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Chapter 5: Problem 112

Different operations with the same rational numbers usually result in different answers. Illustrate some curious exceptions. Show that \(\frac{169}{30}+\frac{13}{15}\) and \(\frac{169}{30} \div \frac{13}{15}\) give the same answer.

Short Answer

Expert verified
The answer to both operations \(\frac{169}{30}+\frac{13}{15}\) and \(\frac{169}{30} \div \frac{13}{15}\) is 6.5 which is a curious exception where different operations on the same fractions result in the same answer.

Step by step solution

01

Addition Operation

First, perform the addition operation: Take \(\frac{169}{30}+\frac{13}{15}\). This will require a common denominator. The common denominator of 30 and 15 is 30, and so \( \frac{13}{15} \) should be converted to \( \frac{26}{30} \). Now add both fractions to get \( \frac{169}{30}+\frac{26}{30} = \frac{195}{30} \). This fraction simplifies to 6.5.
02

Division Operation

Next, perform the division operation: Take \(\frac{169}{30} \div \frac{13}{15}\). Division with fractions involves flipping the second fraction (reciprocal) and converting the operation to multiplication. The reciprocal of \( \frac{13}{15} \) is \( \frac{15}{13} \). Now multiply both fractions to get \( \frac{169}{30} \times \frac{15}{13} = \frac{2535}{390} \). This fraction simplifies to 6.5.
03

Comparing Results

Now, when comparing the results of the two operations, both equal 6.5. This is a curious exception where different operations on the same fractions result in the same answer.

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Most popular questions from this chapter

The bar graph shows changes in the percentage of college graduates for Americans ages 25 and older from 1990 to 2010. Exercises 125-126 involve developing arithmetic sequences that model the data. In \(1990,24.4 \%\) of American men ages 25 and older had graduated from college. On average, this percentage has increased by approximately \(0.3\) each year. a. Write a formula for the \(n\)th term of the arithmetic sequence that models the percentage of American men ages 25 and older who had graduated from college \(n\) years after \(1989 .\) b. Use the model from part (a) to project the percentage of American men ages 25 and older who will be college graduates by \(2019 .\)

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The bar graph shows changes in the percentage of college graduates for Americans ages 25 and older from 1990 to 2010. Exercises 125-126 involve developing arithmetic sequences that model the data. In \(1990,18.4 \%\) of American women ages 25 and older had graduated from college. On average, this percentage has increased by approximately \(0.6\) each year. a. Write a formula for the \(n\)th term of the arithmetic sequence that models the percentage of American women ages 25 and older who had graduated from college \(n\) years after \(1989 .\) b. Use the model from part (a) to project the percentage of American women ages 25 and older who will be college graduates by \(2019 .\)
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