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Chapter 6: Problem 5

Perform the indicated operations and simplify. $$\frac{3}{5}+\frac{6}{5}$$

Short Answer

Expert verified
The result is \(\frac{9}{5}\), which is already simplified.

Step by step solution

01

Check the Denominators

First, notice that both fractions have the same denominator, which is 5. This means we can directly add the numerators since the denominators are already common.
02

Add the Numerators

Add the numerators of the two fractions. The numerators are 3 and 6, so calculate:\(3 + 6 = 9\)
03

Compose the New Fraction

Keep the common denominator which is 5, and combine the result from Step 2 as the numerator to form the new fraction:\(\frac{9}{5}\)
04

Simplify If Possible

Check whether the fraction can be simplified further. Since 9 and 5 have no common factors other than 1, \(\frac{9}{5}\) 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.

Understanding the Numerator
In fractions, the numerator is the number sitting above the line. It's a crucial part of a fraction as it tells us how many parts of the whole we are considering. For instance, in the fraction \( \frac{3}{5} \), the number 3 is the numerator. This describes the portion of the fraction. When adding fractions, you combine their numerators to represent the total parts being counted.
  • The numerator indicates how many parts we have.
  • In \( \frac{3}{5} \), the 3 is the numerator.
  • For addition, you add together the numerators: \(3 + 6 = 9\).
Grasping the Denominator
The denominator is the number beneath the line in a fraction. It tells us into how many equal parts the whole is divided. In \( \frac{3}{5} \), the 5 is the denominator, reflecting that the whole is split into 5 equal parts. When adding fractions, having a common denominator is crucial. This ensures you are adding parts of the same size. Without a common denominator, parts wouldn't align properly.
  • The denominator names the total number of equal parts in a whole.
  • For fractions with a common denominator, only numerators are added.
Simplifying Fractions
Simplifying fractions involves reducing the fraction to its smallest form鈥攚hen the numerator and denominator have no common factors except for one. For example, the fraction \( \frac{9}{5} \) is already simplified because 9 and 5 don't share any common factors other than 1. This simplification ensures we deal with the simplest form for ease in further calculations.
  • To simplify, find the greatest common divisor (GCD) of the numerator and denominator.
  • If the GCD is 1, the fraction is already simplified.
Finding a Common Denominator
A common denominator is shared by two or more fractions. It acts as a unified field where numerators can be safely added or subtracted. In the given problem, both fractions \( \frac{3}{5} \) and \( \frac{6}{5} \) have the same denominator. This allows for direct addition of the numerators, bypassing any need for further calculation of a common denominator.
  • Common denominators ensure fractions represent parts of the same size.
  • For fractions with identical denominators, you can easily combine by adding numerators.

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Most popular questions from this chapter

Solve the given problems. Multiply 82 by 78 by writing \((82)(78)=(80+2)(80-2)\) and using one of the special products.

Use the special products of this section to determine the products. You may need to write down one or two intermediate steps. $$(4-3 x)\left(16+12 x+9 x^{2}\right)$$

Use the special products of this section to determine the products. Each comes from the technical area indicated. \(\frac{1}{2} \pi(R+r)(R-r) \quad\) (architecture)

In Exercises \(29-64,\) reduce each fraction to simplest form. $$\frac{24-3 a^{3}}{a^{2}-4 a+4}$$

Factor the given expressions completely. Each is from the technical area indicated. $$d^{4}-10 d^{2}+16 \quad \text { (magnetic field) }$$
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