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Chapter 1: Problem 48

Add. Do not use the number line except as a check. \(\frac{-2}{7}+\frac{3}{7}\)

Short Answer

Expert verified
\frac{1}{7}

Step by step solution

01

- Identify the operation and denominators

Notice that the problem requires the addition of two fractions: \(\frac{-2}{7}+\frac{3}{7}\). The denominators in both fractions are the same (7).
02

- Add the numerators

Since the denominators are the same, add the numerators directly: \(-2 + 3\).
03

- Simplify the sum of the numerators

Calculate \(-2 + 3\), which equals 1.
04

- Combine the numerators with the common denominator

Write the result as a fraction with the common denominator: \(\frac{1}{7}\).
05

- Verify the result using a number line (optional)

If needed, use a number line to verify that \(\frac{-2}{7}+\frac{3}{7}=\frac{1}{7}\).

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Key Concepts

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

Common Denominator
When adding fractions, it's essential that the denominators (the bottom number of the fractions) are the same. If they aren't, you cannot directly add the fractions together. This common denominator acts like a common ground. In our problem, \(\frac{-2}{7} \frac{3}{7}\), both fractions already have the same denominator, which is 7. That makes our job easier because we can move straight to adding the numerators.
Numerator
The numerator is the top number of a fraction and tells us how many parts we have. In our example, \(\frac{-2}{7} + \frac{3}{7}\), the numerators are -2 and 3. Since the denominators are already the same, we only need to add the numerators together: \(-2 + 3\). This is a simple addition problem, resulting in 1. Therefore, the combined fraction is \(\frac{1}{7}\).
Simplification
Simplification is the process of reducing a fraction to its simplest form. It means that the numerator and the denominator have no common factors other than 1. Though our problem \(\frac{1}{7}\) doesn鈥檛 need further simplification because 1 and 7 have no common factors, this step is crucial in many other fraction problems.
Number Line
A number line can help visualize the addition of positive and negative numbers. Imagine a line with numbers evenly spaced. Negative numbers are to the left, positive numbers to the right. Starting from zero, you can place \(-\frac{2}{7}\) slightly to the left, move right three steps to add \(\frac{3}{7}\), landing on \(\frac{1}{7}\). This visual tool is handy for verification, though in our problem, performing the arithmetic was straightforward.

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Most popular questions from this chapter

Evaluate using a calculator. $$ 1.5+(2 x-y)^{2}, \text { for } x=9.25 \text { and } y=1.7 $$

Simplify. $$ \frac{5^{2}-3^{2}}{2 \cdot 6-4} $$

Simplify. $$ 12 a^{2}-3 a b+5 b^{2}-5\left(-5 a^{2}+4 a b-6 b^{2}\right) $$

Write an equivalent expression without using grouping symbols. $$ -(6 x-7) $$

Simplify using a calculator. Round your answer to the nearest thousandth. $$ \frac{2.5^{2}-10 \cdot 12 \div(-1.5)}{(3+5)^{2}-60} $$
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