Problem 38 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated subtraction. $$\frac{2}{5}-\frac{1}{10}$$

Step-by-Step Solution

Answer

\(\frac{2}{5} - \frac{1}{10} = \frac{3}{10}\)

Step 1: Finding the common denominator

First identify the least common multiple (LCM) of the two denominators (5 and 10). In this case, the LCM is 10, which will be our common denominator.

Step 2: Convert the fractions

Convert each fraction to an equivalent fraction with denominator 10. The first fraction \(\frac{2}{5}\) becomes \(\frac{4}{10}\) (because 2 * 2 = 4 and 5 * 2 = 10) and the second remains the same as \(\frac{1}{10}\), because 10 is already the denominator.

Step 3: Perform the subtraction

Subtract the numerators of the two fractions: 4 (numerator of the first fraction) - 1 (numerator of the second fraction) = 3. Write this answer over the common denominator. The result is \(\frac{3}{10}\).

Key Concepts

Common Denominator

When subtracting fractions, finding a common denominator is crucial. A common denominator is a shared multiple of the denominators of two or more fractions. It allows us to add or subtract fractions by ensuring that the parts we are comparing are the same size. Without a common denominator, it's like trying to combine apples and oranges — they are different units, and you can't combine them until you convert them into a common unit.

In our exercise, we have fractions with denominators of 5 and 10. To find a common denominator, we seek a number that both 5 and 10 can divide into without leaving a remainder. In this case, 10 is the least common multiple of 5 and 10, so it's chosen as the common denominator, allowing us to easily subtract the fractions.

Lowest Common Multiple

The lowest common multiple (LCM), often known as the least common denominator when we are dealing with fractions, is the smallest multiple that is exactly divisible by every number in a set. In the realm of fractions subtraction, the LCM helps us combine fractions by ensuring they have the same denominator.

To calculate the LCM of two numbers, you can list the multiples of each number and find the smallest multiple they share. For instance, the multiples of 5 are 5, 10, 15, 20, ... and the multiples of 10 are 10, 20, 30, ... Therefore, the LCM of 5 and 10 is 10. Knowing how to find the LCM allows us to combine fractions accurately by converting them to equivalent fractions with a common denominator.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same part of a whole. They may look different, but they hold the same value. To find equivalent fractions, you can either multiply or divide both the numerator (the top number) and the denominator (the bottom number) by the same number.

For example, in our exercise, to convert \(\frac{2}{5}\) into a fraction with a denominator of 10, we multiply the numerator and the denominator by 2, yielding \(\frac{2 \times 2}{5 \times 2} = \frac{4}{10}\). Consequently, \(\frac{2}{5}\) and \(\frac{4}{10}\) are equivalent fractions. Understanding equivalent fractions is essential when solving problems requiring you to add or subtract fractions with different denominators.