Chapter 11

When is the LCM of two expressions equal to their product?

A backpacker hiking into a wilderness area walked \(9 \mathrm{mi}\) at a constant rate and then reduced this rate by \(1 \mathrm{mph}\). Another \(4 \mathrm{mi}\) was hiked at the reduced rate. The time required to hike the 4 mi was 1 h less than the time required to walk the 9 mi. Find the rate at which the hiker walked the first \(9 \mathrm{mi}\).

Simplify. $$3+\frac{x-1}{x+1}$$

Multiply. $$\frac{y^{2}+y-20}{y^{2}+2 y-15} \cdot \frac{y^{2}+4 y-21}{y^{2}+3 y-28}$$

Match the polynomials with their LCM. An LCM may be used more than once. a. \(x^{2}-4\) and \(x^{2}+3 x+2\) b. \(x+3\) and \(x^{2}+5 x+6\) c. \(x^{2}-x-2\) and \(x^{2}+2 x+1\) d. \(x-4\) and \(x^{2}-1\) e. \(2-x\) and \(x^{2}+3 x+2\) f. \(4-x\) and \(x^{2}-1\) g. \(x-4\) and \(1-x^{2}\) h. \(2+x-x^{2}\) and \((x+1)^{2}\) i. \(\quad(x+3)(x+2)\) ii. \(\quad(x-4)(x+1)(x-1)\) iii. \((x+2)(x-2)(x+1)\) iv. \((x-2)(x+1)(x+1)\)

An express train traveled \(600 \mathrm{mi}\) in the same amount of time it took a freight train to travel \(360 \mathrm{mi}\). The rate of the express train was \(20 \mathrm{mph}\) faster than the rate of the freight train. Find the rate of each train.

Simplify. $$\frac{x+3}{6 x}-\frac{x-3}{8 x^{2}}$$

Multiply. $$\frac{x^{2}-3 x-4}{x^{2}+6 x+5} \cdot \frac{x^{2}+5 x+6}{8+2 x-x^{2}}$$

A twin-engine plane flies \(800 \mathrm{mi}\) in the same amount of time it takes a single-engine plane to fly 600 mi. The rate of the twin-engine plane is 50 mph faster than the rate of the single-engine plane. Find the rate of the twin-engine plane.

Simplify. $$\frac{x+2}{x y}-\frac{3 x-2}{x^{2} y}$$

Multiply. $$\frac{25-n^{2}}{n^{2}-2 n-35} \cdot \frac{n^{2}-8 n-20}{n^{2}-3 n-10}$$

A small motor on a fishing boat can move the boat at a rate of \(6 \mathrm{mph}\) in calm water. Traveling with the current, the boat can travel \(24 \mathrm{mi}\) in the same amount of time it takes to travel 12 mi against the current. Find the rate of the current.

Simplify. $$\frac{3 x-1}{x y^{2}}-\frac{2 x+3}{x y}$$

Multiply. $$\frac{16+6 x-x^{2}}{x^{2}-10 x-24} \cdot \frac{x^{2}-6 x-27}{x^{2}-17 x+72}$$

A car is traveling at a rate that is 36 mph faster than the rate of a cyclist. The car travels 384 mi in the same amount of time it takes the cyclist to travel 96 mi. Find the rate of the car.

Simplify. $$\frac{4 x-3}{3 x^{2} y}+\frac{2 x+1}{4 x y^{2}}$$

Multiply. $$\frac{x^{2}-11 x+28}{x^{2}-13 x+42} \cdot \frac{x^{2}+7 x+10}{20-x-x^{2}}$$

A commercial jet can fly 550 mph in calm air. Traveling with the jet stream, the plane can fly \(2400 \mathrm{mi}\) in the same amount of time it takes to fly \(2000 \mathrm{mi}\) against the jet stream. Find the rate of the jet stream.

Simplify. $$\frac{5 x+7}{6 x y^{2}}-\frac{4 x-3}{8 x^{2} y}$$

Multiply. $$\frac{2 x^{2}+5 x+2}{2 x^{2}+7 x+3} \cdot \frac{x^{2}-7 x-30}{x^{2}-6 x-40}$$

A cruise ship can sail 28 mph in calm water. Sailing with the Gulf Stream, the ship can sail \(170 \mathrm{mi}\) in the same amount of time it takes to sail \(110 \mathrm{mi}\) against the Gulf Stream. Find the rate of the Gulf Stream.

Simplify. $$\frac{x-2}{8 x^{2}}-\frac{x+7}{12 x y}$$

Multiply. $$\frac{x^{2}-4 x-32}{x^{2}-8 x-48} \cdot \frac{3 x^{2}+17 x+10}{3 x^{2}-22 x-16}$$

Rowing with the current of a river, a rowing team can row \(25 \mathrm{mi}\) in the same amount of time it takes to row 15 mi against the current. The rate of the rowing team in calm water is 20 mph. Find the rate of the current.

Simplify. $$\frac{3 x-1}{6 y^{2}}-\frac{x+5}{9 x y}$$

A plane can fly 180 mph in calm air. Flying with the wind, the plane can fly 600 mi in the same amount of time it takes to fly 480 mi against the wind. Find the rate of the wind.

Simplify. $$\frac{4}{x-2}+\frac{5}{x+3}$$

Use the product \(\frac{x^{a}}{y^{b}} \cdot \frac{y^{t}}{x^{d}},\) where \(a, b, c,\) and \(d\) are all positive integers. If \(a>d\) and \(b>c,\) which variable appears in the denominator of the simplified product?

Work Problem One pipe can fill a tank in \(2 \mathrm{h}\), a second pipe can fill the tank in 4 \(\mathrm{h}\), and a third pipe can fill the tank in \(5 \mathrm{h}\). How long would it take to fill the tank with all three pipes operating?

Simplify. $$\frac{2}{x-3}+\frac{5}{x-4}$$

Work Problem A mason can construct a retaining wall in 10 h. The mason's experienced apprentice can do the same job in 15 h. How long would it take the mason's novice apprentice to do the job if, working together, all three can complete the wall in 5 h?

Simplify. $$\frac{6}{x-7}-\frac{4}{x+3}$$

Divide. $$\frac{4 x^{2} y^{3}}{15 a^{2} b^{3}} \div \frac{6 x y}{5 a^{3} b^{5}}$$

Uniform Motion An Outing Club traveled \(18 \mathrm{mi}\) by canoe and then hiked \(3 \mathrm{mi}\). The rate by canoe was three times the rate on foot. The time spent walking was 1 h less than the time spent canoeing. Find the amount of time spent traveling by canoc.

Simplify. $$\frac{3}{y+6}-\frac{4}{y-3}$$

Divide. $$\frac{9 x^{3} y^{4}}{16 a^{4} b^{2}} \div \frac{45 x^{4} y^{2}}{14 a^{7} b}$$

Uniform Motion A motorist drove 120 mi before running out of gas and walking 4 mi to a gas station. The motorist's driving rate was ten times the walking rate. The time spent walking was 2 h less than the time spent driving. How long did it take for the motorist to drive the \(120 \mathrm{mi}\) ?

Simplify. $$\frac{2 x}{x+1}+\frac{1}{x-3}$$

Divide. $$\frac{6 x-12}{8 x+32} \div \frac{18 x-36}{10 x+40}$$

Uniform Motion Because of bad weather, a bus driver reduced the usual speed along a 150 -mile bus route by \(10 \mathrm{mph}\). The bus arrived only \(30 \mathrm{min}\) later than its usual arrival time. How fast does the bus usually travel?

Simplify. $$\frac{3 x}{x-4}+\frac{2}{x+6}$$

Divide. $$\frac{28 x+14}{45 x-30} \div \frac{14 x+7}{30 x-20}$$

Work Problem A construction project must be completed in 15 days. Twenty-five workers did one-half of the job in 10 days. Working at the same rate, how many workers are needed to complete the job on schedule?

Simplify. $$\frac{4 x}{2 x-1}-\frac{5}{x-6}$$

Divide. $$\frac{6 x^{3}+7 x^{2}}{12 x-3} \div \frac{6 x^{2}+7 x}{36 x-9}$$

Simplify. $$\frac{6 x}{x+5}-\frac{3}{2 x+3}$$

Divide. $$\frac{5 a^{2} y+3 a^{2}}{2 x^{3}+5 x^{2}} \div \frac{10 a y+6 a}{6 x^{3}+15 x^{2}}$$

Divide. $$\frac{x^{2}+4 x+3}{x^{2} y} \div \frac{x^{2}+2 x+1}{x y^{2}}$$

Simplify. $$\frac{4 x}{6-x}+\frac{5}{x-6}$$

Divide. $$\frac{x^{3} y^{2}}{x^{2}-3 x-10} \div \frac{x y^{4}}{x^{2}-x-20}$$