. Add the fractions: Find a common denominator ($6$). $\frac{2}{3} + \frac{1}{2} = \frac{4}{6} + \frac{3}{6} = \frac{7}{6}$
3. Convert $1.5$ to a fraction: $1.5 = 1 \frac{1}{2} = \frac{3}{2} = \frac{9}{6}$
4. Compare the results: $\frac{7}{6} < \frac{9}{6}$

Therefore, $1.5$ is greater.