Problem 39 Add or subtract as indicated. ... [FREE SOLUTION]

Chapter 1: Problem 39

Add or subtract as indicated. $$\frac{4}{9}-\left(\frac{2}{3}+\frac{5}{6}\right)$$

Short Answer

Expert verified

The short answer for the given expression $ \frac{4}{9} - \left( \frac{2}{3} + \frac{5}{6} \right) $ is $\frac{-19}{18}$.

Step by step solution

Add the fractions inside the parentheses

In order to add the fractions $\frac{2}{3}$ and $\frac{5}{6}$, we must first find a common denominator. In this case, the least common multiple of 3 and 6 is 6. So we can write each fraction with 6 as their denominator using equivalent fractions: \[\frac{2}{3} \cdot \frac{2}{2} = \frac{4}{6}\] Now, we can add the two fractions: \[\frac{4}{6} + \frac{5}{6} = \frac{4+5}{6} = \frac{9}{6}\]

Subtract the result from the first fraction

Now, we can subtract the result of the previous step ($\frac{9}{6}$) from $\frac{4}{9}$. First, we also need to find a common denominator for these fractions, 9 and 6. In this case, the least common multiple is 18: \[\frac{4}{9} \cdot \frac{2}{2} = \frac{8}{18}\] \[\frac{9}{6} \cdot \frac{3}{3} = \frac{27}{18}\] Now, we can subtract the two fractions: \[\frac{8}{18} - \frac{27}{18} = \frac{8 - 27}{18} = \frac{-19}{18}\] So, the final result is: \[\frac{4}{9} - \left(\frac{2}{3} + \frac{5}{6}\right) = \frac{-19}{18}\]

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Key Concepts

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

Adding Fractions

Adding fractions might seem tricky at first, but once you know the steps, it becomes much simpler. The first step is to make sure the fractions have the same denominator. This means the bottom numbers of the fractions need to be the same.
Once you have a common denominator, you can add the top numbers of the fractions, known as the numerators. For instance, in the expression $ \frac{2}{3} + \frac{5}{6} $, the common denominator is 6.

Convert $ \frac{2}{3} $ into $ \frac{4}{6} $ to match the denominators.
Add the numerators: $ 4 + 5 $ to get $ 9 $.

You keep the denominator the same, so the result is $ \frac{9}{6} $. This fraction can be simplified if needed, but it's essential to perform addition correctly first.

Subtracting Fractions

When it comes to subtracting fractions, the process is very similar to addition. Just like adding, you need a common denominator. In the problem $ \frac{4}{9} - \frac{9}{6} $, both fractions need the same bottom number again.
After finding the common denominator, you subtract the numerators while keeping the denominator unchanged.

Change $ \frac{4}{9} $ to $ \frac{8}{18} $.
Change $ \frac{9}{6} $ to $ \frac{27}{18} $.
Subtract $ 8 - 27 $ to get $ -19 $.

The result is $ \frac{-19}{18} $. Always check if you can simplify the fraction after subtraction, but sometimes the result is fully simplified.

Common Denominator

The idea of a common denominator is crucial in fraction operations. A common denominator is a shared bottom number between different fractions, allowing you to easily add or subtract them. You find this by determining the least common multiple (LCM) of the denominators.
For example, in the problem $ \frac{2}{3} + \frac{5}{6} $, the LCM of 3 and 6 is 6. This means both fractions can be adjusted to have 6 as their denominator.

Multiply the numerator and denominator of $ \frac{2}{3} $ by 2 to get $ \frac{4}{6} $.
$ \frac{5}{6} $ already has a 6 as a denominator, so it stays the same.

By aligning denominators, the fractions become compatible for direct addition or subtraction.

Equivalent Fractions

Equivalent fractions are different-looking fractions that actually represent the same value. They're key to working with fractions that have different denominators. To create an equivalent fraction, multiply or divide the top and bottom numbers by the same amount.
In our earlier example:

$ \frac{2}{3} $ was converted to $ \frac{4}{6} $ by multiplying both the numerator and the denominator by 2.
This doesn't change the value, just the way it looks.

Understanding equivalent fractions helps ensure you're working with fractions that can be added or subtracted after obtaining a common denominator. This step is essential for making fractions compatible in your calculations.

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91影视

Add or subtract as indicated. $$\frac{4}{9}-\left(\frac{2}{3}+\frac{5}{6}\right)$$

Short Answer

Step by step solution

Add the fractions inside the parentheses

Subtract the result from the first fraction

Key Concepts

Adding Fractions

Subtracting Fractions

Common Denominator

Equivalent Fractions

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