Changing Decimals and Fractions

Changing Decimals and Fractions

This lesson converts in both directions using division and decimal place value. Some fractions terminate as decimals; others repeat forever.

How to solve these problems

To convert a fraction, divide numerator by denominator. For a terminating decimal, write its digits over 10, 100, or 1,000 according to place value, then simplify. Use ≈ when rounding.

Worked Example 1: Convert 3/4 to a decimal

Convert 3/4 to a decimal

Solution: 3 ÷ 4 = 0.75.

Why it works: Multiplying top and bottom by 25 also gives 75/100.

Worked Example 2: Convert 7/8 to a decimal

Convert 7/8 to a decimal

Solution: 7 ÷ 8 = 0.875.

Why it works: The quotient terminates after three decimal places.

Worked Example 3: Convert 1/5 to a decimal

Convert 1/5 to a decimal

Solution: 1 ÷ 5 = 0.2.

Why it works: One fifth equals two tenths.

Worked Example 4: Convert 0.6 to a fraction

Convert 0.6 to a fraction

Solution: 0.6 = 6/10 = 3/5.

Why it works: The final digit is in the tenths place; simplify by dividing by 2.

Worked Example 5: Convert 0.45 to a fraction

Convert 0.45 to a fraction

Solution: 0.45 = 45/100 = 9/20.

Why it works: Divide numerator and denominator by their common factor 5.

Worked Example 6: Convert 0.125 to a fraction

Convert 0.125 to a fraction

Solution: 0.125 = 125/1,000 = 1/8.

Why it works: Three decimal places require a denominator of 1,000 before simplification.

Worked Example 7: Convert 1.25 to a mixed number

Convert 1.25 to a mixed number

Solution: 1.25 = 125/100 = 5/4 = 1¼.

Why it works: The whole-number part must be included in the numerator.

Worked Example 8: Convert 2/3 to a decimal

Convert 2/3 to a decimal

Solution: 2 ÷ 3 = 0.666…; to the nearest hundredth, 2/3 ≈ 0.67.

Why it works: The repeating decimal is exact; the rounded decimal is an approximation.

Worked Example 9: Convert 0.04 to a fraction

Convert 0.04 to a fraction

Solution: 0.04 = 4/100 = 1/25.

Why it works: The zero tenths place matters: this is four hundredths, not four tenths.

Worked Example 10: Convert repeating 0.333… to a fraction

Convert repeating 0.333… to a fraction

Solution: Let x = 0.333… . Then 10x = 3.333… . Subtract: 9x = 3, so x = 1/3.

Why it works: Subtracting aligned repeating tails eliminates them. A terminating 0.333 instead equals 333/1,000.

Try it again

Cover the solutions and solve the ten examples on your own. Explain which rule you used, then compare your method and answer with the worked solution. Check signs, equivalent forms, and units.