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Chapter 5: Problem 98

Perform the indicated operations. If possible, reduce the answer to its lowest terms. \(\left(\frac{1}{2}+\frac{1}{4}\right) \div\left(\frac{1}{2}+\frac{1}{3}\right)\)

Short Answer

Expert verified
\( \frac{9}{10} \)

Step by step solution

01

Add Fractions in the Numerator and Denominator

Add the fractions in the numerator and denominator separately. To do this, one must find the common denominator, which is the least common multiple of the denominators. For the numerator, \( \frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \). For the denominator, \( \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \). This gives the revised expression \( \frac{3}{4} \div \frac{5}{6} \).
02

Divide Fractions

To divide fractions, multiply the first fraction by the reciprocal of the second. Thus, \( \frac{3}{4} \div \frac{5}{6} \) becomes \( \frac{3}{4} \times \frac{6}{5} \).
03

Multiply Fractions

Multiply the numerators together for the new numerator and the denominators together for the new denominator. Hence, \( \frac{3}{4} \times \frac{6}{5} = \frac{18}{20} \).
04

Simplify the Fraction

Simplify the fraction to lowest terms by dividing both the numerator and the denominator by their highest common factor, which in this case is 2. This gives the result \( \frac{9}{10} \).

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The sequence for the number of seats per row in our movie theater as the rows move toward the back is arithmetic with \(d=1\) so people don't block the view of those in the row behind them.

You will develop geometric sequences that model the population growth for California and Texas, the two most populated U.S. states. The table shows the population of California for 2000 and 2010 , with estimates given by the U.S. Census Bureau for 2001 through \(2009 .\) $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \text { Year } & \mathbf{2 0 0 0} & \mathbf{2 0 0 1} & \mathbf{2 0 0 2} & \mathbf{2 0 0 3} & \mathbf{2 0 0 4} & \mathbf{2 0 0 5} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 33.87 & 34.21 & 34.55 & 34.90 & 35.25 & 35.60 \\ \hline \end{array} $$ $$ \begin{array}{|l|l|l|l|l|l|} \hline \text { Year } & \mathbf{2 0 0 6} & \mathbf{2 0 0 7} & \mathbf{2 0 0 8} & \mathbf{2 0 0 9} & \mathbf{2 0 1 0} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 36.00 & 36.36 & 36.72 & 37.09 & 37.25 \\ \hline \end{array} $$ a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that California has a population increase that is approximately geometric. b. Write the general term of the geometric sequence modeling California's population, in millions, \(n\) years after \(1999 .\) c. Use your model from part (b) to project California's population, in millions, for the year 2020 . Round to two decimal places.

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(6,-6,-18,-30, \ldots\)

What is a sequence? Give an example with your description.

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{4}\), when \(a_{1}=4, r=-3\).
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