Problem 19 Find each sum or difference. $... [FREE SOLUTION]

Chapter 0: Problem 19

Find each sum or difference. $$ -\frac{7}{3}+\frac{3}{4} $$

Short Answer

Expert verified

-\frac{19}{12}

Step by step solution

Find a common denominator

To add or subtract fractions, they must have a common denominator. The denominators of $-\frac{7}{3}$ and $\frac{3}{4}$ are 3 and 4. The least common denominator (LCD) for 3 and 4 is 12.

Convert fractions to have the same denominator

Rewrite each fraction with the common denominator of 12. Multiply the numerator and denominator of $-\frac{7}{3}$ by 4: $-\frac{7}{3} \times \frac{4}{4} = -\frac{28}{12}$. Next, multiply the numerator and denominator of $+\frac{3}{4}$ by 3: $+\frac{3}{4} \times \frac{3}{3} = +\frac{9}{12}$.

Add the fractions

Now that both fractions have the same denominator, add their numerators: $-\frac{28}{12} + \frac{9}{12} = \frac{-28 + 9}{12} = \frac{-19}{12}$.

Simplify the result (if needed)

Check if $-\frac{19}{12}$ can be simplified. Since 19 and 12 have no common factors other than 1, $-\frac{19}{12}$ is already in its simplest form.

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Key Concepts

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

Common Denominator

When adding or subtracting fractions, the first step is to locate a **common denominator**. A common denominator is a shared multiple of the denominators of both fractions. In math, a denominator is the bottom number of a fraction that tells us into how many parts the whole is divided. For example, if we need to add $-\frac{7}{3}$ and $\frac{3}{4}$, we need to find a common denominator for 3 and 4 to make the addition possible.

Least Common Denominator

While any common denominator will work, we aim for the **least common denominator (LCD)** to keep our calculations simple. The least common denominator is the smallest number that both denominators can divide into evenly. In our exercise, the denominators are 3 and 4. To find the LCD, list the multiples of each:

Multiples of 3: 3, 6, 9, **12**, ...
Multiples of 4: 4, 8, **12**, ...

The least common multiple is 12. Hence, 12 is the least common denominator for fractions with denominators 3 and 4. Now, both fractions $-\frac{7}{3}$ and $\frac{3}{4}$ will be converted to have 12 as their denominator.

Simplifying Fractions

Now let's discuss why simplifying fractions is essential. After performing operations like addition or subtraction, the result might have a numerator and denominator that could be simplified. Simplifying a fraction means reducing it to its smallest possible form, where the numerator and denominator have no common factors other than 1. In our problem, after adding \-\frac{28}{12}\ and \frac{9}{12}\, we obtain \frac{-19}{12}\. We check if this fraction can be simplified by finding any common factors between 19 and 12. Since they have no common factors other than 1, $-\frac{19}{12}$ is already in its simplest form.

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91影视

Find each sum or difference. $$ -\frac{7}{3}+\frac{3}{4} $$

Short Answer

Step by step solution

Find a common denominator

Convert fractions to have the same denominator

Add the fractions

Simplify the result (if needed)

Key Concepts

Common Denominator

Least Common Denominator

Simplifying Fractions

One App. One Place for Learning.

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