Question

Change each of the following decimals to fractions in lowest terms: (a) \(0.25\) (b) \(0.625\) (c) \(1.67\) (d) \(0.8888\)

Short answer

(a) \(\frac{1}{4}\) (b) \(\frac{5}{8}\) (c) \(\frac{167}{100}\) (d) \(\frac{8}{9}\)

Step-by-step solution

Convert Decimal to Fraction (0.25)

To convert 0.25 to a fraction, recognize it as \(\frac{25}{100}\). Simplify this fraction by dividing both the numerator and the denominator by the greatest common divisor (GCD), which is 25. \(\frac{25 \div 25}{100 \div 25} = \frac{1}{4}\)

Convert Decimal to Fraction (0.625)

To convert 0.625 to a fraction, recognize it as \(\frac{625}{1000}\). Simplify by identifying the GCD of 625 and 1000, which is 125. \(\frac{625 \div 125}{1000 \div 125} = \frac{5}{8}\)

Convert Decimal to Fraction (1.67)

To convert 1.67 to a fraction, first recognize it as \(\frac{167}{100}\) (since 1.67 = 1 + 0.67 and 0.67 = \(\frac{67}{100}\)). Simplify by identifying the GCD, which is 1, so the fraction remains \(\frac{167}{100}\). Hence, it is already in the lowest terms.

Convert Repeating Decimal to Fraction (0.8888...)

To convert 0.8888... (a repeating decimal) to a fraction, let \(x = 0.8888...\). Then, \(10x = 8.8888...\). Subtract the first equation from the second: \(10x - x = 8.8888... - 0.8888...\), which gives \(9x = 8\). Solving for x, \(x = \frac{8}{9}\).

Key concepts

GCD (Greatest Common Divisor)

The Greatest Common Divisor (GCD) is key when simplifying fractions. The GCD of two numbers is the largest number that divides both without leaving a remainder.
To find the GCD of a number, list all the divisors of each number and select the largest one they share. For example, with the fraction \(\frac{625}{1000}\), list the divisors:
- Divisors of 625: 1, 5, 25, 125, 625
- Divisors of 1000: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000
The common divisors are 1, 5, 25, and 125, with the largest being 125. So the GCD is 125.
Simplifying fractions using GCD makes calculations easier and results cleaner.

Repeating decimals

Converting repeating decimals to fractions involves a simple trick. Let's take the example of \(0.8888...\). Start by setting x equal to the repeating decimal: \(x = 0.8888...\).
Then, multiply both sides by 10 (since one digit repeats) to get \(10x = 8.8888...\).
Now, subtract the original equation from this new one.

• \(10x - x = 8.8888... - 0.8888...\),
• which simplifies to \9x = 8\.

Divide both sides by 9 to isolate x:

• \x = \frac{8}{9}\.

Thus, \(0.8888... = \frac{8}{9}\). Repeating decimals can always be converted by following these steps.

Simplifying fractions

Simplifying fractions is crucial for making numbers more manageable. The process involves dividing the numerator and the denominator by their GCD.
Consider the fraction \( \frac{25}{100} \). The GCD of 25 and 100 is 25. Simplify by dividing both terms by this GCD:

• \ \frac{25 \div 25}{100 \div 25} = \frac{1}{4} \.

A more significant number like \( \frac{625}{1000} \) simplifies similarly. Identify the GCD (which is 125) and divide each part:

• \ \frac{625 \div 125}{1000 \div 125} = \frac{5}{8} \.

Simplifying fractions this way helps keep calculations clean and numbers easier to understand and compare.