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Chapter 1: Problem 71

Find each difference. $$ \frac{5}{8}-\left(-\frac{1}{2}-\frac{3}{4}\right) $$

Short Answer

Expert verified
The difference is \( \frac{15}{8} \).

Step by step solution

01

- Simplify inside the parentheses

First, handle the subtraction inside the parentheses: \( -\frac{1}{2} - \frac{3}{4} \). To subtract fractions, find a common denominator. The common denominator for 2 and 4 is 4. Convert \( -\frac{1}{2} \) to \( -\frac{2}{4} \). Now subtract: \( -\frac{2}{4} - \frac{3}{4} = -\frac{5}{4} \).
02

- Simplify the entire expression

Replace the simplified fraction back into the expression: \( \frac{5}{8} - (-\frac{5}{4}) \).
03

- Convert the second fraction

Notice that subtracting a negative is the same as adding. Change the expression to: \( \frac{5}{8} + \frac{5}{4} \). Now, convert \( \frac{5}{4} \) to a fraction with a denominator of 8: \( \frac{5}{4} = \frac{10}{8} \).
04

- Add the fractions

Now add the fractions: \( \frac{5}{8} + \frac{10}{8} = \frac{15}{8} \).

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Key Concepts

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

common denominators
When working with fractions, you need to perform operations such as addition and subtraction. To do this effectively, you'll often need to find a common denominator.

A common denominator is a shared multiple of the denominators of two or more fractions. For example, in our exercise, we need to subtract \(-\frac{1}{2}\) and \(-\frac{3}{4}\). The denominators here are 2 and 4. A common denominator for these numbers is 4, because 4 is a multiple of both 2 and 4.

Here's how to convert:
  • Take \(-\frac{1}{2}\). Multiply both the numerator and the denominator by 2 to get \(-\frac{2}{4}\).


Now, both fractions have a denominator of 4, making them easier to subtract.
simplifying fractions
Simplifying fractions is another key concept to understand. Simplifying means making a fraction as simple as possible for easier computation.
  • If the numerator and the denominator of a fraction have common factors, you can simplify by dividing both by their greatest common divisor (GCD).

For the exercise, after subtracting \(-\frac{2}{4}\) and \(-\frac{3}{4}\), you get \(-\frac{5}{4}\). Here, there are no common factors to simplify, but it's crucial to recognize when you might need to simplify in other problems. This creates a cleaner and more understandable fraction to work with in further steps.
adding fractions
Adding fractions follows a similar process as subtraction, requiring a common denominator. In our exercise, after simplifying the expression, we end up with \ \frac{5}{8} + \frac{5}{4} \.

Convert \ \frac{5}{4} \ to a fraction with an 8 denominator by multiplying both the numerator and denominator by 2: \ \frac{5}{4} = \frac{10}{8} \.
  • With both fractions now having the same denominator, we can simply add the numerators: \ ( \frac{5}{8} + \frac{10}{8} = \frac{15}{8} ) \.
  • It's important to always check the result and see if further simplification is needed. In this case, \( \frac{15}{8} \) is already in its simplest form.

This structured approach ensures accuracy and reduces complexity in fraction arithmetic.

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Most popular questions from this chapter

Use the distributive property to rewrite each expression. $$ -\frac{1}{4}(8 x+3) $$

A manufacturer has fixed costs of \(\$ 1000\) to produce gizmos. Each gizmo costs \(\$ 5\) to make. The fixed cost to produce gadgets is \(\$ 750,\) and each gadget costs \(\$ 3\) to make. Write an expression for the total cost to make \(x\) gizmos and \(y\) gadgets.

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) \(6 x-5 y+4 a\)

Use the distributive property to rewrite each expression. $$ -3(2 x-5) $$

Use the distributive property to rewrite each expression. $$ -5(2 x-5 y+6 z) $$
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