Chapter 7

For exercises \(67-82\), use the five steps and a proportion. In 2010 , there were \(14.9\) cases of syphilis per 100,000 Americans with a total of 45,834 cases of syphilis. Find the population of Americans used to create this ratio. Round to the nearest hundred. (Source: www.cdc.gov, 2011)

The pulp used to make Cascades Moka unbleached toilet paper is \(80 \%\) post- consumer recycled material and \(20 \%\) recovered corrugated boxes. Currently, \(3.4\) million tons of toilet paper are used per year in the United States. Fifty-three percent of this toilet paper is made from virgin (nonrecycled) fiber sources. A switch to \(100 \%\) recycled bath tissue could save \(30.6\) million trees and 68 million gigajoules of energy per year. Find the amount of toilet paper used per year that is made from virgin fiber sources. Write the answer in place value notation. (Source: www.bradenton.com, Jan. 25, 2012)

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 2 c^{2}-2 c-24 ; 6 c^{2}-18 c-24 $$

For exercises 39-82, simplify. $$ \frac{2 d^{2}+7 d}{2 d^{2}+11 d+14} \div \frac{d^{4}-d^{3}}{d^{2}+d-2} $$

$$ \text { For exercises 67-72, simplify. } $$ $$ \frac{k+5}{k^{3}+125} $$

For exercises \(67-82\), use the five steps and a proportion. In \(2010,3.5\) per 100,000 full-time equivalent workers were killed on the job with a total of 547 workers killed on the job. Find the number of full-time equivalent workers used to create this ratio. Round to the nearest whole number. (Source: www.osha.gov)

An American born in February 1950 will reach full retirement age in 2015 . His social security payments are based on his lifetime earnings. His first option is to retire at age 65 years and 10 months and receive \(\$ 1000\) a month for the rest of his life. His second option is to retire at age 62 (46 months before full retirement age) and receive \(\$ 758\) a month for the rest of his life. What age will he be in years and months when the payments he has received from either option are equal? Round up to the nearest month. (Source: www.ssa.gov)

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 28 x^{2} y^{5} ; 84 x y^{3} $$

For exercises 39-82, simplify. $$ \frac{5 b+15}{4 b+4} \div \frac{2 b+6}{7 b+7} $$

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 21 x^{3} y^{5} ; 84 x y^{2} $$

For exercises \(67-82\), use the five steps and a proportion. In 2010 , about \(2,465,940\) Americans died. Find the number of Americans who died without a will. Round to the nearest hundred. (Source: www.cdc.gov, Jan. 11, 2012) Seven out of ten Americans die without a will. (Source: extension.umd.edu)

For exercises \(75-78\), one part of simplifying a rational expression is completed. Problem: To simplify \(\frac{\frac{4}{15 x}}{\frac{8}{15}}\), rewrite the expression as \(\frac{4}{15 x} \div \frac{8}{15}\) and simplify. $$ \text { Incorrect Answer: } \begin{aligned} & \frac{4}{15 x} \div \frac{8}{15} \\ &=\frac{4}{15 x} \cdot \frac{15}{8} \\ &=\frac{4}{15 x} \cdot \frac{15}{4 \cdot 2} \\ &=2 x \end{aligned} $$

For exercises 39-82, simplify. $$ \frac{u^{2}+8 u+15}{u^{2}+2 u+1} \div \frac{u^{2}+7 u+10}{u^{2}+3 u+2} $$

In 2010, about 2,465,940 Americans died. Find the number of these deaths that were from chronic diseases. Round to the nearest hundred. (Source: www.cdc.gov, Jan. 11, 2012) 7 out of 10 deaths among Americans each year are from chronic diseases. (Source: www.cdc.gov, July 7, 2010)

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 60 a b ; 36 x y $$

For exercises 39-82, simplify. $$ \frac{w^{2}+10 w+16}{w^{2}+2 w+1} \div \frac{w^{2}+12 w+32}{w^{2}+5 w+4} $$

Find the number of \(2,200,000\) adults who binge drink about four times a month. Round to the nearest thousand. Binge drinking is a nationwide problem and bigger than previously thought. One in six adults binge drinks about four times a month. Binge drinking is defined as consuming four or more drinks for women or five or more drinks for men over a short period of time. Most binge drinkers are not alcohol-dependent. (Source: www.cdc.gov, Jan. 2012)

For exercises \(75-78\), one part of simplifying a rational expression is completed. Problem: The first step in using Strategy 2 to simplify \(\frac{\frac{1}{x+1}-\frac{1}{x-2}}{\frac{2}{x+1}+\frac{3}{x-2}} \cdot \frac{(x+1)(x-2)}{(x+1)(x-2)}\) is to find the least common denominator of all the individual fractions, multiply by a fraction equal to 1 , use the distributive property, and simplify. Incorrect Answer: $$ \begin{aligned} & \frac{\frac{1}{x+1}-\frac{1}{x-2}}{\frac{2}{x+1}+\frac{3}{x-2}} \cdot \frac{(x+1)(x-2)}{(x+1)(x-2)} \\ =& \frac{\frac{1}{x+1} \cdot(x+1)(x-2)-\frac{1}{x-2}(x+1)(x-2)}{\frac{2}{x+1} \cdot(x+1)(x-2)+\frac{3}{x-2} \cdot(x+1)(x-2)} \\ =& \frac{x-2-x+1}{2 x-4+3 x+3} \end{aligned} $$

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 120 n^{2} p^{2} ; 180 n^{5} p^{2} $$

For exercises 39-82, simplify. $$ \frac{z^{2}+6 z-16}{z-2} \div \frac{z+8}{z+3} $$

For exercises 77-86, find any values of the variable for which this expression is undefined. $$ \frac{c-9}{c+3} $$

Find the number of 80,500 children and adolescents who are obese. Round to the nearest hundred. Approximately one in six children and adolescents are obese. (Source: www.cdc.gov, Nov. 2011)

For exercises \(75-78\), one part of simplifying a rational expression is completed. Problem: To simplify \(\frac{\frac{x^{2}-2 x-15}{x^{2}-7 x+10}}{\frac{x^{2}+2 x-8}{x^{2}+2 x-3}}\), the first step is to factor the polynomials completely. Incorrect Answer: $$ \begin{array}{r} \frac{\frac{x^{2}-2 x-15}{x^{2}-7 x+10}}{\frac{x^{2}+2 x-8}{x^{2}+2 x-3}} \\ =\frac{\frac{(x-5)(x+3)}{(x-2)(x-5)}}{\frac{(x+2)(x-4)}{(x+3)(x-1)}} \end{array} $$

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 168 n^{2} w^{6} ; 252 n^{2} w^{2} $$

For exercises 39-82, simplify. $$ \frac{b^{2}+7 b-18}{b-2} \div \frac{b+9}{b+4} $$

For exercises 77-86, find any values of the variable for which this expression is undefined. $$ \frac{a-7}{a+2} $$

MRI scans of women with the BRCA1 and BRCA2 genetic mutations that were positive for cancer were wrong five out of six times. (These results are "false positives.") If 1500 women with these mutations had MRI scans that indicated cancer, predict how many of these women did not have cancer. (Source: www.telegraph .co.uk, March 26, 2008)

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 315 a^{10} b^{2} c ; 117 a^{5} b^{3} c^{2} $$

For exercises 39-82, simplify. $$ \frac{36 a^{2}+12 a+1}{18 a^{2}+15 a+2} \div \frac{6 a^{2}-17 a-3}{3 a^{2}-16 a-12} $$

For exercises 79-82, (a) clear the fractions and solve. (b) check. $$ \frac{3}{2} u+\frac{3}{4}=\frac{9}{2} $$

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 140 a^{2} b^{11} c^{2} ; 52 a^{3} b^{6} c $$

For exercises 39-82, simplify. $$ \frac{16 x^{2}+8 x+1}{8 x^{2}-10 x-3} \div \frac{4 x^{2}+17 x+4}{2 x^{2}+x-6} $$

For exercises 77-86, find any values of the variable for which this expression is undefined. $$ \frac{y^{2}+4 y}{y^{2}-8 y-20} $$

For exercises 79-82, (a) clear the fractions and solve. (b) check. $$ 1=\frac{7}{6} w+\frac{5}{12} $$

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ x^{3}+6 x^{2}+9 x ; x^{3}+7 x^{2}+12 x $$

For exercises \(67-82\), use the five steps and a proportion. Cyclosporine is an anti-rejection drug given to organ transplant patients. A bottle contains \(50 \mathrm{~mL}\) of liquid. Each milliliter of liquid contains \(100 \mathrm{mg}\) of cyclosporine. A kidney transplant patient needs to take \(850 \mathrm{mg}\) of cyclosporine each day. Find the amount of solution that the patient should take each day.

For exercises 79-82, (a) clear the fractions and solve. (b) check. $$ \frac{5}{6} h+8=12 $$

For exercises 39-82, simplify. $$ \frac{3 p^{2}+15 p+12}{p^{2}+4 p+3} \div \frac{18}{6 p+18} $$

A regulation basketball court in the NBA and the NCAA is \(94 \mathrm{ft}\) long and \(50 \mathrm{ft}\) wide. A regulation high school basketball court is \(84 \mathrm{ft}\) long and \(50 \mathrm{ft}\) wide. Find the percent increase in the area of an NCAA court compared to a high school court. Round to the nearest percent.

When a student with math anxiety is given a test, feelings of anxiety and panic can make the student feel that he or she cannot do a single problem on the test. What do you think a student should do if this happens?

For exercises \(35-86\), simplify. $$ \frac{1}{6 a}+\frac{2}{3 a}-\frac{3}{4 a} $$

For exercises 77-86, find any values of the variable for which this expression is undefined. $$ \frac{5}{81-k^{2}} $$

The height of a triangle is \(3 \mathrm{ft}\) more than the length of its base, and its area is \(54 \mathrm{ft}^{2}\). Use a quadratic equation to find the base and height of this triangle. \(\left(A=\frac{1}{2} b h .\right)\)

For exercises \(35-86\), simplify. $$ \frac{1}{6 n}+\frac{3}{2 n}-\frac{7}{4 n} $$

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 10 x^{4}+20 x^{3}+10 x^{2} ; 8 x^{3}+16 x^{2}+8 x $$

For exercises 77-86, find any values of the variable for which this expression is undefined. $$ \frac{2}{x^{2}+3} $$

The rulebook of the U.S. Lawn Mower Racing Association describes how to award points. 100 points for registration 100 points for starting a race 100 points for finishing a race 300 points for first place 250 points for second place 200 points for third place 150 points for fourth place 100 points for fifth place Source: www.letsmow.com A lawn mower racer registered for a day of racing. She started and completed three races. She placed fourth in the first race, third in the second race, and first in the final race. Find the total number of points she earned.

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 6 x^{4}+24 x^{3}+24 x^{2} ; 9 x^{3}+36 x^{2}+36 x $$

For exercises 77-86, find any values of the variable for which this expression is undefined. $$ \frac{5}{x^{2}+6} $$

For exercises 87-90, 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: Solve: \(\frac{11}{x}+\frac{13}{12}=1\) Incorrect Answer: Least common denominator is \(12 x\). $$ \begin{aligned} 12 x\left(\frac{11}{x}+\frac{13}{12}\right) &=1 \\ 12 x\left(\frac{11}{x}\right)+12 x\left(\frac{13}{12}\right) &=1 \\ 132+13 x &=1 \\ \frac{-132}{0+13 x} &=-131 \\ \frac{13 x}{13} &=\frac{-131}{13} \\ x &=-\frac{131}{13} \end{aligned} $$