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Chapter 7: Problem 26

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{3}+\frac{1}{2}}{\frac{1}{2}+\frac{1}{7}} $$

Short Answer

Expert verified
\( \frac{35}{27} \)

Step by step solution

01

- Simplify the numerator

First, add the fractions in the numerator: \( \frac{1}{3}+\frac{1}{2} \). To do this, find a common denominator. The common denominator of 3 and 2 is 6. Convert the fractions: \( \frac{1}{3} = \frac{2}{6} \) and \( \frac{1}{2} = \frac{3}{6} \). Now add the fractions: \( \frac{2}{6} + \frac{3}{6} = \frac{5}{6} \).
02

- Simplify the denominator

Next, add the fractions in the denominator: \( \frac{1}{2} + \frac{1}{7} \). Find a common denominator. The common denominator of 2 and 7 is 14. Convert the fractions: \( \frac{1}{2} = \frac{7}{14} \) and \( \frac{1}{7} = \frac{2}{14} \). Now add the fractions: \( \frac{7}{14} + \frac{2}{14} = \frac{9}{14} \).
03

- Divide the simplified fractions

Now, divide the simplified numerator by the simplified denominator: \( \frac{\frac{5}{6}}{\frac{9}{14}} \). To divide by a fraction, multiply by its reciprocal: \( \frac{5}{6} \times \frac{14}{9} \).
04

- Simplify the multiplication

Multiply the fractions: \( \frac{5 \times 14}{6 \times 9} = \frac{70}{54} \). Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 2: \( \frac{70 \div 2}{54 \div 2} = \frac{35}{27} \).

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Key Concepts

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

common denominator
To add or compare fractions, they need a common denominator. The common denominator is a shared multiple of each denominator involved. For example, to add the fractions \( \frac{1}{3} \) and \( \frac{1}{2} \), you first need to convert them to have the same denominator. Find the smallest multiple: the least common multiple (LCM) of 3 and 2 is 6. You can convert the fractions to \( \frac{2}{6} \) and \( \frac{3}{6} \). Use this method to make both fractions have the same base before performing operations.
adding fractions
Adding fractions means finding their total. First, ensure both fractions share a common denominator. For instance, if you want to add \( \frac{1}{3} \) and \( \frac{1}{2} \), convert them to have the common denominator 6, making them \( \frac{2}{6} \) and \( \frac{3}{6} \). You can then add their numerators: \( 2 + 3 = 5 \). Thus, the sum is \( \frac{5}{6} \). Remember: only the numerators are added, and the denominator stays the same.
dividing fractions
Dividing one fraction by another involves multiplying by the reciprocal. The reciprocal of a fraction is simply switching its numerator and denominator. For instance, to divide \( \frac{a}{b} \) by \( \frac{c}{d} \), you multiply \( \frac{a}{b} \) by \( \frac{d}{c} \). In the context of the exercise, to divide \( \frac{5}{6} \) by \( \frac{9}{14} \), you multiply \( \frac{5}{6} \) by the reciprocal of \( \frac{9}{14} \), which is \( \frac{14}{9} \). This gives you \( \frac{5}{6} \times \frac{14}{9} = \frac{70}{54} \).
multiplication of fractions
Multiplying fractions involves multiplying their numerators together and their denominators together. For example, multiplying \( \frac{5}{6} \) by \( \frac{14}{9} \) proceeds as follows: multiply the numerators \( 5 \times 14 = 70 \), and then the denominators \( 6 \times 9 = 54 \), resulting in \( \frac{70}{54} \). To simplify this fraction, find the greatest common divisor (GCD) of 70 and 54, which is 2. Divide both the numerator and denominator by 2: \( \frac{70 \div 2}{54 \div 2} = \frac{35}{27} \). The result is the simplified fraction.

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Most popular questions from this chapter

Medical researchers collected data on 272 patients who were hospitalized for at least 24 hours with the 2009 H1N1 influenza in the United States from April 2009 to mid-June 2009. One out of four of these patients were admitted to an intensive care unit. About 9 out of 20 patients were children under the age of 18 years. Find the number of patients who were children. Round to the nearest whole number. (Source: www.nejm.org, Nov. 12, 2009)

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