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Chapter 7: Problem 90

Perform the operations. Simplify, if possible. $$ \frac{21 y}{12}-\frac{7 y}{6} $$

Short Answer

Expert verified
The simplified form of the expression is \( \frac{7y}{12} \).

Step by step solution

01

Find a Common Denominator

To perform the subtraction of two fractions, it's easiest if they share a common denominator. The denominators here are 12 and 6. The least common denominator (LCD) of 12 and 6 is 12. We will convert the second fraction to have a denominator of 12.
02

Adjust the Second Fraction

Convert \( \frac{7y}{6} \) to have the denominator of 12. Multiply both the numerator and the denominator by 2: \[\frac{7y}{6} = \frac{7y \times 2}{6 \times 2} = \frac{14y}{12}\]
03

Subtract the Fractions

Now that both fractions share the same denominator, subtract the numerators while keeping the denominator constant:\[\frac{21y}{12} - \frac{14y}{12} = \frac{21y - 14y}{12} = \frac{7y}{12}\]
04

Simplification

The fraction \( \frac{7y}{12} \) is already in its simplest form because 7 and 12 have no common factors other than 1. Therefore, no further simplification is necessary.

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Key Concepts

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

Common Denominator
When subtracting fractions, it's essential that they have the same denominator, which is called a *common denominator*. This makes it possible to directly subtract the numerators while maintaining the denominator.
This can be a little confusing at first, but here's how you can think about it:
  • Just like you would convert feet to inches to measure lengths with the same units, you need to convert fractions so they measure in the same size 'pieces'.
  • Without a common denominator, subtracting fractions is like trying to subtract apples from oranges 鈥� it just doesn't work unless they're in the same format.
To get a common denominator, you can sometimes simply make them the same by adjusting one of the denominators. In other cases, both denominators might need adjusting to become the same. In our example, however, only one fraction needed changing.
Least Common Denominator (LCD)
The *least common denominator* (LCD) is the smallest number that can be used as a common denominator for two or more fractions. Finding the LCD is like finding the smallest common multiple of the denominators.
This minimizes the amount of work in rewriting the fractions. Here's how to find it:
  • List the multiples of each denominator.
  • Find the smallest multiple shared between them.
In the exercise, the denominators were 12 and 6. The multiples of 12 are 12, 24, 36, etc., and the multiples of 6 are 6, 12, 18, etc.
Notice that 12 is the first multiple they share, so it's the LCD. Using 12 avoids unnecessary complication when dealing with larger numbers.
Simplifying Fractions
After performing the operations on fractions, your result may not be in its simplest form. *Simplifying fractions* means to reduce them to their lowest terms.
This process makes them easier to understand and compare:
  • Divide the numerator and the denominator by their greatest common factor (GCF).
  • The greatest common factor is the largest number that divides both the numerator and the denominator without leaving a remainder.
In our worked example, the fraction \( \frac{7y}{12} \) is already simplified. Why? Because there are no numbers other than 1 that divide into both 7 and 12 evenly.
This means you can rest assured that this fraction is as "small" as it can be, making it easy to work with in further calculations.

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Most popular questions from this chapter

Change \(\frac{9}{10}\) to a percent.

Perform the operations. Simplify, if possible. $$ b-\frac{3}{a^{2}} $$

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