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Chapter 4: Problem 2

Perform the indicated operations. Write the answers as fractions. $$\frac{3}{16}+\frac{7}{12}$$

Short Answer

Expert verified
\(\frac{37}{48}\)

Step by step solution

01

Identify the fractions

The given fractions are \(\frac{3}{16}\) and \(\frac{7}{12}\).
02

Find a common denominator

To add fractions, find a common denominator. The denominators are 16 and 12. The least common multiple (LCM) of 16 and 12 is 48.
03

Convert fractions to have the same denominator

Convert each fraction to an equivalent fraction with the denominator of 48:\(\frac{3}{16} = \frac{3 \times 3}{16 \times 3} = \frac{9}{48}\)\(\frac{7}{12} = \frac{7 \times 4}{12 \times 4} = \frac{28}{48}\)
04

Add the fractions

Now that the fractions have the same denominator, add them together:\(\frac{9}{48} + \frac{28}{48} = \frac{9 + 28}{48} = \frac{37}{48}\)
05

Simplify the fraction (if necessary)

Check if the fraction \(\frac{37}{48}\) can be simplified. In this case, 37 and 48 have no common factors other than 1, so \(\frac{37}{48}\) 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.

Adding Fractions with a Common Denominator
Adding fractions involves several steps, but the key is to make sure the denominators are the same. A common denominator ensures that fractions have a uniform base for accurate addition. To add fractions like \(\frac{3}{16}\) and \(\frac{7}{12}\), follow these steps: Identify the denominators, find their least common multiple (LCM), convert each fraction to an equivalent fraction with the common denominator, and combine the numerators while keeping the common denominator.
Finding the Least Common Multiple (LCM)
To add fractions, it鈥檚 crucial to find a common base for their denominators. This is done using the Least Common Multiple (LCM). To find the LCM, list the multiples of each number and identify the smallest number that appears in both lists. For 16 and 12, the multiples are:

16: 16, 32, 48, 64, 80, ...
12: 12, 24, 36, 48, 60, ...
The LCM here is 48 because it鈥檚 the smallest number appearing in both lists. This means we convert both fractions to have 48 as the denominator.
Simplifying Fractions
After adding fractions, you need to check if the resultant fraction can be simplified. Simplifying a fraction means reducing it to its lowest terms where the numerator and denominator have no common factors other than one. For instance, after adding \(\frac{3}{16}\) and \(\frac{7}{12}\), we get \(\frac{37}{48}\). To simplify, find the greatest common divisor (GCD) of the numerator and denominator. If it鈥檚 1, the fraction is already in its simplest form. In this example, since 37 and 48 have no common factors other than 1, the fraction \(\frac{37}{48}\) is already simplified.

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Most popular questions from this chapter

Raymond read 720 pages of a 792 -page book. His roommate, Travis, read 540 pages from a 660 -page book. a. What fractional part of the book did each person read? b. Which of the roommates read a greater fraction of his book?

Convert the improper fraction to a mixed number. $$\frac{13}{7}$$

Which expressions are equivalent to \(\frac{-9}{10} ?\) \(\begin{array}{cccc}\text { a. } \frac{9}{10} & \text { b. }-\frac{9}{10} & \text { c. } \frac{9}{-10} & \text { d. } \frac{-9}{-10}\end{array}\)

Simplify. $$\left|-\frac{8}{7}\right|$$

Simplify to lowest terms by first reducing the powers of 10. $$\frac{5100}{30,000}$$
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