1. Write the amount that a person can do in one time period as a fraction.
2. Multiply that amount by the x.
(You’ve multiplied each fraction that each person can do by the time it takes to do the whole job.)
3. Add the portions of the job that are completed in one time period together and set the sum equal to 1.
(Setting the amount to 1 is 100% of the job.)
Q. Meg can clean out the garage in five hours. Mike can clean out the same garage in three hours. How long will the job take if they work together?
A. Slightly less than two hours. Let x represent the amount of time it takes to do the cleaning when Meg and Mike work together. Meg can do of the job in one hour, and Mike can do of the job in one hour. The equation to use is . Multiply both sides of the equation by the common denominator and add the two fractions together: , or 3x + 5x = 15, giving you 8x = 15. Divide by 8, and you have , just under two hours.
Q. Carlos can wash the bus in seven hours, and when Carlos and Carol work together, they can wash the bus in three hours. How long would it take Carol to wash the bus by herself?
A. 5 hours and 15 minutes. Let x represent the amount of time it takes Carol to wash the bus by herself. This time, you have the time that it takes working together, so your equation is . The common denominator of the two fractions on the left is 7x. Multiply both sides of the equation by 7x, simplify, and solve for x.
11. Alissa can do the job in three days, and Alex can do the same job in four days. How long will it take if they work together?
Solve It
12. George can paint the garage in five days, Geanie can paint it in eight days, and Greg can do the job in ten days. How long will painting the garage take if they all work together?
Solve It
13. Working together, Sam and Helene wrote a company organizational plan in days. Working alone, it would have taken Sam four days to write that plan. How long would it have taken Helene if she had written it alone?
Solve It
14. Rancher Biff needs his new fence put up in four days — before the herd arrives. Working alone, it’ll take him six days to put up all the fencing. He can hire someone to help. How fast does the hired hand have to work in order for the team to complete the job before the herd arrives?
Solve It
15. When hose A is running full-strength to fill the swimming pool, it takes 8 hours; when hose B is running full-strength, filling the pool takes 12 hours. How long would it take to fill the pool if both hoses were running at the same time?
Solve It
16. Elliott set up hose A to fill the swimming pool, planning on it taking eight hours. But he didn’t notice that water was leaking out of the pool through a big crack in the bottom. With just the water leaking, the pool would be empty in 12 hours. How long would filling the pool take with Elliott adding water with hose A and the leak emptying the pool at the same time?
Solve It
Answers to Relating Values in Story Problems
The following are the answers (in bold) to the practice problems in this chapter.
1. Jack is three times as old as Chloe. Ten years ago, Jack was five times as old as Chloe. How old are Jack and Chloe now? Chloe is 20, and Jack is 60.
Let x = Chloe’s present age and 3x = Jack’s present age. Ten years ago, Chloe’s age was , and Jack’s was . Also ten years ago, Jack’s age was 5 times Chloe’s age. You can write this equation as . Distribute the 5 to get . Subtract 3x from each side and add 50 to each side, and the equation becomes 40 = 2x. Divide by 2 to get .
2. Linda is ten years older than Luke. In ten years, Linda’s age will be thirty years less than twice Luke’s age. How old is Linda now? Linda is 40.
Let x = Luke’s age now. That makes Linda’s present age = x + 10. In ten years, Luke will be , and Linda will be . But at these new ages, Linda’s age will be thirty years less than twice Luke’s age. This is written as . Distribute the 2 to get . Simplifying, . Subtract x from each side and add 10 to each side to get . Luke is 30, and Linda is 40. In ten years, Luke will be 40, and Linda will be 50. Twice Luke’s age then, minus 30, is . It checks.
3. Avery is six years older than Patrick. In four years, the sum of their ages will be 26. How old is Patrick? Patrick is 6.
Let x = Patrick’s age now. Then Avery’s age = . In four years, Patrick will be years old, and Avery will be years old. Write that the sum of their ages in four years: . Simplify on the left to get . Subtract 14 from each side: . Divide by 2, and you get . Patrick is 6, and Avery is 12. In four years, Patrick will be 10, and Avery will be 16. The sum of 10 and 16 is 26.
4. Jon is three years older than Jim, and Jim is two years older than Jane. Ten years ago, the sum of their ages was 40. How old is Jim now? Jim is 23.
Let x represent Jane’s age now. Jim is two years older, so Jim’s age is . Jon is three years older than Jim, so Jon’s age is . Ten years ago Jane’s age was , Jim’s age was , and Jon’s age was . Add the ages ten years ago together to get 40: . Simplifying on the left, . Add 23 to each side to get . Divide by 3, . Jane’s age is 21, so Jim’s age is .
5. The sum of three consecutive integers is 57. What are they? The integers are 18, 19, and 20.
Let x = the smallest of the three consecutive integers. Then the other two are and . Adding the integers together to get 57, , which simplifies to . Subtract 3 from each side to get Dividing by 3, . , .
6. The sum of four consecutive even integers is 52. What is the largest of the four? 16. The four integers are 16, 14, 12, and 10.
Let that largest integer = x. The other integers will be 2, 4, and 6 smaller, so they can be written with . Adding them, you get . Simplifying gives you . Add 12 to each side to get . Divide each side by 4, and . .
7. The sum of three consecutive odd integers is 75. What is the middle number? 25.
You can add and subtract 2 from that middle integer. Let that integer = x. Then the other two are and . Add them together: . Simplifying, . Divide by 3 to get . That’s the middle number. The other two are 23 and 27: .
8. The sum of five consecutive multiples of 4 is 20. What are they? The numbers are –4, 0, 4, 8, and 12.
Let x = the first of the consecutive multiples of 4. Then the other four are , , , and . Add them together: . Simplifying on the left, . Subtract 40 from each side to get . Dividing by 5, .
9. The sum of the smallest and largest of three consecutive integers is 126. What is the middle number of those consecutive integers? 63.
Let x = the smallest of the consecutive integers. Then the other two are and . Because the sum of the smallest and largest is 126, you can write it as . Simplifying the equation, . Subtract 2 from each side to get . Dividing by 2, , which is the smallest integer. The middle one is one bigger, so it’s 63.
10. The product of two consecutive integers is 89 more than their sum. What are they? The numbers are 10 and 11 or –9 and –8.
Let the smaller of the integers = x. The other one is then . Their product is written and their sum is . Now, to write that their product is 89 more than their sum, the equation is . Distributing the x on the left and simplifying on the right, . Subtract 2x and 90 from each side to set the quadratic equation equal to 0: . The trinomial on the left side of the equation factors to give you . or . If , then . The product of 10 and 11 is 110. That’s 89 bigger than their sum, 21. What if ? The next bigger number is then –8. Their product is 72. The difference between their product of 72 and sum of –17 is . So this problem has two possible solutions.
11. Alissa can do the job in three days, and Alex can do the same job in four days. How long will it take if they work together? It will take a little less than 2 days working together ( days).
Let x = the number of days to do the job together. Alissa can do of the job in one day and of the job in x days. In x days, as they work together, they’re to do 100% of the job. , which is 100%. Multiply by 12: . Divide by 7: days to do the job.
Alissa’s share is , and Alex�
��s share is . Together .
12. George can paint the garage in five days, Geanie can paint it in eight days, and Greg can do the job in ten days. How long will painting the garage take if they all work together? days.
Let x = the number of days to complete the job together. In x days, George will paint of the garage, Geanie of the garage, and Greg of the garage. The equation for completing the job is . Multiplying through by 40, which is the least common denominator of the fractions, you get . Simplifying, you get . Dividing by 17 gives you days to paint the garage. Checking the answer, George’s share is , Geanie’s share is , and Greg’s share is . Together, , or 100%.
13. Working together, Sam and Helene wrote a company organizational plan in days. Working alone, it would have taken Sam four days to write that plan. How long would it have taken Helene, if she had written it alone? 2 days.
Let x = the number of days for Helene to write the plan alone. So Helene writes of the plan each day. Sam writes of the plan per day. In days, they complete the job together. The equation is . Multiplying each side by 12x, the least common denominator, gives you . Subtract 4x from each side to get .
Dividing by 8, . Helene will complete the job in 2 days. To check this, in days, Sam will do of the work, and Helene will do . Together, or 100%.
14. Rancher Biff needs his new fence put up in four days — before the herd arrives. Working alone, it’ll take him six days to put up all the fencing. He can hire someone to help. How fast does this person have to work in order for him or her to complete the job before the herd arrives? The hired hand must be able to do the job alone in 12 days.
Let x = the number of days the hired hand needs to complete the job alone. The hired hand does of the fencing each day, and Biff puts up of the fence each day. In four days, they can complete the project together, so . Multiply by the common denominator 6x: . Simplifying, . Subtracting 4x from each side, .
Dividing by 2, . The hired hand must be able to do the job alone in 12 days. To check this, in four days, Biff does of the fencing, and the hired hand of the job.
15. When hose A is running full strength to fill the swimming pool, it takes 8 hours; when hose B is running full strength, it takes 12 hours to fill the swimming pool. How long would it take to fill the pool if both hoses were running at the same time? 4.8 hours
Let x represent the amount of time to complete the job. Your equation is . Multiplying both sides of the equation by 24, you get 3x + 2x = 24. Simplifying and then dividing by 5, you get 5x = 24, x = 4.8.
16. Elliott set up hose A to fill the swimming pool, planning on it taking 8 hours. But he didn’t notice that water was leaking out of the pool through a big crack in the bottom. With just the water leaking, the pool would be empty in 12 hours. How long would filling the pool take with Elliott adding water with hose A and the leak emptying the pool at the same time? 24 hours
This time, one of the terms is negative. The leaking water takes away from the completion of the job, making the time longer. Using the equation , multiply both sides of the equation by 24, giving you 3x – 2x = 24 or x = 24.
Chapter 20
Measuring Up with Quality and Quantity Story Problems
In This Chapter
Making the most of mixtures
Understanding the strength of a solution
Keeping track of money in piggy banks and interest problems
The story problems in this chapter have a common theme to them; they deal with quality (the strength or worth of an item) and quantity (the measure or count) and adding up to a total amount. (Chapters 17, 18, and 19 have other types of story problems.) You encounter quantity and quality problems almost on a daily basis. For instance, if you have four dimes, you know that you have forty cents. How do you know? You multiply the quantity, four dimes, times the quality, ten cents each, to get the total amount of money. And if you have a fruit drink that’s 50 percent real juice, then a gallon contains one-half gallon of real juice (and the rest is who-knows-what) — again, multiplying quality times quantity.
In this chapter, take time to practice with these story problems. Just multiply the amount of something, the quantity, times the strength or worth of it, quality, in order to solve for the total value.
Achieving the Right Blend with Mixtures Problems
Mixtures include what goes in granola, blends of coffee, or the colors of sugarcoated candies in a candy dish. In mixture problems, you often need to solve for some sort of relationship about the mixture: its total value or the amount of each item being blended. Just be sure that the variable represents some number — an amount or value. Bring along an appetite. Many of these problems deal with food.
Q. A health store is mixing up some granola that has many ingredients, but three of the basics are oatmeal, wheat germ, and raisins. Oatmeal costs $1 per pound, wheat germ costs $3 per pound, and raisins cost $2 per pound. The store wants to create a base granola mixture of those three ingredients that will cost $1.50 per pound. (These items serve as the base of the granola; the rest of the ingredients and additional cost will be added later.) The granola is to have nine times as much oatmeal as wheat germ. How much of each ingredient is needed?
A. Every pound of mixed granola will need pound of wheat germ, pound ofoatmeal, and or pound of raisins.
To start this problem, let x represent the amount of wheat germ in pounds. Because you need nine times as much oatmeal as wheat germ, you have 9x pounds of oatmeal. How much in raisins? The raisins can have whatever’s left of the pound after the wheat germ and oatmeal are taken out: 1 – (x + 9x) or 1 – 10x pounds. Now multiply each of these amounts by their respective price: $3(x) + $1(9x) + $2(1 – 10x). Set this equal to the $1.50 price multiplied by its amount, as follows: $3(x) + $1(9x) + $2(1 – 10x) = $1.50(1). Simplify and solve for x: 3x + 9x + 2 – 20x = –8x + 2 = $1.50. Subtracting 2 from each side and then dividing each side by 8, gives you –8x = –0.50, .
1. Kathy’s Kandies features a mixture of chocolate creams and chocolate-covered caramels that sells for $9 per pound. If creams sell for $6.75 per pound and caramels sell for $10.50 per pound, how much of each type of candy should be in a 1-pound mix?
Solve It
2. Solardollars Coffee is trying new blends to attract more customers. The premium Colombian costs $10 per pound, and the regular blend costs $4 per pound. How much of each should the company use to make 100 pounds of a coffee blend that costs $5.50 per pound?
Solve It
3. Peanuts cost $2 per pound, almonds cost $3.50 per pound, and cashews cost $6 per pound. How much of each should you use to create a mixture that costs $3.40 per pound, if you have to use twice as many peanuts as cashews?
Solve It
4. A mixture of jellybeans is to contain twice as many red as yellow, three times as many green as yellow, and twice as many pink as red. Red jelly beans cost $1.50 per pound, yellow cost $3.00 per pound, green cost $4.00, and pink only cost $1.00 per pound. How many pounds of each color jellybean should be in a 10-pound canister that costs $2.20 per pound?
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5. A Very Berry Smoothie calls for raspberries, strawberries, and yogurt. Raspberries cost $3 per cup, strawberries cost $1 per cup, and yogurt costs $0.50 per cup. The recipe calls for twice as much strawberries as raspberries. How many cups of strawberries are needed to make a gallon of this smoothie that costs $10.10? (Hint: 1 gallon = 16 cups)
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6. A supreme pizza contains five times as many ounces of cheese as mushrooms, twice as many ounces of peppers as mushrooms, twice as many ounces of onions as peppers, and four more ounces of sausage than mushrooms. Mushrooms and onions cost 10 cents per ounce, cheese costs 20 cents per ounce, peppers are 25 cents per ounce, and sausage is 30 cents per ounce. If the toppings are to cost no more than a total of $5.80, then how many ounces of each ingredient can be used?
Solve It
Concocting the Correct Solution One Hundred Percent of the Time
Solutions problems are sort of like mix
tures problems (explained in the preceding section). The main difference is that solutions usually deal in percents — 30%, , 0%, or even 100%. These last two numbers indicate, respectively, that none of that ingredient is in the solution (0%) or that it’s pure for that ingredient (100%). You’ve dealt with these solutions if you’ve had to add antifreeze or water to your radiator. Or how about adding that frothing milk to your latte mixture?
Algebra I Workbook For Dummies Page 19