Math 115 Word Problem Solutions
3.2.41: Twelve added to a certain number is 21. What is the number?
a. Choose a variable and indicate what it represents in the problem.
Let x = a certain number.
b. Set up an equation that represents the situation described.
c. Solve the equation.
3.2.43: Nine subtracted from a certain number is 13. Find the number.
a. Choose a variable and indicate what it represents in the problem.
Let x = a certain number.
b. Set up an equation that represents the situation described.
c. Solve the equation.
3.2.55: Dress socks cost $2.50 a pair more than athletic socks. Randall purchased one pair of dress socks and six pairs of athletic socks for $21.75. Fine the price of a pair of dress socks.
a. Choose a variable and indicate what it represents in the problem.
Cost of Dress socks = x + 2.50
Cost of Athletic Socks = x
b. Set up an equation that represents the situation described.
c. Solve the equation.
Cost of Athletic Socks = $2.75
Cost of Dress Socks = 2.75 + 2.50 = $5.25
3.3.35: Find two consecutive odd numbers whose sum is 72.
Set up and solve an algebraic equation.
First odd number = x
Second odd number = x + 2
First odd number = 35
Second odd number = 35 + 2 = 37
3.3.37: Find three consecutive even numbers whose sum is 114.
Set up and solve an algebraic equation.
First even number = x
Second even number = x + 2
Third even number = x + 4
First even number = 36
Second even number = 36 + 2 = 38
Third even number = 36 + 4 = 40
3.3.45: If two angles are supplementary and the larger angle is 20° less than three times the smaller angle, find the measure of each angle.
Set up and solve an algebraic expression.
First angle = x
Second angle = 3x – 20
NOTE: Two angles for which the sum of their measure is 180° are called supplementary angles.
First angle = 50°
Second angle = 3(50) – 20 = 130°
3.3.53: At a university-sponsored concert, there were three times as many women as men. A total of 600 people attended the concert. How many men and how man women attended?
Set up and solve an algebraic equation.
Number of men = m
Number of women = 3m
Number of men = 150
Number of women = 3(150) = 450
3.3.57: At a local restaurant, $275 in tips is to be shared between the server, bartender, and busboy. The server gets $25 more than three times the amount the busboy receives. The bartender gets $50 more than the amount the busboy receives. How much will the server receive?
Set up and solve an algebraic equation.
Busboy’s share = b
Bartender’s share = 50 + b
Server’s share = 3b + 25
Server’s share = 3(40) + 25 = $145
3.4.63: Find three consecutive whole numbers such that twice the sum of the two smallest numbers is 10 more than three times the largest number.
Set up and solve an algebraic equation.
First number = x
Second number = x + 1
Third number = x + 2
First number = 14
Second number = 14+1 = 15
Third number = 14+2 = 16
3.4.67: Find a number such that 20 more than one-third of the number equals three-fourths of the number.
Set up and solve an algebraic expression.
Let x = a certain number.
3.4.75: Max has a collection of 210 coins consisting of nickels, dimes, and quarters. He has twice as many dimes as nickels, and 10 more quarters than dimes. How many coins of each kind does he have?
Set up and solve an algebraic expression.
Amount of dimes = d
Amount of nickels =
Amount of quarters =
Amount of dimes = 80
Amount of nickels = = 40
Amount of quarters = 80 + 10 = 90
3.4.83: In triangle ABC, the measure of angle A is 2° less than one-fifth of the measure of angle C. The measure of angle B is 5° less than one-half of the measure of angle C. Find the measure of the three angles of the triangle.
Set up and solve an algebraic expression.
Measure of angle C =
Measure of angle A =
Measure of angle B =
NOTE: The sum of the measures of the three angles of a triangle is 180°.
Measure of angle C = 110°
Measure of angle A =
Measure of angle B =
3.4.85: The supplement of an angle is 10° smaller than three times its complement. Find the size of the angle.
Set up and solve an algebraic expression.
Let x = an angle.
Supplement of the angle =
Complement of an angle =
The measure of angle x = 40°.
3.6.69: Suppose that the perimeter of a rectangle is to be no greater than 70 inches, and the length of the rectangle must be 20 inches. Find the largest possible value for the width of the rectangle.
Set up and solve an appropriate inequality.
Let P = perimeter
Let l = length.
Let w = width
The largest possible width, w, of the rectangle is 15 inches.
3.T.22: Suppose that a triangular plot of ground is enclosed with 20 meters of fencing. The longest side of the lot is two times the length of the shortest side, and the third side is 10 meters longer than the shortest side. Find the length of each side of the plot.
Set up and solve an algebraic expression.
Let s = the shortest side.
Longest side = 2s
Third side = s + 10
Shortest side = 15 meters
Longest side = = 30 meters
Third side = meters
4.1.47: 15% of what number is 6.3?
Set up and solve an appropriate equation.
Let n = a number.
4.1.49: 76 is what percent of 95?
Set up and solve an appropriate equation.
Let p = the percentage.
4.1.51: What is 120% of 50?
Set up and solve an appropriate equation.
Let n = a number.
4.1.55: 160% of what number is 144?
Set up and solve an appropriate equation.
Let n = a number.
4.1.59: Suppose that a car can travel 264 miles using 12 gallons of gasoline. How far will it go on 15 gallons?
Solve using a proportion.
Let m = miles traveled.
4.1.67: A preelection poll indicated that three out of every seven eligible voters were going to vote in an upcoming election. At this rate, how many people are expected to vote in a city of 210,000?
Solve using a proportion.
Let a = the number of actual voters.
4.1.73: An inheritance of $180,000 is to be divided between a child and the local cancer fund in the ratio of 5 to 1. How much money will the child receive?
Solve using a proportion.
Let i = the child’s inheritance.
4.2.23: Tom bought an electric drill at a 30% discount sale for $35. What was the original price of the drill?
Let p = original price.
4.2.25: Find the cost of a $4800 wide-screen plasma television that is on sale for 25% off.
Let d = the discounted selling price.
4.2.29: Pierre bought a coat for $126 that was listed for $180. What rate of discount did he receive?
Let r = the rate of discount.
4.2.31: A retailer has some toe rings that cost him $5 each. He wants to sell them at a profit of 70% of the cost. What should be the selling price of the toe rings?
Let s = selling price.
4.2.35: Jewelry has a very high markup rate. If a ring costs a jeweler $400, at what price should it be sold to gain a profit of 60% of the selling price.
Let s = selling price
4.2.37: If the cost of a pair of shoes for a retailer is $32 and he sells them for $44.80, what is his rate of profit based on the cost?
Let r = rate of profit.
4.2.39: Find the annual interest rate if $560 in interst is earned when $3500 is invested for 2 years.
4.2.43: What will be the interest earned on a $5000 certificate of deposit invested at 3.8% annual interest for 10 years?
4.2.45: How much is a month’s interest on a mortgage balance of $145,000 at a 6.5% annual interest rate?
4.3.15: A dirt path 4 feet wide surrounds a rectangular garden that is 38 feet long and 17 feet wide. Find the area of the dirt path.
4.3.19: Find the length of an altitude of a trapezoid with bases of 8 inches and 20 inches and an area of 98 square inches.
4.3.31: If the total surface area of a right circular cylinder is 104π square meters, and a radius of the base is 4 meters long, find the height of the cylinder.
4.4.21: The width of a rectangle is 3 inches less than one-half of its length. If the perimeter of the rectangle is 42 inches, find the area of the rectangle.
4.4.25: The second side of a triangle is 1 centimeter longer than three times the first side. The third side is 2 centimeters longer than the second side. If the perimeter is 46 centimeters, find the length of each side of the triangle.
First Side = F
Second Side = 3F+1
Third Side = (3F+1)+2
Perimeter = First Side + Second Side + Third Side
First Side = 6 cm
Second Side =
Third Side =
4.4.33: The distance between Jacksonville and Miami is 325 miles. A freight train leaves Jacksonville and travels toward Miami at 40 miles per hour. At the same time, a passenger train leaves Miami and travels toward Jacksonville at 90 miles per hour. How long will it take the two trains to meet?
Freight Train Leaving Jacksonville:
Passenger Train Leaving Miami:
Freight Train’s Distance + Passenger Train’s Distance = Total Distance
4.4.35: A car leaves a town traveling at 40 miles per hour. Two hours later a second car leaves the town traveling the same route and overtakes the first care in 5 hours and 20 minutes. How fast was the second car traveling?
5 hours and 20 minutes =
D / r / tFirst Car / 40 / +2
Second Car / r
4.4.37: Two trains leave at the same time, one traveling east and the other traveling west. At the end of hours they are 1292 miles apart. If the rate of the train traveling east is 8 miles per hour faster than the rate of the other train, find their rates.
Train Traveling East
Train Traveling West / r
Rate of Eastbound Train = 64+8 = 72 miles per hour
Rate of Westbound Train = 64 miles per hour.
4.4.39: Jeff leaves home and rides his bicycle out into the country for 3 hours. On his return trip along the same route, it takes him three-quarters of an hour longer. If his rate on the return trip was 2 miles per hour slower than on the trip out into the country, find the total roundtrip distance.
D / r / tLeaving Home / 3r / r / 3
Returning Home
The distance he traveled away from home = 3(10) 30 miles. Thus, the roundtrip distance is 2(30) = 60 miles.
4.5.13: How many milliliters of pure acid must be added to 100 milliliters of a 10% acid solution to obtain a 20% solution?
Let l = milliliters of pure acid.
4.5.15: How many centiliters of distilled water must be added to 10 centiliters of a 50% acid solution to obtain a 20% acid solution?
Let c = centiliters of distilled water
A solution that is 50% acid is 50% water.
A solution that is 20% acid is 80% water.
4.5.17: Suppose that we want to mix some 30% alcohol solution with some 50% alcohol solution to obtain 10 quarts of a 35% solution. How many quarts of each kind should we use?
If we let q = quarts of 30% alcohol solution
Then, 10 – q = quarts of 50% alcohol solution.
Amount of 30% alcohol solution needed = 7.5 quarts
Amount of 50% alcohol solution needed =
4.5.19: How much water needs to be removed from 30 liters of a 20% salt solution to change it to a 50% salt solution.
Let g = gallons of water.
A 30% salt solution has 70% water.
A 40% salt solution has 60% water.
4.5.21: Suppose that a 12-quart radiator contains a 20% solution of antifreeze. How much solution needs to be drained out and replaced with pure antifreeze to obtain a 50% solution?
Let q = quarts of antifreeze to be drained out and replaced.
4.5.25: Thirty ounces of punch that contains 10% grapefruit juice is added to 50 ounces of a punch that contains 20% grapefruit juice. Find the percent of grapefruit juice in the resulting mixture.
Let p = percent of grapefruit juice.
4.5.27: Suppose that the perimeter of a square equals the perimeter of a rectangle. The width of the rectangle is 9 inches less than twice the side of the square, and the length of the rectangle is 3 inches less than twice the side of the square. Find the dimensions of the square and the rectangle.
4.5.31: Pam is half as old as her brother Bill. Six years ago Bill was four times older than Pam. How old is each sibling now?
Bill’s age now = b
Pam’s age now =
Bill’s age 6 years ago =
Pam’s age 6 years ago =
Bill’s age now = 18 years.
Pam’s age now =
4.5.33: Nina received an inheritance of $12,000 from her grandmother. She invested part of it at 6% interest, and she invested the remainder at 8% interest. If the total yearly interest from both investments was $860, how much did she invest at each rate?
If we let x = the amount invested at 6%
Then, the amount invested at 8%.
The amount invested at 6% = $5,000
The amount invested at 8% = 12,000-5000 = $7,000.
4.5.35: Sally invested a certain sum of money at 9%, twice that sum at 10%, and three times that sum at 11%. Her total yearly interest from all three investments was $310. How much did she invest at each rate?