How do you write $0.5 \overline{3}$ as a fraction?
$\frac{7}{9}$
$\frac{7}{16}$
$\frac{3}{11}$
$\frac{8}{15}$
Answer (1)
Let x = 0.5 3 .
Multiply by 10 and 100 to get 10 x = 5. 3 and 100 x = 53. 3 .
Subtract 10 x from 100 x to eliminate the repeating part: 90 x = 48 .
Solve for x and simplify the fraction: x = 90 48 = 15 8 . The answer is 15 8 .
Explanation
Understanding the Problem We are asked to convert the repeating decimal 0.5\[\]\overline{3} into a fraction. This means we want to express 0.53333... as a fraction.
Setting up the Equation Let x = 0.53333... . Our goal is to find a fraction that is equal to x .
Multiplying by 10 Multiply x by 10: 10 x = 5.3333...
Multiplying by 100 Multiply x by 100: 100 x = 53.3333...
Subtracting the Equations Subtract 10 x from 100 x to eliminate the repeating decimal part: 100 x − 10 x = 53.3333... − 5.3333... This simplifies to: 90 x = 48
Solving for x Solve for x : x = 90 48
Simplifying the Fraction Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 48 and 90 is 6. Dividing both by 6, we get: x = 90 ÷ 6 48 ÷ 6 = 15 8
Final Answer Therefore, the repeating decimal 0.5 3 can be written as the fraction 15 8 .
Examples
Repeating decimals can be converted into fractions, which is useful in various real-life situations. For example, when dealing with precise measurements or financial calculations, it's often necessary to convert a repeating decimal into a fraction to ensure accuracy. Imagine you're calculating the total cost of an item with a price of 0.5 3 dollars per unit, and you need to buy 15 units. Converting 0.5 3 to 15 8 allows you to easily calculate the total cost as 15 × 15 8 = 8 dollars.
