Problem 58 - Algebra A Combined Function

Question

Perform each indicated operation. $$ \frac{9}{10}-\frac{3}{5} $$

Step-by-Step Solution

Answer

The result of the subtraction is \( \frac{3}{10} \).

Step 1: Identify the Operation and Find a Common Denominator

The problem requires us to subtract the fractions \( \frac{9}{10} \) and \( \frac{3}{5} \). To perform this subtraction, we need a common denominator. The denominators in this case are 10 and 5, and 10 is the least common multiple of these two denominators.

Step 2: Convert the Fractions to have a Common Denominator

The fraction \( \frac{9}{10} \) already has the common denominator 10. We need to convert \( \frac{3}{5} \) to a fraction with denominator 10. To do this, multiply both the numerator and the denominator by 2: \( \frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10} \).

Step 3: Perform the Subtraction

Subtract the fractions by taking the difference of their numerators, using the common denominator: \( \frac{9}{10} - \frac{6}{10} = \frac{9 - 6}{10} = \frac{3}{10} \).

Step 4: Simplify the Result if Necessary

The fraction \( \frac{3}{10} \) is already in its simplest form because 3 and 10 have no common factors other than 1.

Key Concepts

Common Denominator

When subtracting fractions, having a common denominator is crucial because it ensures that the fractions are in comparable terms. A common denominator is a shared multiple of the denominators of the fractions you need to operate on. This is similar to ensuring that you're comparing apples to apples rather than apples to oranges. To find a common denominator, here’s what you can do:

• Identify the denominators of each fraction. In our example, these are 10 and 5.
• Determine a number that both denominators can divide evenly into. This is the common denominator. In this case, 10 is such a number, as both 10 and 5 can divide into it without leaving a remainder.

By finding the common denominator, you can then proceed to convert both fractions into equivalent forms that can be directly subtracted.

Least Common Multiple (LCM)

To find a common denominator, one effective way is to use the least common multiple (LCM). The LCM is the smallest number that is a multiple of two or more numbers. It helps simplify the process of putting fractions over a common denominator. Here's how you find the LCM:

• List the multiples of each number. For 5, the multiples are 5, 10, 15, etc. For 10, the multiples are 10, 20, 30, etc.
• Identify the smallest multiple that appears in both lists. Here, it is 10.

Using the LCM of 5 and 10, which is 10, you can comfortably convert and subtract fractions knowing they are expressed with a standard denominator. Finding the LCM prevents unnecessary calculations and ensures a smooth subtraction process.

Fraction Simplification

After performing the subtraction, you often need to simplify your result to its lowest terms. Fraction simplification involves reducing the fraction to a form where the numerator and the denominator have no common factors other than 1. Here's the step-by-step:

• After subtracting, our result was \( \frac{3}{10} \).
• Check for common factors between the numerator (3) and the denominator (10). They only share the factor 1, indicating that \( \frac{3}{10} \) is already in its simplest form.

In cases where simplification is possible, divide the numerator and the denominator by their greatest common divisor (GCD). Simplification helps in understanding the result better and makes it easier to work with in future calculations.