Fractions and Decimal II

Grade 12 Math - Fractions: Converting Between Fractions and Decimals

Lesson Objectives

Convert fractions (proper, improper, mixed) to decimals using division.
Convert terminating and recurring decimals to fractions.
Simplify and rationalize complex fractional expressions involving decimals.
Apply conversions in real-world contexts and exam-standard problems.

Lesson Introduction

Fractions and decimals are two different ways of expressing the same idea—parts of a whole. Converting between them is crucial for handling measurements, finance, and problem-solving. In this lesson, you’ll explore efficient techniques to move between fractions and decimals and deal with recurring decimals more confidently.

Core Lesson Content

Worked Example

Converting Fractions to Decimals

This is done by dividing the numerator by the denominator.

Example: $\frac{3}{4} = 3 \div 4 = 0.75$

If the division ends after a few digits, it is a terminating decimal. If it continues in a repeating pattern, it is a recurring decimal.

Example 1: Convert

\frac{7}{8}

to a decimal.

7 \div 8 = 0.875

(terminating)

Example 2: Convert

\frac{1}{3}

to a decimal.

1 \div 3 = 0.\overline{3}

(recurring)

Converting Terminating Decimals to Fractions

Write the decimal as a fraction and simplify.

Example: $0.75 = \frac{75}{100} = \frac{3}{4}$

Example 3: Convert

0.125

to a fraction.

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

Converting Recurring Decimals to Fractions

Let $x$ equal the repeating decimal, then multiply both sides to eliminate the recurring part, subtract and solve for $x$ .

Example 4: Convert

0.\overline{6}

to a fraction.
Let

x = 0.\overline{6}

10x = 6.\overline{6}

Subtract:

10x - x = 6.6... - 0.6... = 6

9x = 6 \Rightarrow x = \frac{2}{3}

Example 5: Convert

0.\overline{123}

to a fraction.
Let

x = 0.\overline{123}

1000x = 123.\overline{123}

Subtract:

1000x - x = 999x = 123

x = \frac{123}{999} = \frac{41}{333}

Mixed Repeating Decimals

Example: $0.16\overline{3}$ has a non-repeating part (16) and a repeating part (3).

Example 6: Convert

0.16\overline{3}

to a fraction.
Let

x = 0.163333...

100x = 16.3333...

1000x = 163.3333...

Subtract:

1000x - 100x = 163.333... - 16.333... = 147

900x = 147 \Rightarrow x = \frac{147}{900} = \frac{49}{300}

Exercises

Convert $\frac{5}{6}$ to a decimal.
[NECO] Convert $0.625$ to a fraction in its simplest form. (Past Question)
Convert $0.\overline{27}$ to a fraction.
[WAEC] Convert $0.5\overline{83}$ to a fraction. (Past Question)
Express $\frac{13}{99}$ as a decimal.
[JAMB] Convert $0.\overline{142857}$ to a fraction. (Past Question)
Simplify and express $3 + 0.\overline{45}$ as a fraction.
Convert $1.2\overline{1}$ to a fraction.
[WAEC] Given $x = 0.0\overline{6}$ , express $5x$ as a simplified fraction. (Past Question)
Convert $2.\overline{9}$ to a fraction and verify your result.

Conclusion

Converting between fractions and decimals is essential for simplifying and comparing values, solving equations, and interpreting results accurately. Mastery of recurring decimals gives you an advantage in exams and real-life applications involving precision.