Chapter 2

Last week, the ratio of the number of hours that Joseph worked to the number of hours that Nicole worked was \(2 : 3 .\) This week Joseph worked 4 hours more than last week and Nicole worked twice as many hours as last week. This week the ratio of the hours Joseph worked to the number of hours Nicole worked is \(1 : 2 .\) How many hours did each person work each week?

In \(13-24,\) divide and express each quotient in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{4 b+12}{b} \div(b+3) $$

The ratio of the length to the width of a rectangle is 7 : 3. The area of the rectangle is 336 square centimeters. What are the dimensions of the rectangle?

Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. \(\frac{3 x y}{9 x y+6 x^{2} y^{3}}\)

In \(13-22,\) write each decimal as a common fraction. $$ 0.1 \overline{36} $$

Simplify each expression. In each case, list any values of the variables for which the fractions are not defined. \(\frac{\frac{a}{4}+\frac{7 a}{8}}{\frac{6 a^{2}}{5}-\frac{3 a^{2}}{10}}+\frac{3}{a}\)

In \(21-24,\) the length and width of a rectangle are expressed in terms of a variable. a. Express each perimeter in terms of the variable. b. Express each area in terms of the variable. $$ l=3 x+3 \text { and } w=\frac{1}{3} $$

Anthony rode his bicycle to his friend's house, a distance of 1 mile. Then his friend's mother drove them to school, a distance of 12 miles. His friend's mother drove at a rate that is 25 miles per hour faster than Anthony rides his bike. If it took Anthony \(\frac{3}{5}\) of an hour to get to school, at what average rate does he ride his bicycle? (Use distance \(=\) time for each part of the trip to school.)

The basketball team has played 21 games. The ratio of wins to losses is 5 : 2. How many games has the team won?

Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. \(\frac{2 a+10}{3 a+15}\)

In \(13-22,\) write each decimal as a common fraction. $$ 0.15 \overline{90} $$

Simplify each expression. In each case, list any values of the variables for which the fractions are not defined. \(\left(\frac{3}{a}+\frac{5}{a^{2}}\right) \div\left(\frac{10}{a}+6\right)+\frac{3}{4}\)

In \(21-24,\) the length and width of a rectangle are expressed in terms of a variable. a. Express each perimeter in terms of the variable. b. Express each area in terms of the variable. $$ l=\frac{x}{x-1} \text { and } w=\frac{3}{x-1} $$

Amanda drove 40 miles. Then she increased her rate of speed by 10 miles per hour and drove another 40 miles to reach her destination. If the trip took 1\(\frac{4}{5}\) hours, at what rate did Amanda drive?

In \(13-24,\) divide and express each quotient in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ (2 x+7) \div \frac{1}{2 x^{2}+5 x-7} $$

In the chess club, the ratio of boys to girls is 6 : 5. There are 3 more boys than girls in the club. How many members are in the club?

Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. \(\frac{4 a^{2}-16}{4 a+8}\)

Simplify each expression. In each case, list any values of the variables for which the fractions are not defined. \(\left(6+\frac{12}{b}\right) \div\left(3 b-\frac{12}{b}\right)+\frac{b}{2-b}\)

In \(21-24,\) the length and width of a rectangle are expressed in terms of a variable. a. Express each perimeter in terms of the variable. b. Express each area in terms of the variable. $$ l=\frac{x}{x+1} \text { and } w=\frac{x}{x+2} $$

Last week, Emily paid \(\$ 8.25\) for \(x\) pounds of apples. This week she paid \(\$ 9.50\) for \((x+1)\) pounds of apples. The price per pound was the same each week. How many pounds of apples did Emily buy each week and what was the price per pound? (Use \(\frac{\text { total cost }}{\text { number of pounds }}=\) cost per pound for each week.)

In \(13-24,\) divide and express each quotient in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \left(a^{2}-1\right) \div \frac{2 a+2}{a} $$

Every year, Javier makes a total contribution of \(\$ 125\) to two local charities. The two donations are in the ratio of \(3 : 2 .\) What contribution does Javier make to each charity?

Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. \(\frac{x^{2}-7 x+12}{x^{2}+2 x-15}\)

In \(25-30,\) perform the indicated operations and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{3}{5} \times \frac{5}{9} \div \frac{4}{3} $$

A cookie recipe uses flour and sugar in the ratio of 9 : 4. If Nicholas uses 1 cup of sugar, how much flour should he use?

Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. \(\frac{5 y^{2}-20}{y^{2}+4 y+4}\)

In \(25-30,\) perform the indicated operations and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{3 x}{x-1} \cdot \frac{x^{2}-1}{x} \div \frac{x+1}{3} $$

The directions on a bottle of cleaning solution suggest that the solution be diluted with water. The ratio of solution to water is 1 : 7. How many cups of solution and how many cups of water should Christopher use to make 2 gallons (32 cups) of the mixture?

Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. \(\frac{-7+7 a}{21 a^{2}-21}\)

In \(25-30,\) perform the indicated operations and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{2 a}{a+2} \cdot \frac{a^{2}-4}{4 a^{2}} \div \frac{a-2}{a} $$

Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. \(\frac{a^{3}-a^{2}-a+1}{a^{2}-2 a+1}\)

In \(25-30,\) perform the indicated operations and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \left(x^{2}-2 x+1\right) \div \frac{x-1}{3} \cdot \frac{x+4}{3 x} $$

Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. \(\frac{3-(b+1)}{4-b^{2}}\)

In \(25-30,\) perform the indicated operations and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ (3 b)^{2} \div \frac{3 b}{b+2} \cdot \frac{2 b+4}{b} $$

Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. \(\frac{4-2(x-1)}{x^{2}-6 x+9}\)

In \(25-30,\) perform the indicated operations and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{x^{2}-3 x+2}{4 x} \cdot \frac{12 x^{2}}{x^{2}-2 x} \div \frac{x-1}{x} $$

Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined. \(\frac{5(1-b)+15}{b^{2}-16}\)