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Chapter 7: Problem 99

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

Short Answer

Expert verified
\(\frac{31}{45}\)

Step by step solution

01

Identify the Common Denominator

The denominators of the two fractions are 15 and 45. The least common multiple of 15 and 45 is 45. This is the smallest number that both 15 and 45 divide evenly into, hence it is our common denominator.
02

Convert the Fractions to Have the Common Denominator

To convert \(\frac{13}{15}\) to have a denominator of 45, you multiply both the numerator and denominator by 3, resulting in \(\frac{39}{45}\). We don't need to convert \(\frac{8}{45}\) as it already has the same denominator.
03

Subtract the Fractions

Once the fractions are written with the same denominator, they can be subtracted. Subtract the numerators while leaving the denominator the same. This can be written as \(\frac{39}{45}-\frac{8}{45} = \frac{31}{45}\).

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Key Concepts

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

Least Common Multiple
When dealing with fractions, especially ones that need subtraction, finding the least common multiple (LCM) of the denominators is crucial. The LCM is the smallest number that both denominators can divide into without a remainder. For example, with denominators 15 and 45, the LCM is 45. This is because 45 is the smallest number that both 15 and 45 can evenly divide into. Knowing the LCM helps simplify the subtraction process by ensuring the fractions have equivalent denominators.
Common Denominator
A common denominator is essential when subtracting fractions. It allows us to write fractions with the same base for easy subtraction. In our exercise, the common denominator was 45 (the LCM of 15 and 45). In this case, both fractions need to be expressed with this common denominator. This ensures that we're dealing with parts of the same whole, allowing straightforward arithmetic operations, like subtraction.
Numerator and Denominator Conversion
Converting fractions involves changing them to have the same denominator without altering their values. For instance, \(\frac{13}{15}\) needs to be converted to a fraction with 45 as the denominator. We achieve this by multiplying both the numerator and the denominator by the same number, which retains the fraction's value. Here, multiplying by 3 will achieve this: \(\frac{13 \times 3}{15 \times 3} = \frac{39}{45}\). The fraction \(\frac{8}{45}\) doesn鈥檛 need conversion, as it already shares the common denominator.
Fraction Subtraction Steps
Subtracting fractions with the same denominators involves simple arithmetic. After conversion, write both fractions under the shared common denominator. In our example, we had \(\frac{39}{45} - \frac{8}{45}\). This is then handled like a regular subtraction problem with numerators. So, subtract 8 from 39 to get 31. The solution is then \(\frac{31}{45}\). This method keeps the operation straightforward and ensures that the fractions remain properly aligned.

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Most popular questions from this chapter

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{y}{y-1}-\frac{1}{1-y}$$

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{(y+1)(2 y-1)}{(y-2)(y-3)}+\frac{(y+2)(y-1)}{(y-2)(y-3)}-\frac{(y+5)(2 y+1)}{(3-y)(2-y)}$$

Use a proportion to solve each problem. According to the authors of Number Freaking, in a global village of 200 people, 9 get drunk every day. How many of the world's 6.9 billion people \((2010\) population) get drunk everyday? Round to the nearest hundredth of a billion.

Add or subtract as indicated. Simplify the result, if possible. $$\frac{2}{x-1}+\frac{3}{x+2}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{2}{y+5}+\frac{3}{4 y}$$
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