Converting the decimal 1.33333 to a fraction involves understanding its composition and applying mathematical techniques to express it accurately.
Understanding 1.33333 as a Fraction
The decimal 1.33333 represents a value slightly greater than 1, with the digit ‘3’ repeating indefinitely. This repeating decimal can be expressed as a fraction through the following steps:
- Set Up the Equation: Let x=1.33333…x = 1.33333\ldotsx=1.33333… (where the digit ‘3’ repeats indefinitely).
- Multiply to Isolate the Repeating Part: Multiply both sides of the equation by 10 (since one digit repeats) to shift the decimal point: 10x=13.33333…10x = 13.33333\ldots10x=13.33333…
- Subtract the Original Equation: Subtract the original xxx from this new equation to eliminate the repeating decimal: 10x−x=13.33333…−1.33333…10x – x = 13.33333\ldots – 1.33333\ldots10x−x=13.33333…−1.33333… 9x=129x = 129x=12
- Solve for xxx: Divide both sides by 9: x=129x = \frac{12}{9}x=912
- Simplify the Fraction: Reduce 129\frac{12}{9}912 by dividing both the numerator and denominator by their greatest common divisor (3): 12÷39÷3=43\frac{12 \div 3}{9 \div 3} = \frac{4}{3}9÷312÷3=34
Therefore, 1.33333…1.33333\ldots1.33333… as a fraction is 43\frac{4}{3}34, which is an improper fraction. It can also be expressed as the mixed number 1131 \frac{1}{3}131.
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Alternative Method Using Fraction Conversion:
Another approach involves recognizing that 1.33333…1.33333\ldots1.33333… can be split into 1 plus 0.33333…0.33333\ldots0.33333…. The repeating decimal 0.33333…0.33333\ldots0.33333… is known to equal 13\frac{1}{3}31. Adding this to 1 gives: 1+13=33+13=431 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3}1+31=33+31=34
FAQ
- What is 1.33333 as a fraction?
The decimal 1.33333 (with the digit ‘3’ repeating) converts to the fraction 43\frac{4}{3}34. - How do you convert a repeating decimal to a fraction?
To convert a repeating decimal to a fraction, set the decimal equal to a variable, multiply to shift the decimal point past the repeating part, subtract to eliminate the repeating portion, and solve for the variable. - Is 43\frac{4}{3}34 in its simplest form?
Yes, 43\frac{4}{3}34 is in its simplest form. - Can 43\frac{4}{3}34 be expressed as a mixed number?
Yes, 43\frac{4}{3}34 can be expressed as the mixed number 1131 \frac{1}{3}131. - Why is it important to convert repeating decimals to fractions?
Converting repeating decimals to fractions provides an exact representation of the value, which is useful in mathematical calculations and analysis.
Understanding how to convert repeating decimals like 1.33333 to fractions enhances numerical literacy and aids in precise mathematical computations.
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