Chapter 3

Identify an unknown and rewrite each expression as an algebraic expression. Ten more than three times a number.

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$x+9=16$$

Treat the percents given in this exercise as exact numbers, and work to three significant digits. Two different mixtures of gasohol are available, one with \(5 \%\) alcohol and the other containing \(12 \%\) alcohol. How many gallons of the \(12 \%\) mixture must be added to 252 gal of the \(5 \%\) mixture to produce a mixture containing \(9 \%\) alcohol?

A horizontal beam of negligible weight is \(18.0 \mathrm{ft}\) long and is supported by columns at either end. A vertical load of 14,500 lb is applied to the beam at a distance \(x\) from the left end. (a) Find \(x\) so that the reaction at the left column is 10,500 lb. (b) Find the reaction at the right column.

Two planes start from the same city at the same time. One travels at \(252 \mathrm{mi} / \mathrm{h}\) and the other at \(266 \mathrm{mi} / \mathrm{h},\) in the opposite direction. How long will it take for them to be 1750 miles apart?

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$3 x-2=10$$

Treat the percents given in this exercise as exact numbers, and work to three significant digits. How many metric tons of chromium must be added to 2.50 metric tons of stainless steel to raise the percent of chromium from \(11 \%\) to \(18 \% ?\)

A company has \(\$ 86,500\) invested in bonds and earns \(\$ 6751\) in interest annually. Part of the money is invested at \(7.4 \%,\) and the remainder at \(8.1 \%,\) both simple interest. How much is invested at each rate?

The pointer of a certain meter can travel to the right at the rate of \(10.0 \mathrm{cm} / \mathrm{s}\) What must be the minimum return rate if the total time for the pointer to traverse the full 12.0 -cm scale and return to zero must not exceed 2.00 seconds?

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$30+5 x=20 x$$

Treat the percents given in this exercise as exact numbers, and work to three significant digits. How many kilograms of nickel silver alloy containing \(18 \%\) zinc and how many kilograms of nickel silver alloy containing \(31 \%\) zinc must be melted together to produce \(706 \mathrm{kg}\) of a new nickel silver alloy containing \(22 \%\) zinc?

If a carpenter can roof a house in 10 days and another can do the same in 14 days, how many days will it take if they work together?

How much, to the nearest dollar, must a company earn in order to have \(\$ 895,000\) left after paying \(27 \%\) in taxes?

A train travels from \(P\) to \(Q\) at a rate of \(22.5 \mathrm{km} / \mathrm{h}\). After it has been gone 2.75 hours, an express train leaves \(P\) for \(Q\) traveling at \(85.5 \mathrm{km} / \mathrm{h},\) and reaches \(Q\) 1.50 hours ahead of the first train. Find the distance from \(P\) to \(Q,\) and the time taken by the express train.

Treat the percents given in this exercise as exact numbers, and work to three significant digits. A certain bronze alloy containing \(4 \%\) tin is to be added to 351 lb of bronze containing \(18 \%\) tin to produce a new bronze containing \(15 \%\) tin. How many pounds of the \(4 \%\) bronze are required?

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$7 x-29=6$$

A technician can assemble an instrument in 9.5 h. After working for \(2.0 \mathrm{h},\) she is joined by another technician who, alone, could do the job in 7.5 h. How many additional hours are needed to finish the job?

A freight train leaves \(A\) for \(B, 175\) miles away, and travels at the rate of 31.5 mi/h. After 1.50 hours, a train leaves \(B\) for \(A,\) traveling at 21.5 mi/h. How many miles from \(B\) will they meet?

Identify an unknown and rewrite each expression as an algebraic expression. A fraction whose denominator is 4 more than 6 times its numerator.

Treat the percents given in this exercise as exact numbers, and work to three significant digits. How many kilograms of brass containing \(63 \%\) copper must be melted with \(1120 \mathrm{kg}\) of brass containing \(72 \%\) copper to produce a new brass containing \(67 \%\) copper?

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$4 t+9=11 t-3 t$$

A uniform horizontal beam is 9.74 ft long and weighs 386 lb. It is supported by columns at either end. A vertical load of 3814 lb is applied to the beam at a distance \(x\) from the left end. Find \(x\) so that the reaction at the right column is \(2000 \mathrm{lb}\)

A certain screw machine can produce a box of parts in \(3.3 \mathrm{h}\). A new machine is to be ordered having a speed such that both machines working together would produce a box of parts in 1.4 h. How long would it take the new machine alone to produce a box of parts?

Two submarines start from the same spot and travel in opposite directions, one at \(115 \mathrm{km} /\) day and the other at \(182 \mathrm{km} /\) day. How long will it take for the submarines to be \(1470 \mathrm{km}\) apart?

Treat the percents given in this exercise as exact numbers, and work to three significant digits. A certain chain saw requires a fuel mixture of \(5.5 \%\) oil and the remainder gasoline. How many liters of \(2.5 \%\) mixture and how many of \(9.0 \%\) mixture must be combined to produce 40.0 liters of \(5.5 \%\) mixture?

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$20-y=13$$

A uniform horizontal beam is \(19.80 \mathrm{ft}\) long and weighs 1360 lb. It is supported at either end. A vertical load of 13,510 lb is applied to the beam 8.450 ft from the left end. Find the reaction at each end of the beam.

A tank can be filled by a pipe in \(3.0 \mathrm{h}\) and emptied by another pipe in \(4.0 \mathrm{h}\). How much time will be required to fill an empty tank if both are running?

A bus travels \(87.5 \mathrm{km}\) to another town at a speed of \(72.0 \mathrm{km} / \mathrm{h} .\) What must be its return rate if the total time for the round trip is to be 2.50 hours?

Identify an unknown and rewrite each expression as an algebraic expression. The number of gallons of antifreeze in a radiator containing \(x\) gallons of a mixture that is \(11 \%\) antifreeze.

Treat the percents given in this exercise as exact numbers, and work to three significant digits. A certain automobile cooling system contains 11.0 liters of coolant that is \(15 \%\) antifreeze. How many liters of mixture must be removed so that, when it is replaced with pure antifreeze, a mixture of \(25 \%\) antifreeze will result?

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$x+9=5$$

Two pipes empty into a tank. One pipe can fill the tank in \(8.0 \mathrm{h}\), and the other in 9.0 h. How long will it take both pipes together to fill the tank?

A student sold used skis and boots for \(\$ 210,\) getting 4 times as much for the boots as for the skis. What was the price of each?

An oil slick from a runaway offshore oil well is advancing toward a beach 354 miles away at the rate of 10.5 mi/day. Two days after the spill, cleanup ships leave the beach and steam toward the slick at a rate of 525 milday. At what distance from the beach will they reach the slick?

Identify an unknown and rewrite each expression as an algebraic expression. The distance traveled in \(x\) hours by a car going \(78 \mathrm{km} / \mathrm{h}\).

Treat the percents given in this exercise as exact numbers, and work to three significant digits. A vat contains 4110 liters of wine with an alcohol content of \(10 \% .\) How much of this wine must be removed so that, when it is replaced with wine with a \(17 \%\) alcohol content, the alcohol content in the final mixture will be \(12 \% ?\)

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$-6 y-4=2 y$$

A used truck and a snowplow attachment are worth \(\$ 7200,\) the truck being worth 7 times as much as the plow. Find the value of each.

Treat the percents given in this exercise as exact numbers, and work to three significant digits. A certain paint mixture weighing 315 lb contains \(20 \%\) solids suspended in water. How many pounds of water must be allowed to evaporate to raise the concentration of solids to \(25 \% ?\)

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$5 x+8=9 x$$

A person spends \(1 / 4\) of her annual income for board, \(1 / 12\) for clothes, and \(1 / 2\) for other expenses, and saves \(\$ 10,000 .\) What is her income?

Spacecraft \(A\) is over Houston at noon on a certain day and traveling at a rate of \(275 \mathrm{km} / \mathrm{h}\). Spacecraft \(B,\) attempting to overtake and dock with \(A\) is over Houston at 1: 15 P.M. and is traveling in the same direction as \(A\), at \(444 \mathrm{km} / \mathrm{h} .\) At what time will \(B\) overtake \(A ?\) At what distance from Houston?

Solve each problem for the required quantity. Four less than 6 times a number is \(32 .\) Find the number.

Treat the percents given in this exercise as exact numbers, and work to three significant digits. Fifteen liters of fuel containing \(3.2 \%\) oil is available for a certain two- cycle engine. This fuel is to be used for another engine requiring a \(5.5 \%\) oil mixture. How many liters of oil must be added?

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$4 x=3 x-6$$

At what rate must liquid be drained from a tank in order to empty it in \(1.50 \mathrm{h}\) if the tank takes \(4.70 \mathrm{h}\) to fill at the rate of \(3.50 \mathrm{m}^{3} / \mathrm{min} ?\)

Solve each problem for the required quantity. Find a number such that the sum of 16 and that number is 3 times that number.

Treat the percents given in this exercise as exact numbers, and work to three significant digits. A concrete mixture is to be made which contains \(35 \%\) sand by weight, and 642 lb of mixture containing \(29 \%\) sand is already on hand. How many pounds of sand must be added to this mixture to arrive at the required \(35 \% ?\)

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$10-5 y=1-y$$