Chapter 7

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 12 ; 20 $$

For exercises 39-82, simplify. $$ \frac{2 a^{2}+2 a}{9} \div \frac{a+1}{3 a^{2}} $$

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 15 ; 20 $$

For exercises 39-82, simplify. $$ \frac{5 b^{2}+10 b}{8} \div \frac{b+2}{2 b^{3}} $$

For exercises 1-66, simplify. $$ \frac{10 a^{2}+20 a-30}{5 a^{2}+20 a+15} $$

When the top of a cone is removed, the formula for the volume of the remaining cone (the frustrum) is \(V=\frac{1}{3} \pi\left(R^{2}+R r+r^{2}\right) h\), where \(r\) is the radius of the circle at the top of the frustrum and \(R\) is the radius of the circle at the bottom of the frustrum. In 1856, an American army officer, Henry Hopkins Sibley, invented and received a patent for the design of a conical tent that could sleep 12 soldiers. (The apex is the diameter of the top of the frustrum.) Find the volume of the tent in cubic feet. Use \(\pi \approx 3.14\). Round to the nearest whole number. Be it known that I, H.H. Sibley, United States Army, have invented a new and improved Conical Tent ... the tent is in shape the frustrum of a cone; the base 18 feet; the height 12 feet; the apex 1 foot 6 inches [1.5 ft]. (Source: patimg1.uspto.gov)

For exercises 43-58, (a) solve. (b) check. $$ \frac{d+1}{3}=\frac{d-3}{6} $$

A student overdraws a bank account about five times each month. Predict the total overdraft fees the student will pay in 1 year. Chase's overdraft fees are \(\$ 25\) for the first fee each year, \(\$ 32\) for the next four and \(\$ 35\) after that. (Source: www.nytimes .com, Sept. 23, 2009)

For exercises 43-58, (a) solve. (b) check. $$ \frac{z+2}{4}=\frac{z-8}{12} $$

For exercises \(25-68\), evaluate or simplify. $$ \frac{x-y}{\frac{1}{y}-\frac{1}{x}} $$

For exercises 39-82, simplify. $$ 9 k \div \frac{27 k^{4}}{4} $$

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)

For exercises 59-66, use the five steps. Assume that the rate of work does not change if done individually or together. A worker can prune one row of grapevines in \(44 \mathrm{~min}\). Another worker can prune one row in \(33 \mathrm{~min}\). Find the time for these workers to do the job together. Round to the nearest whole number.

For exercises 1-66, simplify. $$ \frac{x^{3}-x^{2}-72 x}{x^{4}+5 x^{3}-24 x^{2}} $$

In 2011, the total property tax millage rate for Fort Lauderdale, Florida, was \(20.1705\). (For every \(\$ 1000\) in taxable property, an owner owes a tax of \(\$ 20.1705\).) If a property owner pays the tax in four installments, a discount is applied to the first three installments. Find the total amount of tax paid by installments on taxable property of \(\$ 175,000\). Round to the nearest hundredth. $$ \begin{array}{|c|c|} \hline \text { Installment due date } & \text { Discount on the payment } \\ \hline \text { June 30 } & 6 \% \\ \text { September 30 } & 4.5 \% \\ \text { December 31 } & 3 \% \\ \text { March 31 } & \text { None } \\ \hline \end{array} $$ Sources: www.broward.org; www.bcpa.net.millage.asp

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{y}+\frac{1}{x}}{y+x} $$

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 12 a^{2} b ; 18 a b^{2} $$

For exercises 39-82, simplify. $$ \frac{40 n^{5}}{21} \div 8 n $$

For exercises 1-66, simplify. $$ \frac{y^{3}-y^{2}-56 y}{y^{4}+5 y^{3}-14 y^{2}} $$

For exercises 61-64, the completed problem has one mistake. (a) Describe the mistake in words or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: The relationship of the number of gallons of gas, \(x\), and the total cost of the gas, \(y\), is a direct variation. If 8 gallons of gas costs \(\$ 24\), find the constant of proportionality. Incorrect Answer: $$ \begin{aligned} &k=x y \\ &k=(8 \mathrm{gal})(\$ 24) \\ &k=\$ 192 \mathrm{gal} \end{aligned} $$

For exercises 59-66, use the five steps. Assume that the rate of work does not change if done individually or together. The water from a garden hose turned on at full pressure fills a hot tub in \(45 \mathrm{~min}\). If the drain is open, the hot tub empties in \(62 \mathrm{~min}\). Find the amount of time to fill the hot tub with the drain open. Round to the nearest whole number.

For exercises 1-66, simplify. $$ \frac{2 a^{3}-4 a^{2}-6 a}{4 a^{3}-16 a^{2}-20 a} $$

For exercises 61-64, the completed problem has one mistake. (a) Describe the mistake in words or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: The relationship of the number of weeks a box of garbage bags is used, \(x\), and the number of bags left in the box, \(y\), is an inverse variation. When \(x\) is 8 weeks, \(y\) is 168 bags. Find the constant of proportionality, \(k\). Incorrect Answer: \(k=\frac{y}{x}\) $$ k=\frac{168 \text { bags }}{8 \text { weeks }} $$

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

For exercises 1-66, simplify. $$ \frac{2 c^{3}-2 c^{2}-4 c}{4 c^{3}-8 c^{2}-12 c} $$

For exercises 39-82, simplify. $$ \frac{2 z+6}{z^{2}+3 z+2} \div \frac{z+3}{z+2} $$

For exercises 61-64, the completed problem has one mistake. (a) Describe the mistake in words or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: In the formula \(A=\frac{10}{B}\), is the relationship between \(A\) and \(B\) a direct variation or an inverse variation? Incorrect Answer: Since as \(B\) increases, \(A\) also increases, this is a direct variation.

For exercises 39-82, simplify. $$ \frac{3 a+6}{a^{2}+4 a+4} \div \frac{a+4}{a+2} $$

For exercises \(65-68\), evaluate. $$ \sqrt{16} $$

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

For exercises 39-82, simplify. $$ \frac{3 p-1}{8 p} \div \frac{3 p^{2}+14 p-5}{6 p^{2}} $$

For exercises \(67-82\), use the five steps and a proportion. About five of 100 pregnant women have pre-eclampsia, a condition that results in high blood pressure. About 300,000 pregnant women per year in the United States have pre-eclampsia. Find the number of pregnant women in the United States used to create this ratio. (Source: www.nytimes.com, March 17, 2009)

For exercises \(35-86\), simplify. $$ \frac{x}{x-2}-\frac{4 x}{x^{2}-4} $$

For exercises 39-82, simplify. $$ \frac{p^{2}-64}{p+4} \div \frac{p+8}{-p-4} $$

$$ \text { For exercises 67-72, simplify. } $$ $$ \frac{x^{3}+8}{x^{2}-4} $$

For exercises \(67-82\), use the five steps and a proportion. Find the number of adults used to create the ratio "four out of five." Four out of five adults now use the Internet. 184 million adults are online from their homes, offices, schools or other locations. (Source: www.harrisinteractive.com, Nov. 17, 2008)

For exercises 39-82, simplify. $$ \frac{k^{2}-36}{k+3} \div \frac{k+6}{-k-3} $$

$$ \text { For exercises 67-72, simplify. } $$ $$ \frac{x^{3}+27}{x^{2}-9} $$

For exercises \(67-82\), use the five steps and a proportion. Find the number of 725,000 women in their mid \(-40 \mathrm{~s}\) with a history of normal pregnancy who would be expected to have a heart attack or stroke some 10 years later. Of 100 women in their mid-40's with a history of normal pregnancy, about 4 would be expected to have a heart attack or stroke some 10 years later. (Source: www.nytimes.com, March 17, 2009)

For exercises 69-70, resistors restrict the flow of electrons in an electric circuit. The unit of resistance is an ohm. describes the total resistance \(R\) in ohms of a parallel circuit with two resistors, \(R_{1}\) and \(R_{2}\). In a parallel circuit, Resistor \(R_{1}\) has a resistance of \(5 \mathrm{ohms}\), and Resistor \(R_{2}\) has a resistance of \(15 \mathrm{ohms}\). Find the total resistance of the circuit.

For exercises 39-82, simplify. $$ \frac{z^{2}+18 z+81}{z^{2}+7 z-18} \div \frac{z^{2}-5 z+6}{z^{2}-4 z+4} $$

$$ \text { For exercises 67-72, simplify. } $$ $$ \frac{a^{3}-125}{a^{2}-10 a+25} $$

For exercises \(67-82\), use the five steps and a proportion. A survey asked 505 companies whether they would continue to match their employees' contributions to their \(401 \mathrm{k}\) retirement plans. Find the number of companies that will continue to match the contributions. Three out of five employers maintain \(401(\mathrm{k})\) match despite economic crisis. (Source: www.americanbenefitscouncil.org, March 17, 2009)

For exercises 69-70, resistors restrict the flow of electrons in an electric circuit. The unit of resistance is an ohm. describes the total resistance \(R\) in ohms of a parallel circuit with two resistors, \(R_{1}\) and \(R_{2}\). In a parallel circuit, Resistor \(R_{1}\) has a resistance of \(8 \mathrm{ohms}\), and Resistor \(R_{2}\) has a resistance of \(12 \mathrm{ohms}\). Find the total resistance of the circuit.

For exercises 39-82, simplify. $$ \frac{a^{2}+12 a+36}{a^{2}-3 a-54} \div \frac{a^{2}+11 a+30}{a^{2}-a-72} $$

$$ \text { For exercises 67-72, simplify. } $$ $$ \frac{p^{3}-8}{p^{2}-4 p+4} $$

For exercises \(67-82\), use the five steps and a proportion. In 2010 , there were \(426.0\) cases of chlamydia per 100,000 Americans with a total of \(1,307,893\) cases of chlamydia. Find the population of Americans used to create this ratio. Round to the nearest hundred. (Source: www.cdc .gov, 2011)

The wholesale price of a display with 24 flashlights is \(\$ 16.40\). The company wants to make a profit of \(45 \%\) on each flashlight. Find the retail price that the company should charge for one flashlight. Round to the nearest hundredth.

For exercises \(35-86\), simplify. $$ \frac{4}{v^{2}-1}-\frac{2}{v-1} $$

For exercises 39-82, simplify. $$ \frac{2 k^{2}+3 k}{2 k^{2}-13 k-24} \div \frac{k^{4}-k^{3}}{k^{2}-9 k+8} $$