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Chapter 0: Problem 37

Write each fraction in lowest terms. $$ \frac{18}{90} $$

Short Answer

Expert verified
\( \frac{1}{5} \)

Step by step solution

01

- Find the Greatest Common Divisor (GCD)

First, find the greatest common divisor (GCD) of the numerator (18) and the denominator (90). List the factors of each number. Factors of 18 are 1, 2, 3, 6, 9, 18. Factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90. The greatest common factor is 18.
02

- Divide Both Numerator and Denominator by the GCD

Divide both the numerator (18) and the denominator (90) by the GCD (18). \( \frac{18 \text{ (numerator)}}{18} = 1 \) and \( \frac{90 \text{ (denominator)}}{18} = 5 \).
03

- Write the Fraction in Lowest Terms

Now, rewrite the fraction with the simplified numerator and denominator: \( \frac{1}{5} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Divisor
The Greatest Common Divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder.
To find the GCD, list all the factors of the numerator and the denominator.
  • For example, for 18, the factors are 1, 2, 3, 6, 9, and 18.
  • For 90, the factors are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.
In this case, the greatest factor that appears in both lists is 18, making it the GCD. Knowing how to find the GCD is crucial for simplifying fractions, as it allows you to determine the largest number by which both the numerator and denominator can be divided evenly. This process will lead you to the simplest form of the original fraction.
Factors
Factors are numbers that can be multiplied together to produce another number. They are essential in breaking down numbers into their basic components.
For instance:
  • The factors of 18 are 1, 2, 3, 6, 9, and 18.
  • The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.
Using these lists, we can determine the common factors to find the GCD.
Prime factors, a subset of factors, are particularly important. They are factors that are prime numbers, like 3 for both 18 and 90 in this example.
Being able to list factors and find common ones helps in many areas of math, especially when simplifying fractions or finding GCDs.
Lowest Terms
A fraction is in its lowest terms if the numerator and the denominator have no common divisors other than 1.
To reduce a fraction to its lowest terms:
  • First, find the GCD of the numerator and the denominator.
  • Then, divide both the numerator and the denominator by the GCD.
For example, with the fraction \(\frac{18}{90}\), the GCD is 18.
Divide both the numerator and the denominator by the GCD:
\(\frac{18 \div 18}{90 \div 18} = \frac{1}{5}\).
Therefore, \(\frac{1}{5}\) is the fraction in its lowest terms.
Understanding how to simplify fractions to their lowest terms ensures that fractions are easily comparable and calculations are more straightforward.

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