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Chapter 7: Problem 12

Add or subtract. Write answer in lowest terms. \(\frac{6}{t+3}-\frac{3}{t+3}\)

Short Answer

Expert verified
\( \frac{3}{t+3} \)

Step by step solution

01

Identify Common Denominators

Recognize that both fractions have the same denominator: \( t+3 \). This allows for direct addition or subtraction of the numerators.
02

Subtract the Numerators

Since the denominators are the same, subtract the numerators directly: \( 6 - 3 = 3 \).
03

Write the Resulting Fraction

Combine the result from Step 2 with the common denominator: \( \frac{3}{t+3} \).
04

Simplify if Possible

Ensure the fraction is in its lowest terms. In this case, \( \frac{3}{t+3} \) 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.

Common Denominators
When working with fractions, it鈥檚 crucial to have a common denominator. The denominator is the bottom number of a fraction, which shows how many equal parts the whole is divided into. Having a common denominator means the denominators of the fractions you are working with are the same. This makes addition or subtraction of the fractions straightforward, as you only need to work with the numerators (the top numbers).

In the exercise \(\frac{6}{t+3} - \frac{3}{t+3}\) , both fractions already have the denominator \(t+3\). This common denominator allows us to directly perform operations on the numerators.
Subtracting Fractions
Subtracting fractions is simple when the denominators are the same. Here鈥檚 what you need to do:

  • Keep the common denominator.
  • Subtract the numerators.

For example, in the problem \(\frac{6}{t+3} - \frac{3}{t+3}\) , since the denominators \(t+3\) are the same, you only subtract the numerators:
\(\frac{6}{t+3} - \frac{3}{t+3} = \frac{6-3}{t+3}= \frac{3}{t+3}\).
Simplifying Fractions
Simplifying a fraction means reducing it to its lowest terms. This makes the fraction easier to understand and work with. To simplify a fraction, you need to divide both the numerator and the denominator by their greatest common divisor (GCD).

Here鈥檚 a quick guide to simplify a fraction:
  • Find the GCD of the numerator and the denominator.
  • Divide both the numerator and the denominator by the GCD.

In the example \(\frac{3}{t+3}\), the fraction is already in its simplest form because there are no common factors between 3 and \(t+3\) besides 1. Thus, there鈥檚 no further simplification needed.

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Most popular questions from this chapter

Solve each problem involving direct or inverse variation. If \(x\) varies inversely as \(y,\) and \(x=3\) when \(y=8,\) find \(y\) when \(x=4\)

Determine whether each equation represents direct or inverse variation. $$ y=50 x $$

Solve each problem involving direct or inverse variation. If \(z\) varies inversely as \(x,\) and \(z=50\) when \(x=2,\) find \(z\) when \(x=25\)

Solve each variation problem. If the temperature is constant, the pressure of a gas in a container varies inversely as the volume of the container. If the pressure is 10 lb per \(\mathrm{ft}^{2}\) in a container with volume \(3 \mathrm{ft}^{3}\), what is the pressure in a container with volume \(1.5 \mathrm{ft}^{3} ?\)

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{r^{2}+r-6}{r^{2}+4 r-12} \div \frac{r+3}{r-1}$$
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