∑∫ax²+bx+c=0π√x
Grade: 6-7 | Topic: Arithmetic
What You Will Learn
After this guide, you will be able to convert any fraction to a decimal using division, and convert any terminating or repeating decimal back to a fraction. You will also learn to recognize which fractions produce terminating decimals and which produce repeating decimals, giving you flexibility to work with numbers in whichever form is most convenient.
Theory
Fractions and decimals are two forms of the same number
Every fraction is a division problem waiting to happen. The fraction bar means "divided by":
[ \frac{a}{b} = a \div b ]
So ( \frac{3}{4} = 3 \div 4 = 0.75 ). The fraction and the decimal are two different ways to write the same value. Being able to switch between them is essential for comparing numbers, solving equations, and working with real-world measurements.
Converting fractions to decimals
The division method: Divide the numerator by the denominator using long division (or a calculator). This always works.
[ \frac{7}{8} = 7 \div 8 = 0.875 ]
The equivalent-fraction method (when the denominator is a factor of 10, 100, or 1000): If you can rewrite the fraction with a denominator of 10, 100, or 1000, the decimal is immediate:
[ \frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10} = 0.6 ]
[ \frac{7}{25} = \frac{7 \times 4}{25 \times 4} = \frac{28}{100} = 0.28 ]
This shortcut is fast but only works when the denominator has no prime factors other than 2 and 5.
Terminating vs. repeating decimals
Terminating decimals end after a finite number of digits: [ \frac{1}{4} = 0.25, \quad \frac{3}{8} = 0.375, \quad \frac{9}{20} = 0.45 ] A fraction (in lowest terms) produces a terminating decimal when its denominator has only the prime factors 2 and/or 5.
Repeating decimals have one or more digits that cycle infinitely: [ \frac{1}{3} = 0.\overline{3} = 0.333… ] [ \frac{5}{6} = 0.8\overline{3} ] [ \frac{2}{7} = 0.\overline{285714} ] A fraction produces a repeating decimal when its denominator (in lowest terms) has a prime factor other than 2 or 5.
Converting terminating decimals to fractions
Step 1: Write the decimal as a fraction over the appropriate power of 10.
- 1 decimal place: denominator is 10
- 2 decimal places: denominator is 100
- 3 decimal places: denominator is 1000
For example: 0.35 = ( \frac{35}{100} )
Step 2: Simplify by dividing numerator and denominator by their GCD. [ \frac{35}{100} = \frac{35 \div 5}{100 \div 5} = \frac{7}{20} ]
Converting repeating decimals to fractions
For a single repeating digit like 0.( \overline{3} ): Let ( x=0.\overline{3} ) [ 10x=3.\overline{3} ] Subtract: [ 10x−x=3.\overline{3}−0.\overline{3} ] [ 9x=3 ] [ x=\frac{1}{3} ]
Common fraction-decimal equivalents worth memorizing
| Fraction | Decimal | Fraction | Decimal |
|---|---|---|---|
| ( \frac{1}{2} ) | 0.5 | ( \frac{1}{3} ) | 0.\overline{3} |
| ( \frac{1}{4} ) | 0.25 | ( \frac{2}{3} ) | 0.\overline{6} |
| ( \frac{3}{4} ) | 0.75 | ( \frac{1}{5} ) | 0.2 |
| ( \frac{1}{8} ) | 0.125 | ( \frac{1}{6} ) | 0.16\overline{6} |
| ( \frac{3}{8} ) | 0.375 | ( \frac{5}{6} ) | 0.8\overline{3} |
Worked Examples
Example 1: Fraction to decimal using division (easy)
Problem: Convert ( \frac{3}{8} ) to a decimal. Step 1: Divide 3 by 8 using long division. [ 3 \div 8 = 0.375 ] Answer: 0.375
Example 2: Fraction to decimal using equivalent fractions (easy)
Problem: Convert ( \frac{9}{25} ) to a decimal. Step 1: Rewrite with a denominator of 100: [ \frac{9}{25} = \frac{9 \times 4}{25 \times 4} = \frac{36}{100} ] Step 2: Write as a decimal. [ \frac{36}{100} = 0.36 ] Answer: 0.36
Example 3: Terminating decimal to fraction (medium)
Problem: Convert 0.625 to a fraction in lowest terms. Step 1: Write over 1000 (three decimal places). [ 0.625 = \frac{625}{1000} ] Step 2: Find the GCD of 625 and 1000. [ \text{gcd} = 125 ] Step 3: Simplify: [ \frac{625 \div 125}{1000 \div 125} = \frac{5}{8} ] Answer: ( \frac{5}{8} )
Example 4: Repeating decimal to fraction (medium)
Problem: Convert 0.45( \overline{45} ) to a fraction. Step 1: Let ( x=0.45\overline{45} ). Step 2: Since two digits repeat, multiply by 100. [ 100x=45.45\overline{45} ] Step 3: Subtract the original equation. [ 100x−x=45.45−0.45 ] Step 4: Solve and simplify. [ x=\frac{5}{11} ] Answer: ( \frac{5}{11} )
Example 5: Mixed number to decimal and back (challenging)
Problem: Convert ( 2\frac{5}{12} ) to a decimal, then confirm by converting back. Step 1: Convert the fraction part to a decimal. [ \frac{5}{12} = 0.41\overline{6} ] Step 2: Add the whole number. [ 2 + 0.41\overline{6} = 2.41\overline{6} ] Answer: 2.41\overline{6}
Common Mistakes
Mistake 1: Dividing the denominator by the numerator instead of the other way.
Mistake 2: Forgetting to simplify after converting a decimal to a fraction.
Mistake 3: Using the wrong power of 10 for the denominator.
Practice Problems
Try these on your own before checking the answers:
- Convert ( \frac{5}{16} ) to a decimal.
- Convert 0.875 to a fraction in lowest terms.
- Convert ( \frac{4}{11} ) to a decimal.
- Convert 0.72( \overline{72} ) to a fraction in lowest terms.
- Which fractions produce terminating decimals: ( \frac{3}{15} ), ( \frac{7}{12} ), ( \frac{9}{40} )?
Summary
- To convert a fraction to a decimal, divide the numerator by the denominator.
- A fraction in lowest terms produces a terminating decimal when the denominator has only the prime factors 2 and 5, and a repeating decimal otherwise.
- To convert a terminating decimal to a fraction, write it over the appropriate power of 10 and simplify.
- To convert a repeating decimal to a fraction, use the algebraic method: multiply by 10^n, subtract, and solve.
- Memorize common fraction-decimal equivalents to speed up your work.