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Chapter 7: Problem 99

Subtract: $$\frac{13}{15}-\frac{8}{45} .$$ (Section 1.2, Example 9)

Short Answer

Expert verified
The result of the subtraction \(\frac{13}{15}-\frac{8}{45}\) is \(\frac{31}{45}\).

Step by step solution

01

Identify the lowest common denominator

The denominator of the first fraction is 15, and the denominator of the second fraction is 45. The lowest common denominator for these two numbers is 45 since it is the smallest number which both 15 and 45 can divide evenly into. In this case, we can take 45 as the common denominator because 15 divides into 45 exactly 3 times.
02

Adjust the fractions to have the common denominator

When the denominators are the same, subtraction can be performed in a straightforward way. We need to multiply the first fraction (both numerator and denominator) by the factor which makes the denominator equal to 45. Thus, \(\frac{13}{15}\) becomes \(\frac{13*3}{15*3} = \frac{39}{45}\). Now, the problem expression is \(\frac{39}{45} - \frac{8}{45}\).
03

Perform the subtraction

Now, both fractions have the same denominator, we can subtract the numerators, \(39 - 8 = 31\). Therefore, the result is \(\frac{31}{45}\).

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Key Concepts

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

Common Denominator
In the world of fractions, a common denominator is a must-have for adding or subtracting them. If you've ever thought of fractions as pieces of a pie, think of the denominator as the number of equally-sized slices each pie is cut into. To perform operations like subtraction, both pies need to have the same number of slices. That's what we mean by finding a common denominator.
  • The common denominator is the number you use when you align the slices in a pie.
  • For the fractions \(\frac{13}{15}\) and \(\frac{8}{45}\), we align them by transforming \(\frac{13}{15}\) to have the same denominator as \(\frac{8}{45}\).
  • This requires both fractions to have denominators that can be equally divided by the same number.
Once you've adjusted both fractions to have the same slices鈥攐r denominator鈥攜ou can easily perform the operation, just like stacking identical-sized blocks.
Fraction Subtraction
Subtracting fractions becomes a breeze once they share the same denominator. The initial challenge of different denominators can be handled by converting the fractions to share a common one, allowing for direct subtraction. Fraction subtraction involves a few simple steps:
  • Ensure both fractions have the same denominator.
  • Subtract the numerators, as you would subtract any two numbers.
  • Keep the denominator the same.
In our example, \(\frac{39}{45} - \frac{8}{45}\), we first aligned the denominators. This allowed us to subtract the numerators directly鈥擻(39 - 8 = 31\). The result was \(\frac{31}{45}\), showing how easy fraction subtraction is once you have a common denominator.
Lowest Common Multiple
Finding the lowest common multiple (LCM) is vital when dealing with different denominators in fraction subtraction. The LCM is the smallest number that both denominators divide into without leaving a remainder. It helps standardize fractions so that subtraction becomes straightforward.
  • Identify the denominators of the fractions you want to subtract.
  • Calculate the LCM of these denominators.
  • Convert each fraction to use this LCM as the common denominator.
In the problem \(\frac{13}{15} - \frac{8}{45}\), the denominators 15 and 45 have an LCM of 45. By transforming \(\frac{13}{15}\) to \(\frac{39}{45}\), we make both denominators identical, simplifying the subtraction process considerably.

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Most popular questions from this chapter

In Exercises \(87-90\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. To solve \(\frac{5}{3 x}+\frac{3}{x}=1,\) we must first add the rational expressions on the left side.

Add or subtract as indicated. Simplify the result, if possible. $$4+\frac{1}{x-3}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x-7}{x+4}+\frac{x+4}{x-7}$$

Determine whether each statement 鈥渕akes sense鈥� or 鈥渄oes not make sense鈥� and explain your reasoning. The fastest way for me to find \(\frac{4}{x-5}+\frac{9}{5-x}\) is by using \((x-5)(5-x)\) as the \(\mathrm{LCD}\)

Add or subtract as indicated. Simplify the result, if possible. $$7-\frac{4 y}{y+5}$$
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