Mada za sehemu hiiUse numerical skills in different contextsMada 6
- Explain the basic concepts of Mathematics (meaning of mathematics, branches of mathematics, relationship between mathematics and other subjects, importance of mathematics)
- Explain the concept of rational, irrational and real numbers
- Convert repeating/recurring decimals into fractions and vice versa
- Represent rational numbers on a number line
- Explain the concept of inequalities and absolute values of real numbers
- Describe the importance of numbers in real-life situations
Converting Repeating Decimals to Fractions and Vice Versa
A repeating decimal (also called a recurring decimal) is a decimal number in which one or more digits repeat forever. For example, 0.333... has the digit 3 repeating, and 0.1666... has the digit 6 repeating. Every repeating decimal can be written as a fraction, which is why it is a rational number.
We use a bar (called a vinculum) over the digit or digits that repeat:
- 0.333... = 0.3̅ — the digit 3 repeats
- 0.1666... = 0.16̅ — the digit 6 repeats
- 0.142857142857... = 0.142857̅ — the block "142857" repeats
- 0.1234444... = 0.1234̅ — the digit 4 repeats (note: only the 4 repeats, not the 3)
We use a simple algebraic method:
Example 1: Convert 0.3̅ to a fraction
- Let x = 0.333...
- Multiply by 10 (since 1 digit repeats): 10x = 3.333...
- Subtract the original equation: 10x − x = 3.333... − 0.333...
- 9x = 3
- x = 3/9 = 1/3
Example 2: Convert 0.16̅ to a fraction
- Let x = 0.1666...
- Multiply by 100 (since 2 digits repeat): 100x = 16.666...
- Subtract: 100x − x = 16.666... − 0.1666...
- 99x = 16
- x = 16/99 — this fraction cannot be simplified further.
Example 3: Convert 0.1234̅ to a fraction
- Let x = 0.1234444...
- Multiply by 10,000 (4 digits after decimal, 1 repeats): 10000x = 1234.4444...
- Multiply by 10 (for the non-repeating part): 10x = 1.2344444...
- Subtract: 10000x − 10x = 1234.4444... − 1.2344444...
- 9990x = 1233
- x = 1233/9990
- Simplify: divide by 3 → 411/3330 → divide by 3 again → 137/1110
To convert a fraction to a decimal, divide the numerator by the denominator and observe which digits repeat.
Example 1: Convert 2/9 to a decimal
Divide 2 by 9: 2 ÷ 9 = 0.2222... The digit 2 repeats forever, so 2/9 = 0.2̅
Example 2: Convert 5/12 to a decimal
Divide 5 by 12: 5 ÷ 12 = 0.41666... Only the digit 6 repeats, so 5/12 = 0.416̅
Example 3: Convert 1/7 to a decimal
1 ÷ 7 = 0.142857142857... The block "142857" repeats, so 1/7 = 0.142857̅
- A repeating decimal is a rational number (it can be written as a fraction).
- The number of digits that repeat determines the power of 10 to multiply by.
- Always simplify the final fraction.
- To convert a fraction to a decimal, perform long division and watch for the repeating pattern.
When calculating interest on a savings account or a loan from a bank in Tanzania, the result is sometimes a repeating decimal. For example, if you divide 1000 TZS interest by 3 months, you get 333.333... TZS per month. Knowing how to convert this to the fraction 1/3 helps you understand that the total interest is exactly 1000 TZS, not an approximation. This skill is also useful when dividing money or goods equally among a group of people.
Swali
What fraction is equal to the repeating decimal ?
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