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Chapter 4: Problem 41

Rewrite each fraction with the indicated denominators. $$-\frac{3}{4}=-\frac{ }{16}$$

Short Answer

Expert verified
-\(\frac{12}{16}\)

Step by step solution

01

Understand the Problem

Given the fraction \(-\frac{3}{4}\), it needs to be rewritten so that it has a denominator of 16.
02

Find the Multiplication Factor

To rewrite the fraction with a denominator of 16, figure out what number you need to multiply 4 by to get 16. \(\frac{16}{4} = 4\). Thus, the multiplication factor is 4.
03

Multiply the Numerator and the Denominator

Multiply both the numerator and the denominator of the original fraction by 4: -\(\frac{3}{4} = -\frac{3\cdot4}{4\cdot4} = -\frac{12}{16}\).
04

Verify the Fraction

Confirm that the rewritten fraction -\(\frac{12}{16}\) is equivalent to the original -\(\frac{3}{4}\). Simplify -\(\frac{12}{16}\) back to -\(\frac{3}{4}\) to assure correctness.

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Key Concepts

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

Multiplication Factor
When you're given a fraction and asked to rewrite it with a different denominator, the first step is often to figure out the right multiplication factor. To do this, you divide the new denominator by the old one.
For example, in the problem, you have \(-\frac{3}{4}\) and you need to find a denominator of 16. By dividing 16 by the existing denominator, 4, you get: \(\frac{16}{4} = 4\).This means that the **multiplication factor** is 4.
So, you'll use this factor to change both the numerator and the denominator in the next steps.
Denominator
The **denominator** is the number at the bottom of a fraction and tells you how many equal parts the whole is divided into.
In \(-\frac{3}{4}\), the denominator is 4, meaning the whole is split into 4 parts.
To change \(-\frac{3}{4}\) to something over 16, we need to adjust the denominator. This adjustment must be done by the multiplication factor (which we found to be 4).
Thus: 4 \(\times 4 = 16\). So, \(-\frac{3}{4} = -\frac{3 \times 4}{4\times4} = -\frac{12}{16}\).
Simplifying Fractions
After rewriting or working with a fraction, it's often good practice to **simplify** it.
Simplifying means making the fraction as simple as possible while keeping the same value. To simplify, divide both the numerator and the denominator by their greatest common divisor (GCD).
In this exercise, we started with \(-\frac{3}{4}\) and rewrote it as \(-\frac{12}{16}\). To prove equivalence, simplify \(-\frac{12}{16}\) back. The GCD of 12 and 16 is 4.
So, \(-\frac{12}{16} \div 4 = -\frac{3}{4}\). Now, it's simplified and confirms our original fraction.

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Most popular questions from this chapter

Mr. Bishop and Ms Waymire both gave exams today. By mid-afternoon, Mr. Bishop had finished grading 16 out of 36 exams, and Ms. Waymire had finished grading 15 out of 27 exams. a. What fractional part of her total has Ms. Waymire completed? b. What fractional part of his total has Mr. Bishop completed?

How many thirds are in 10?

Simplify to lowest terms by first reducing the powers of 10. $$-\frac{3000}{1800}$$

Convert the improper fraction to a mixed number. $$-\frac{39}{5}$$

Which expressions are equivalent to \(-\frac{4}{5} ?\) \(\begin{array}{lll}\text { a. } \frac{-4}{5} & \text { b. } \frac{-4}{-5} & \text { c. } \frac{4}{-5} & \text { d. }-\frac{5}{4}\end{array}\)
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