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Chapter 4: Problem 52

Add or subtract the fractions, as indicated, and simplify your result. $$\frac{1}{3}-\frac{1}{8}$$

Short Answer

Expert verified
\( \frac{5}{24} \)

Step by step solution

01

Identify the Denominators

The fractions given are \( \frac{1}{3} \) and \( \frac{1}{8} \). The denominators are 3 and 8, respectively.
02

Find the Least Common Denominator (LCD)

To subtract these fractions, we need a common denominator. The least common denominator of 3 and 8 is 24.
03

Convert the Fractions to Have the Common Denominator

Convert \( \frac{1}{3} \) to \( \frac{8}{24} \) by multiplying both the numerator and the denominator by 8. Convert \( \frac{1}{8} \) to \( \frac{3}{24} \) by multiplying both the numerator and the denominator by 3.
04

Subtract the Fractions

Now that the fractions have the same denominator, subtract the numerators: \( \frac{8}{24} - \frac{3}{24} = \frac{5}{24} \).
05

Simplify the Result, if Possible

Verify if \( \frac{5}{24} \) can be simplified. Since 5 and 24 do not have a common factor other than 1, \( \frac{5}{24} \) 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.

Least Common Denominator
When dealing with fraction subtraction, such as with \( \frac{1}{3} - \frac{1}{8} \), the first concept to grasp is the **least common denominator** or LCD. The LCD is the smallest number that both denominators can divide into without leaving a remainder.

Finding the LCD is a key step because it allows us to rewrite fractions with a common denominator, essential for performing addition or subtraction.
  • First, identify the denominators of the fractions you are working with. Here, they are 3 and 8.
  • List the multiples of each denominator. Multiples of 3 include 3, 6, 9, 12, 15, 18, 21, 24... and multiples of 8 include 8, 16, 24, 32...
  • The LCD is the first number that appears in both lists, which in this case is 24.
Now, with an LCD of 24, both fractions can be adjusted to have the same denominator, making subtraction straightforward.
Simplifying Fractions
Once you've successfully subtracted fractions, like turning \( \frac{8}{24} - \frac{3}{24} \) into \( \frac{5}{24} \), it's important to determine if the result can be simplified further. **Simplifying fractions** means reducing them to their most basic form, so that no common factor exists between the numerator and the denominator besides 1.

To simplify:
  • Check if both the numerator and the denominator have any common factors.
  • In \( \frac{5}{24} \), check the factors of 5 (which is just 1 and 5) and 24 (which are 1, 2, 3, 4, 6, 8, 12, and 24).
  • Since 5 is a prime number and doesn't share any common factors with 24 other than 1, \( \frac{5}{24} \) is already in simplest form.
Remember, simplification is essential for clear mathematical communication and making calculations easier.
Common Denominator
In the context of subtracting fractions, having a **common denominator** is essential. This concept is crucial because without a common denominator, you cannot add or subtract fractions directly.

But what is a common denominator? It is a shared multiple of the denominators of the fractions involved. Having each fraction with the same denominator aligns them on a similar scale, enabling straightforward arithmetic.
  • Take \( \frac{1}{3} \) and \( \frac{1}{8} \). To find a common denominator, you convert them both using the least common denominator, which is 24 here.
  • Multiply the numerator and denominator of \( \frac{1}{3} \) by 8 to get \( \frac{8}{24} \).
  • Multiply the numerator and denominator of \( \frac{1}{8} \) by 3 to get \( \frac{3}{24} \).
  • Now, with the common denominator of 24, subtract the numerators to get \( \frac{5}{24} \).
This strategy of using a common denominator allows you to handle addition and subtraction of fractions seamlessly.

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Most popular questions from this chapter

Simplify the complex rational expression. $$\frac{-\frac{3}{7}-\frac{1}{3}}{\frac{1}{3}-\frac{6}{7}}$$

Is \(1 / 3\) a solution of the equation \(x+\frac{3}{4}=\frac{13}{12}\) ?

Solve the equation and simplify your answer. $$x-\frac{1}{9}=-\frac{3}{2}$$

Solve the equation and simplify your answer. $$\frac{5}{9} x-\frac{7}{2}=\frac{1}{4}$$

Is \(8 / 5\) a solution of the equation \(\frac{11}{14} x=\frac{44}{35}\) ?
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