Chapter 7

For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{4}{5}}{\frac{3}{10}} $$

For exercises 1-12, use prime factorization to find the least common denominator. $$ \frac{1}{40 x y^{2} z} ; \frac{1}{42 x y z^{3}} $$

For exercises 7-32, simplify. $$ \left(\frac{8 p-24}{9 p+18}\right)\left(\frac{27}{32}\right) $$

For exercises 1-66, simplify. $$ \frac{3 x-12}{15} $$

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=2, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=4\).

For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{5 x}{4}}{\frac{2 x^{2}}{15}} $$

For exercises \(5-48\), simplify. $$ \frac{4 n}{n+3}+\frac{n}{n+3} $$

For exercises 7-32, simplify. $$ \left(\frac{x^{2}+5 x}{x^{2}}\right)\left(\frac{3 x}{x+5}\right) $$

For exercises 1-66, simplify. $$ \frac{2 x-8}{10 x} $$

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=3, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=9\).

For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{4 p}{9}}{\frac{2 p^{2}}{3}} $$

For exercises \(5-48\), simplify. $$ \frac{6 w}{w+2}+\frac{w}{w+2} $$

For exercises 7-32, simplify. $$ \left(\frac{y^{2}+8 y}{y^{2}}\right)\left(\frac{9 y}{y+8}\right) $$

For exercises 1-66, simplify. $$ \frac{3 x-12}{15 x} $$

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=2, y=10\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=5\).

For exercises 13-24, rewrite each expression as an equivalent expression with the given denominator. $$ \frac{3}{7} ; 56 $$

For exercises 7-32, simplify. $$ \left(\frac{h^{2}}{h^{2}+3 h}\right)\left(\frac{h^{2}-9}{h}\right) $$

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=4, y=5\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=10\).

For exercises 11-30, (a) solve. (b) check. $$ \frac{13}{d}-\frac{5}{9}=\frac{1}{6} $$

For exercises 13-24, rewrite each expression as an equivalent expression with the given denominator. $$ \frac{5}{7} ; 42 $$

For exercises 7-32, simplify. $$ \left(\frac{r^{2}}{r^{2}+2 r}\right)\left(\frac{r^{2}-4}{r}\right) $$

For a fixed number of windows, the number of windows washed per hour, \(x\), and the number of hours it takes to wash the windows, \(y\), is an inverse variation. If a person can wash 20 windows per hour, it takes \(9 \mathrm{hr}\) to wash the windows. a. Find the constant of variation, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. If a person can wash 30 windows per hour, find the time needed to wash the windows.

For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{5 p-5}{4 p+12}}{\frac{10 p+10}{7 p+21}} $$

For exercises 13-24, rewrite each expression as an equivalent expression with the given denominator. $$ \frac{2}{9 a} ; 27 a^{2} b $$

For exercises \(5-48\), simplify. $$ \frac{r^{2}-12 r}{r+2}-\frac{28}{r+2} $$

For exercises 1-66, simplify. $$ \frac{4 z^{2}+20 z}{32 z} $$

For a fixed number of hotel rooms, the number of rooms cleaned per hour, \(x\), and the number of hours it takes to clean the rooms, \(y\), is an inverse variation. If a person can clean 8 rooms per hour, it takes 15 hr to clean the rooms. a. Find the constant of variation, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. If a person can clean 6 rooms per hour, find the time needed to clean the rooms.

For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{2 w-6}{4 w+16}}{\frac{12 w+36}{5 w+20}} $$

For exercises 7-32, simplify. $$ \frac{y^{2}-y}{y+7} \cdot \frac{3 y+21}{y^{2}+y} $$

For exercises 1-66, simplify. $$ \frac{5 m^{2}+30 m}{75 m} $$

The relationship of the radius of a circle, \(x\), and the circumference of the circle, \(y\), is a direct variation. The radius of a circle is \(10 \mathrm{~cm}\), and the circumference is \(62.8 \mathrm{~cm}\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this relationship. c. Find the circumference of a circle with a radius of \(20 \mathrm{~cm}\).

For exercises 11-30, (a) solve. (b) check. $$ \frac{3}{10}+\frac{7}{m}=\frac{14}{m}+\frac{1}{15} $$

For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{a^{2}+6 a+5}{6 a}}{\frac{a^{2}-1}{24 a^{4}}} $$

For exercises \(5-48\), simplify. $$ \frac{n^{2}}{n^{2}+3 n+2}-\frac{1}{n^{2}+3 n+2} $$

For exercises 7-32, simplify. $$ \frac{z^{2}-7 z-18}{z^{2}+4 z+4} \cdot \frac{z^{2}-4 z-12}{z^{2}-11 z+18} $$

For exercises 1-66, simplify. $$ \frac{y+9}{y^{2}+9 y} $$

The relationship of the diameter of a circle, \(x\), and the circumference of the circle, \(y\), is a direct variation. The diameter of a circle is \(20 \mathrm{~cm}\), and the circumference is \(62.8 \mathrm{~cm}\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this relationship. c. Find the circumference of a circle with a diameter of \(40 \mathrm{~cm}\).

For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{r^{2}+11 r+24}{9 r}}{\frac{r^{2}-64}{27 r^{3}}} $$

For exercises 13-24, rewrite each expression as an equivalent expression with the given denominator. $$ \frac{4}{15 x+6} ; 12(5 x+2) $$

For exercises \(5-48\), simplify. $$ \frac{u^{2}}{u^{2}+6 u+8}-\frac{4}{u^{2}+6 u+8} $$

For exercises 7-32, simplify. $$ \frac{n^{2}-9 n+18}{n^{2}-6 n+9} \cdot \frac{n^{2}-2 n-3}{n^{2}-9 n+18} $$

If the price per share of a company's stock is constant, the relationship of the earnings per share, \(x\), and the financial ratio price to earnings, \(y\), is an inverse variation. The earnings per share of a company is \(\$ 3.50\), and its price to earnings ratio is 16 . a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the price to earnings ratio when the earnings per share is \(\$ 2\).

For exercises 11-30, (a) solve. (b) check. $$ \frac{15}{4 z}+\frac{2}{3}=\frac{1}{24} $$

For exercises 7-32, simplify. $$ \frac{p^{2}+11 p+18}{p^{2}-2 p-15} \cdot \frac{p^{2}+3 p-40}{p^{2}+10 p+16} $$

For exercises 1-66, simplify. $$ \frac{y^{2}+9 y}{y+9} $$

If the annual credit sales are constant, the relationship of the accounts receivable, \(x\), and the financial ratio receivables turnover, \(y\), is an inverse variation. The accounts receivable of a company are \(\$ 150\) million, and its receivables turnover ratio is 12 . a. Find the constant of proportionality. Include the units of measurement. b. Write an equation that represents this relationship. c. Find the receivables turnover ratio when the accounts receivable are \(\$ 200\) million.

For exercises 13-24, rewrite each expression as an equivalent expression with the given denominator. $$ \frac{3}{x^{3}-7 x^{2}-8 x} ; 6 x^{2}(x-8)(x+1) $$

For exercises 7-32, simplify. $$ \frac{k^{2}+12 k+27}{k^{2}+7 k-18} \cdot \frac{k^{2}+5 k-14}{k^{2}+7 k+12} $$

For exercises 1-66, simplify. $$ \frac{z^{2}+8 z}{z+8} $$

For a fixed length of household copper wire, the relationship of the cross- sectional area, \(x\), and the resistance, \(y\), is an inverse variation. When the cross-sectional area is \(3.14 \times 10^{-6} \mathrm{~m}^{2}\), the resistance is \(5.4 \times 10^{-3} \mathrm{ohm}\). a. Find the constant of proportionality, \(k\). Use scientific notation. Include the units of measurement. b. Write an equation that represents this relationship. c. Find the resistance when the cross-sectional area is \(2.05 \times 10^{-6} \mathrm{~m}^{2}\). Round the mantissa to the nearest tenth.