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Q:

How long will it take $$\ 1500$$ to double itself at $$6 \%$$ sim- ple interest?

Q:

A pharmacist makes eye drops with saline solution. A $$10 \%$$ saline solution mixed with an $$80 \%$$ saline solution makes $$7$$ cups of a $$60 \%$$ saline solution. How many cups of the $$80 \%$$ solution should be used?

Q:

Action Toy Company makes action figures once a month. Producing the cricket action figure requires $$3$$ hours of time on Machine I and $$9$$ hours on Machine II. Producing the beetle action figure requires $$8$$ hours of time on Machine I and $$4$$ hours on Machine II. Machine I is only available for $$240$$ hours per month, but Machine II is available for $$360$$ hours per month. The cricket action figure can be sold at a profit of $$\ 2.90$$ , and the beetle action figure can be sold at a profit of $$\ 3.50$$ . How many of each action figure should the company produce each month to maximize its profit?

Q:

The square of a number equals seven times the num- ber. Find the number.

Q:

If $$\ 3500$$ is invested at $$7.5 \%$$ interest compounded quar- terly, how much money has accumulated at the end of $$8$$ years?

Q:

A box contains $$\ 17.70$$ in nickels, dimes, and quarters. The number of dimes is $$8$$ less than twice the number of nickels. The number of quarters is $$2$$ more than the sum of the numbers of nickels and dimes. How many coins of each kind are there in the box?

Q:

Suppose that Maria has $$150$$ coins consisting of pen- nies, nickels, and dimes. The number of nickels she has is $$10$$ less than twice the number of pennies; the num- ber of dimes she has is $$20$$ less than three times the number of pennies. How many coins of each kind does she have?

Q:

A strip of uniform width is to be cut off both sides and both ends of a sheet of paper that is $$8$$ inches by $$14$$ inches to reduce the size of the paper to an area of $$72$$ square inches. Find the width of the strip.

Q:

Two planes leave Kansas City at the same time and fly in opposite directions. If one travels at $$450$$ miles per hour and the other travels at $$400$$ miles per hour, how long will it take for them to be $$3400$$ miles apart?

Q:

Suppose that the equation $$p ( x ) = - 2 x ^ { 2 } + 280 x - 1000$$ , where $$x$$ represents the number of items sold, describes the profit function for a certain business. How many items should be sold to maximize the profit?

Q:

One angle of a triangle has a measure of $$60 ^ { \circ }$$ , and the measures of the other two angles are in the ratio of $$2$$ to 3. Find the measures of the other two angles.

Q:

The sum of the present ages of Angie and her mother is $$64$$ years. In eight years Angie will be three-fifths as old as her mother at that time. Find the present ages of Angie and her mother.

Q:

Suppose that a certain radioactive substance has a half- life of $$40$$ days. If there are presently $$750$$ grams of the substance, how much, to the nearest gram, will remain after $$100$$ days?

Q:

The measure of the smallest angle in a triangle is $$5 ^ { \circ }$$ less than the measure of the middle angle. Three times the measure of the middle angle is $$15 ^ { \circ }$$ more than twice the measure of the largest angle. Find the measure of each angle of the triangle.

Q:

Two trains leave the same depot at the same time, one traveling east and the other west. At the end of $$4 \frac { 1 } { 2 }$$ hours, the trains are $$639$$ miles apart. If the rate of the train trav- eling east is $$10$$ miles per hour faster than the rate of the other train, find their rates.

Q:

Marsha bowled $$142$$ and $$170$$ in her first two games. What must she bowl in the third game to have an aver- age of at least $$160$$ for the three games?

Q:

The width of a rectangle is $$2$$ meters more than one- third of the length. The perimeter of the rectangle is $$44$$ meters. Find the length and width of the rectangle.

Q:

A pharmacist has a $$6 \%$$ solution of cough syrup and a $$14 \%$$ solution of the same cough syrup. How many ounces of each must be mixed to make $$16$$ ounces of a $$10 \%$$ solution of cough syrup?

Q:

Suppose that a highway rises $$200$$ feet in a horizontal distance of $$3000$$ feet. Express the grade of the highway to the nearest tenth of a percent.

Q:

A crime-scene investigator has $$3.4$$ ounces of a sample. He needs to conduct four tests that each require $$0.6$$ ounces of the sample, and one test that requires $$0.8$$ ounces of the sample. How much of the sample remains after he uses it for the five tests?

Q:

A sum of $$\ 95,000$$ is split between two investments, one paying $$3 \%$$ and the other $$5 \%$$ . If the total yearly interest amounted to $$\ 3910$$ , how much was invested at $$5 \%$$ ?

Q:

Strontium- $$90$$ has a half-life of $$29$$ years. If there are $$400$$ grams of strontium- $$90$$ initially, how much, to the nearest gram, will remain after $$87$$ years? After $$100$$ years?

Q:

The combined area of a square and a rectangle is $$57$$ square feet. The width of the rectangle is $$3$$ feet more than the length of a side of the square, and the length of the rectangle is $$5$$ feet more than the length of a side of the square. Find the length of the rectangle.

Q:

On a $$195$$ -mile trip from Pensacola to Tallahassee, Florida, Shanna drove $$10$$ miles per hour slower than she did on her $$100$$ -mile trip from Ocala to Orlando, Florida. The Tallahassee trip took $$1$$ hour $$40$$ minutes longer than the Orlando trip. How fast did Shanna drive on her Tallahassee trip?

Q:

Use the formula $$A = P ( 1 + \frac { r } { n } ) ^ { n t }$$ to calculate the amount of money accumulated when investing $$\ 1000$$ for $$5$$ years at $$4 \%$$ interest compounded:

(a) Annually

(b) Semi-annually

(c) Quarterly

(d) Monthly (e) Continuously (Hint: Use $$A = P e ^ { r t }$$ )

Q:

If a ring costs a jeweler $$\ 1200$$ , at what price should it be sold to yield a profit of $$50 \%$$ on the selling price?

Q:

The selling price of a ring is $$\ 750$$ . This represents $$\ 150$$ less than three times the cost of the ring. Find the cost of the ring.

Q:

Suppose that the width of a certain rectangle is $$1$$ inch more than one-fourth of its length. The perimeter of the rectangle is $$42$$ inches. Find the length and width of the rectangle.

Q:

Carolyn has $$19$$ bills consisting of ten-dollar bills, twenty- dollar bills, and fifty-dollar bills. The number of twenties is three times the number of tens, and the number of fifties is one less than the number of tens. How many of each bill does Carolyn have? How much money does she have?

Q:

Suppose that the present population of a city is $$50,000$$ . Use the equation $$P ( t ) = P _ { 0 } e ^ { 0.02 t }$$ (in which $$P _ { 0 }$$ represents an initial population) to estimate future populations, and estimate the population of that city in $$10$$ years, $$15$$ years, and $$20$$ years.

Q:

The measure of the largest angle of a triangle is $$20 ^ { \circ }$$ more than the sum of the measures of the other two angles. The difference in the measures of the largest and smallest angles is $$65 ^ { \circ }$$ . Find the measure of each angle.

Q:

If the ratio of rise to run is to be $$\frac { 2 } { 3 }$$ for some steps, and the run is $$28$$ centimeters, find the rise to the nearest centimeter.

Q:

A $$10$$ -quart radiator contains a $$50 \%$$ solution of antifreeze. How much needs to be drained out and replaced with pure antifreeze to obtain a $$70 \%$$ antifreeze solution?

Q:

A car wash charges $$\ 5.00$$ for an express wash and $$\ 15.00$$ for a full wash. On a recent day there were $$75$$ car washes of these two types, which brought in $$\ 825.00$$ . Find the number of express washes.

Q:

Sydney's present age is one-half of Marcus's present age. In $$12$$ years, Sydney's age will be five-eighths of Marcus's age. Find their present ages.

Q:

How long will it take $$\ 1000$$ to be worth $$\ 3500$$ if it is invested at $$4.5 \%$$ interest compounded quarterly?

Q:

Aura took three biology exams and has an average score of $$88$$ . Her second exam score was $$10$$ points bet- ter than her first, and her third exam score was $$4$$ points better than her second exam. What were her three exam scores?

Q:

A sum of $$\ 2450$$ is to be divided between two people in the ratio of $$3$$ to $$4$$ . How much does each person receive?

Q:

A parabolic arch spans a stream $$200$$ feet wide. How high above the stream must the arch be to give a mini- mum clearance of $$40$$ feet over a channel in the center that is $$120$$ feet wide?

Q:

Debbie rode her bicycle out into the country for a dis- tance of $$24$$ miles. On the way back, she took a much shorter route of $$12$$ miles and made the return trip in one-half hour less time. If her rate out into the country was $$4$$ miles per hour greater than her rate on the return trip, find both rates.

Q:

The weight of a body above the surface of the earth varies inversely as the square of its distance from the center of the earth. Assuming the radius of the earth to be $$4000$$ miles, determine how much a man would weigh $$1000$$ miles above the earth's surface if he weighs $$200$$ pounds on the surface.

Q:

A video rental service has $$1000$$ subscribers, each of whom pays $$\ 15$$ per month. On the basis of a survey, the company believes that for each decrease of $$\ 0.25$$ in the monthly rate, it could obtain $$20$$ additional subscribers. At what rate will the maximum revenue be obtained, and how many subscribers will there be at that rate?

Q:

From the list $$0 , \sqrt { 2 } , \frac { 3 } { 4 } , - \frac { 5 } { 6 } , \frac { 25 } { 3 } , - \sqrt { 3 } , - 8,0.34,0.2 \overline { 3 }$$ , $$67$$ , and $$\frac { 9 } { 7 }$$ , identify each of the following.

(a) The natural numbers

(b) The integers

(c) The nonnegative integers

(d) The rational numbers (e) The irrational numbers For Problems $$2 - 10$$ , state the property of equality or the property of real numbers that justifies each of the state- ments. For example, $$6 ( - 7 ) = - 7 ( 6 )$$ because of the com- mutative property of multiplication; and if $$2 = x + 3$$ , then $$x + 3 = 2$$ is true because of the symmetric property of equality.

Q:

The equation $$P ( a ) = 14.7 e ^ { - 0.21 a }$$ , in which $$a$$ is the alti- tude above sea level measured in miles, yields the atmospheric pressure in pounds per square inch. If the atmospheric pressure at Cheyenne, Wyoming, is approximately $$11.53$$ pounds per square inch, find that city's altitude above sea level. Express your answer to the nearest hundred feet.

Q:

The equation $$A ( r ) = \pi r ^ { 2 }$$ expresses the area of a circu- lar region as a function of the length of a radius $$( r )$$ . Compute $$A ( 2 ) , A ( 3 ) , A ( 12 )$$ , and $$A ( 17 )$$ and express your answers to the nearest hundredth.

Q:

Working together, Sue and Dean can complete a task in $$1 \frac { 1 } { 5 }$$ hours. Dean can do the task by himself in $$2$$ hours. How long would it take Sue to complete the task by herself?

Q:

The number of bacteria present in a certain culture after $$t$$ hours is given by the equation $$Q = Q _ { 0 } e ^ { 0.3 t }$$ , where $$Q _ { 0 }$$ represents the initial number of bacteria. If $$6640$$ bacte- ria are present after $$4$$ hours, how many bacteria were present initially?

Q:

We have used the following two multiplication patterns. $$( a + b ) ^ { 2 } = a ^ { 2 } + 2 a b + b ^ { 2 }$$ $$( a + b ) ^ { 3 } = a ^ { 3 } + 3 a ^ { 2 } b + 3 a b ^ { 2 } + b ^ { 3 }$$

Q:

Melinda invested three times as much money at $$6 \%$$ yearly interest as she did at $$4 \%$$ . Her total yearly inter- est from the two investments was $$\ 110$$ . How much did she invest at each rate?

Q:

A company uses $$7$$ pounds of fertilizer for a lawn that measures $$5000$$ square feet and $$12$$ pounds for a lawn that measures $$10,000$$ square feet. Let $$y$$ represent the pounds of fertilizer and $$x$$ the square footage of the lawn.

Q:

Mitsuko's salary for next year is $$\ 44,940$$ . This rep- resents a $$7 \%$$ increase over this year's salary. Find Mitsuko's present salary.

Q:

Victor can rake the lawn in $$20$$ minutes, and his sister Lucia can rake the same lawn in $$30$$ minutes. How long will it take them to rake the lawn if they work together?

Q:

René can ride her bike $$60$$ miles in $$1$$ hour less time than it takes Sue to ride $$60$$ miles. René's rate is $$3$$ miles per hour faster than Sue's rate. Find René's rate.

Q:

An apartment complex contains $$230$$ apartments, each having one, two, or three bedrooms. The number of two-bedroom apartments is $$10$$ more than three times the number of three-bedroom apartments. The number of one-bedroom apartments is twice the number of two- bedroom apartments. How many apartments of each kind are in the complex?

Q:

The sum of the two smallest angles of a triangle is $$40 ^ { \circ }$$ less than the other angle. The sum of the smallest and largest angles is twice the other angle. Find the measures of the three angles of the triangle.

Q:

How many pints of a $$1 \%$$ hydrogen peroxide solution should be mixed with a $$4 \%$$ hydrogen peroxide solution to obtain $$10$$ pints of a $$2 \%$$ hydrogen peroxide solution?

Q:

Eric has a collection of $$63$$ coins consisting of nickels, dimes, and quarters. The number of dimes is $$6$$ more than the number of nickels, and the number of quarters is $$1$$ more than twice the number of nickels. How many coins of each kind are in the collection?

Q:

Suppose that ABC Car Rental agency charges a fixed amount per day plus an amount per mile for renting a car. Heidi rented a car one day and paid $$\ 80$$ for $$200$$ miles. On another day she rented a car from the same agency and paid $$\ 117.50$$ for $$350$$ miles. Determine the linear function the agency uses to calculate its daily rental charges.

Q:

Suppose that the length of a certain rectangle is $$2$$ meters less than four times its width. The perimeter of the rec- tangle is $$56$$ meters. Find the length and width of the rec- tangle.

Q:

Natasha recorded the daily gains or losses for her compa- ny stock for a week. On Monday it gained $$1.25$$ dollars; on Tuesday it gained $$0.88$$ dollar; on Wednesday it lost $$0.50$$ dollar; on Thursday it lost $$1.13$$ dollars; on Friday it gained $$0.38$$ dollar. What was the net gain (or loss) for the week?

Q:

The sum of the areas of two squares is $$89$$ square centimeters. The length of a side of the larger square is $$3$$ centimeters more than the length of a side of the smaller square. Find the dimensions of each square.

Q:

The freight charged by a trucking firm for a parcel under $$200$$ pounds depends on the distance it is being shipped. To ship a $$150$$ -pound parcel $$300$$ miles, it costs $$\ 40$$ . If the same parcel is shipped $$1000$$ miles, the cost is $$\ 180$$ . Assume the relationship between the cost and distance is linear. Find the equation for the relationship. Let $$y$$ be the cost and $$x$$ be the miles. Write the equation in slope-in- tercept form.

Q:

A sum of $$\ 68,000$$ is to be divided between two partners in the ratio of $$\frac { 1 } { 4 }$$ . How much does each person receive?

Q:

A manufacturer of electric ice cream freezers makes a profit of $$\ 4.50$$ on a one-gallon freezer and a profit of $$\ 5.25$$ on a two-gallon freezer. The company wants to produce at least $$75$$ one-gallon and at least $$100$$ two- gallon freezers per week. However, the weekly produc- tion is not to exceed a total of $$250$$ freezers. How many freezers of each type should be produced per week in order to maximize the profit?

Q:

Sophie leaves Camp Tesomas paddling a kayak down- river, with the current, at the same time that Finn leaves the camp paddling a canoe upstream, against the current. Finn paddles for $$1$$ hour at half Sophie's rate for $$1$$ mile, and Sophie paddles for $$3$$ hours for $$15$$ miles. Find the rate of the current, Sophie's rate, and Finn's rate.

Q:

"All Items $$20 \%$$ Off Marked Price" is a sign at a local golf pro shop. Create a function and then use it to determine how much one has to pay for each of the fol- lowing marked items: a $$\ 9.50$$ hat, a $$\ 15$$ umbrella, a $$\ 75$$ pair of golf shoes, a $$\ 12.50$$ golf glove, a $$\ 750$$ set of golf clubs.

Q:

Suppose that an investor wants to invest up to $$\ 10,000$$ . She plans to buy one speculative type of stock and one conservative type. The speculative stock is paying a $$12 \%$$ return, and the conservative stock is paying a $$9 \%$$ return. She has decided to invest at least $$\ 2000$$ in the conservative stock and no more than $$\ 6000$$ in the spec- ulative stock. Furthermore, she does not want the spec- ulative investment to exceed the conservative one. How much should she invest at each rate to maximize her return?

Q:

A gift store is making a mixture of almonds, pecans, and peanuts, which sells for $$\ 6.50$$ per pound, $$\ 8.00$$ per pound, and $$\ 4.00$$ per pound, respectively. The store- keeper wants to make $$20$$ pounds of the mix to sell at $$\ 5.30$$ per pound. The number of pounds of peanuts is to be three times the number of pounds of pecans. Find the number of pounds of each to be used in the mixture.

Q:

Part of $$\ 3000$$ is invested at $$4 \%$$ , another part at $$5 \%$$ , and the remainder at $$6 \%$$ yearly interest. The total yearly income from the three investments is $$\ 160$$ . The sum of the amounts invested at $$4 \%$$ and $$5 \%$$ equals the amount invested at $$6 \%$$ . How much is invested at each rate?

Q:

One of two supplementary angles is $$4 ^ { \circ }$$ more than one- third of the other angle. Find the measure of each of the angles.

Q:

The denominator of a rational number is $$9$$ less than three times the numerator. The number in simplest form is $$\frac { 3 } { 8 }$$ . Find the number.

Q:

The time needed to install computer cables has a linear relationship with the number of feet of cable being installed. It takes $$1 \frac { 1 } { 2 }$$ hours to install $$300$$ feet, and $$1050$$ feet can be installed in $$4$$ hours. Find the equation for the relationship. Let $$y$$ be the feet of cable installed and $$x$$ be the time in hours. Write the equation in slope- intercept form.

Q:

A $$20$$ -foot board is to be cut into two pieces whose lengths are in the ratio of $$7$$ to $$3$$ . Find the lengths of the two pieces.

Q:

It takes Jodi three times as long to deliver papers as it does Jannie. Together they can deliver the papers in $$15$$ minutes. How long would it take Jodi by herself?

Q:

It takes Amy twice as long to deliver papers as it does Nancy. How long would it take each girl to deliver the papers by herself if they can deliver the papers together in $$40$$ minutes?

Q:

A collection of $$70$$ coins consisting of dimes, quarters, and half-dollars has a value of $$\ 17.75$$ . There are three times as many quarters as dimes. Find the number of each kind of coin.

Q:

The cost of labor varies jointly as the number of work- ers and the number of days that they work. If it costs $$\ 900$$ to have $$15$$ people work for $$5$$ days, how much will it cost to have $$20$$ people work for $$10$$ days?

Q:

Kent drives his Mazda $$270$$ miles in the same time that it takes Dave to drive his Nissan $$250$$ miles. If Kent averages $$4$$ miles per hour faster than Dave, find their rates.

Q:

The area of a triangular sheet of paper is $$28$$ square inches. One side of the triangle is $$2$$ inches more than three times the length of the altitude to that side. Find the length of that side and the altitude to the side.

Q:

The perimeter of a rectangle is $$32$$ meters, and its area is $$48$$ square meters. Find the length and width of the rectangle.

Q:

Suppose that a highway rises a distance of $$215$$ feet in a horizontal distance of $$2640$$ feet. Express the grade of the highway to the nearest tenth of a percent.

Q:

What rate of interest compounded continuously is needed for an investment of $$\ 500$$ to grow to $$\ 900$$ in $$10$$ years?

Q:

A sum of $$\ 1750$$ is to be divided between two people in the ratio of $$3$$ to $$4$$ . How much does each person receive?

Q:

Suppose that an arch is shaped like a parabola. It is $$20$$ feet wide at the base and $$100$$ feet high. How wide is the arch $$50$$ feet above the ground?

Q:

Three-fourths of the sum of a number and $$12$$ For Problems $$55 - 64$$ , answer the question with an algebraic expression.

Q:

Neglecting air resistance, the height of a projectile fired vertically into the air at an initial velocity of $$96$$ feet per second is a function of time $$x$$ and is given by the equa- tion $$f ( x ) = 96 x - 16 x ^ { 2 }$$ . Find the highest point reached by the projectile.

Q:

A motel rents double rooms at $$\ 100$$ per day and single rooms at $$\ 75$$ per day. If $$23$$ rooms were rented one day for a total of $$\ 2100$$ , how many rooms of each kind were rented?

Q:

A rectangle is twice as long as it is wide, and its area is $$50$$ square meters. Find the length and the width of the rectangle.

Q:

For week $$1$$ of a weight loss competition, Team A had three members lose $$8$$ pounds each, two members lose $$5$$ pounds each, one member loses $$4$$ pounds, and two members gain $$3$$ pounds. What was the total weight loss for Team A in the first week of the competition?

Q:

A pump removes $$\frac { 1 } { 2 }$$ of the liquid in a container with each stroke. What fraction of the liquid is left in the container after seven strokes?

Q:

Qihong paddled a canoe $$10$$ mi upstream and then paddled back to the starting point. If the rate of the current was $$3 mph$$ and the entire trip took $$3 \frac { 1 } { 2 } hr$$ , what was Qihong's rate?

Q:

In a recent year, approximately $$76$$ million people from other countries visited the United States. The circle graph shows the distribution of these interna- tional visitors by country or region. Use the graph to work each problem. 127. How many travelers visited the United States from Canada? 128. How many travelers visited the United States to the from Mexico? 129. What percent of travelers visited the United States from places other than Canada, Mexico, Europe, and Asia? (Hint: The sum of the parts of the graph must equal $$1$$ whole, that is, 100\%.) 130. How many travelers visited the United States from places other than Canada, Mexico, Europe, and Asia?

Q:

The product of the lesser two of three consecutive integers is equal to $$23$$ plus the greatest. Find the integers.

Q:

(a) What is the equation of the line on which segment $$P Q$$ lies?

(b) Let $$x = 2015$$ in the equation from part (a), and solve for $$y$$ . How does the result compare with the actual figure of $$\ 8.43$$ ?

Q:

A one-cup serving of orange juice contains $$3$$ mg less than four times the amount of vitamin $$C$$ as a one-cup serving of pineapple juice. Servings of the two juices contain a total of $$122 mg$$ of vitamin C. How many milligrams of vitamin C are in a serving of each type of juice? (Data from U.S. Agriculture Department.)

Q:

In one day, a store sold $$\frac { 2 } { 3 }$$ as many DVDs as Blu-ray discs. The total number of DVDs and Blu-ray discs sold that day was $$280$$ . How many DVDs were sold?

Q:

For a body falling freely from rest (disregarding air resistance), the distance the body falls varies directly as the square of the time. If an object is dropped from the top of a tower $$576 ft$$ high and hits the ground in $$6 sec$$ , how far did it fall in the first $$4 sec$$ ?

Q:

The U.S. Postal Service requires that any box sent by Priority Mail $${ } ^ { B }$$ have length plus girth (distance around) totaling no more than $$108$$ in. The maxi- mum volume that meets this condition is contained by a box with a square end $$18$$ in. on each side. What is the length of the box? What is the max- imum volume? (Data from United States Postal Service.)

Q:

The player who gave Billy the correct answer solved the problem as follows: Using the simple formula a times $$b$$ over a plus $$b$$ , we get our answer of one and seven-eighths. Show that if it takes one person $$a$$ hours to complete a job and another $$b$$ hours to complete the same job, then the expression stated by the player, $$\frac { a \cdot b } { a + b }$$

Q:

A machine purchased for business use depreciates, or loses value, over a period of years. The value of the machine at the end of its useful life is its scrap value. By one method of depreciation, the scrap value, $$S$$ , is given by $$S = C ( 1 - r ) ^ { n }$$ ,