Algebra I Workbook For Dummies
Page 20
The general format for these solutions problems is
(% A × amount A) + (% B × amount B) = (% C × amount C)
Q. How many gallons of 60% apple juice mix need to be added to 20 gallons of mix that’s currently 25% apple juice to bring the new mix up to 32% apple juice?
A. Five gallons. To solve this problem, let x represent the unknown amount of 60% apple juice. Using the format of all the percents times the respective amounts, you get (60% × x) + (25% × 20 gallons) = [32% × (x + 20 gallons)]. Change the percents to decimals, and the equation becomes . (If you don’t care for decimals, you could multiply each side by 100 to change everything to whole numbers.) Now distribute the 0.32 on the right and simplify so you can solve for x:
If you’re adding pure alcohol or pure antifreeze or something like this, use 1 (which is 100%) in the equation. If there’s no alcohol, chocolate syrup, salt, or whatever in the solution, use 0 (which is 0%) in the equation.
Q. How many quarts of water do you need to add to 4 quarts of lemonade concentrate in order to make the drink 25% lemonade?
A. 12 quarts. To solve this problem, let x represent the unknown amount of water,which is 0% lemonade. Using the format of all the percents times the respective amounts, you get (0% × x) + (100% × 4 quarts) = [25% × (x + 4) quarts]. Change the percents to decimals, and the equation becomes . When the equation gets simplified, the first term disappears.
7. How many quarts of 25% solution do you need to add to 4 quarts of 40% solution to create a 31% solution?
Solve It
8. How many gallons of a 5% fertilizer solution have to be added to 2 gallons of a 90% solution to create a fertilizer solution that has 15% strength?
Solve It
9. How many quarts of pure antifreeze need to be added to 8 quarts of 30% antifreeze to bring it up to 50%?
Solve It
10. How many cups of chocolate syrup need to be added to 1 quart of milk to get a mixture that’s 25% syrup?
Solve It
11. What concentration should 4 quarts of salt water have so that, when it’s added to 5 quarts of 40% solution salt water, the concentration goes down to ?
Solve It
12. What concentration and amount of solution have to be added to 7 gallons of 60% alcohol to produce 16 gallons of alcohol solution?
Solve It
Dealing with Money Problems
Story problems involving coins, money, or interest earned all are solved with a process like that used in solutions problems: You multiply a quantity times a quality. In these cases, the qualities are the values of the coins or bills, or they’re the interest rate at which money is growing.
Q. Gabriella is counting the bills in her cash drawer before the store opens for the day. She has the same number of $10 bills as $20 bills. She has two more $5 bills than $10 bills, and ten times as many $1 bills as $5 bills. She has a total of $300 in bills. How many of each does she have?
A. Gabriella has 6 $10 bills, 6 $20 bills, 8 $5 bills, and 80 $1 bills for a total of $300. You can compare everything, directly or indirectly, to the $10 bills. Let x represent the number of $10 bills. The number of $20 bills is the same, so it’s x, also. The number of $5 bills is two more than the number of $10 bills, so let the number of fives be represented by x + 2. Multiply x + 2 by 10 for the number of $1 bills, 10(x + 2). Now take each number of bills and multiply by the quality or value of that bill. Add them to get $300:
Using x=6, the number of $10 bills and $20 bills, you get x+2=8 for the number of $5 bills, and 10(x+2)=80 for the number of $1 bills.
Q. Jon won the state lottery and has $1 million to invest. He invests some of it in a highly speculative venture that earns 18% interest. The rest is invested more wisely, at 5% interest. If he earns $63,000 in simple interest in one year, how much did he invest at 18%?
A. He invested $100,000 at 18% interest. Let x represent the amount of money invested at 18%. Then the remainder, 1,000,000 – x, is invested at 5%. The equation to use is 0.18x + 0.05(1,000,000 – x) = 63,000. Distributing the 0.05 on the left and combining terms, you get 0.13x + 50,000 = 63,000. Subtract 50,000 from each side, and you get 0.13x = 13,000. Dividing each side by 0.13 gives you x = 100,000. It’s kind of mind boggling.
13. Carlos has twice as many quarters as nickels and has a total of $8.25. How many quarters does he have?
Solve It
14. Gregor has twice as many $10 bills as $20 bills, five times as many $1 bills as $10 bills, and half as many $5 bills as $1 bills. He has a total of $750. How many of each bill does he have?
Solve It
15. Stella has 100 coins in nickels, dimes, and quarters. She has 18 more nickels than dimes and a total of $7.40. How many of each coin does she have?
Solve It
16. Betty invested $10,000 in two different funds for one year. She invested part at 2% and the rest at 3%. She earned $240 in simple interest. How much did she invest at each rate? (Hint: Use the simple interest formula: I = Prt.)
Solve It
Answers to Problems on Measuring Up with Quality and Quantity
The following are the answers (in bold) to the practice problems in this chapter.
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 one-pound mix? 0.4 pounds of creams and 0.6 pounds of caramels.
Let x represent the amount of chocolate creams in pounds. Then is the pounds of chocolate caramels. In a pound of the mixture, creams cost $6.75x and caramels . Together, the mixture costs $9. So
Subtract 10.5 from each side and then divide by –3.75:
To get the amount of caramels, pounds of caramels. Checking this, the cost of creams is . The cost of caramels is . Adding these together, you get .
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? The company needs 25 pounds of Colombian and 75 pounds of regular blend.
Let x = the pounds of Colombian coffee at $10 per pound. Then pounds of regular blend at $4 per pound. The cost of 100 pounds of the mixture blend is . Use
Subtract 400 from each side and then divide each side by 6:
To check, multiply $10(25) and $4(75) to get
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? pounds almonds, pounds peanuts, pounds cashews.
Let x = the pounds of cashews at $6 per pound. Then 2x = pounds of peanuts at $2 per pound. The almonds are then pounds at $3.50 per pound. Combine all this and solve:
Use this amount for x and substitute in to get the other weights. You get pounds of peanuts and pounds of almonds.
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? 1 pound yellow, 2 pounds red, 3 pounds green, and 4 pounds pink jellybeans.
Let x represent the number of pounds of yellow jellybeans. Then 2x is the pounds of red jellybeans, 3x is the pounds of green jellybeans, and is the pounds of pink jellybeans. Multiplying each quantity times its price and solve:
With 1 pound yellow jellybeans, you then know that the mixture has 2(1) = 2 pounds red jellybeans, 3(1) pounds green jellybeans, and 4(1) pounds pink jellybeans. The 1 + 2 + 3 + 4 pounds adds up to 10 pounds jellybeans.
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.) 1.2 cups.
Let x = the cups of raspberries at $3 per cup. Then 2x = the cups of strawberries at $1 per cup. The remainder of the 16 cups of mixture are for yogurt, which comes out to be cups of yogurt at $0.50 per cup. The equation you need is
The mixture has 0.6 cups of raspberries and 2(0.6) = 1.2 cups of strawberries.
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? The mixture has 2 ounces of mushrooms, 10 ounces of cheese, 4 ounces of peppers, 8 ounces of onions, and 6 ounces of sausage.
Let x represent the number of ounces of mushrooms. Then 5x is the ounces of cheese, 2x is the ounces of peppers, is the ounces of onions, and is the ounces of sausage. Multiplying each quantity times its cost (quality) and setting that equal to 580 cents, you get . Simplifying, you get . Solving for x gives you 230x + 120 = 580, or 230x = 460. Divide each side by 230 to get x = 2. So you need 2 ounces of mushrooms, 5(2) = 10 ounces of cheese, 2(2) = 4 ounces of peppers, 4(2) = 8 ounces of onions, and 2 + 4 = 6 ounces of sausage.
7. How many quarts of 25% solution do you need to add to 4 quarts of 40% solution to create a 31% solution? 6 quarts.
Let x = the quarts of 25% solution needed. Add x quarts of 25% solution to 4 quarts of 40% solution to get quarts of 31% solution. Write this as . Multiply through by 100 to get rid of the decimals: . Distribute the 31 and simplify on the left: . Subtract 25x and 124 from each side to get . Divide by 6 to get . The answer is 6 quarts of 25% solution.
8. How many gallons of a 5% fertilizer solution have to be added to 2 gallons of a 90% solution to create a fertilizer solution that has 15% strength? 15 gallons.
Let x = the gallons of 5% solution. Then for the gallons. Multiply through by 100: . Simplify each side: . Subtract 5x and 30 from each side: gallons of 5% solution.
9. How many quarts of pure antifreeze need to be added to 8 quarts of 30% antifreeze to bring it up to 50%? 3.2 quarts.
Let x = the quarts of pure antifreeze to be added. Pure antifreeze is 100% antifreeze . So . Simplify on the left and distribute on the right: . Subtract 0.5x and 2.4 from each side: . Divide by 0.5 to get quarts of pure antifreeze.
10. How many cups of chocolate syrup need to be added to 1 quart of milk to get a mixture that’s 25% syrup? cups.
Let x = the cups of chocolate syrup needed. (I only use the best quality chocolate syrup, of course, so you know that the syrup is pure chocolate.) 1 quart = 4 cups, and the milk has no chocolate syrup in it. So
11. What concentration should the 4 quarts of salt water have so that, when it’s added to 5 quarts of 40% solution salt water, the concentration goes down to ? 25%.
Let x% = the percent of the salt solution in the 4 quarts:
The 4 quarts must have a 25% salt solution.
12. What concentration and amount solution have to be added to 7 gallons of 60% alcohol to produce 16 gallons of alcohol solution? 9 gallons of 20% solution.
To get 16 gallons, 9 gallons must be added to the 7 gallons. Let x% = the percent of alcohol in the 9 gallons:
13. Carlos has twice as many quarters as nickels and has a total of $8.25. How many quarters does he have? 30 quarters.
Let x = the number of nickels. Then 2x = the number of quarters. These coins total $8.25 or 825 cents. So, in cents,
Because Carlos has 15 nickels, he must then have 30 quarters.
14. Gregor has twice as many $10 bills as $20 bills, five times as many $1 bills as $10 bills, and half as many $5 bills as $1 bills. He has a total of $750. How many of each bill does he have? 10 $20 bills, 20 $10 bills, 100 $1 bills, and 50 $5 bills.
Let x = the number of $20 bills. Then 2x = the number of $10 bills, the number of $1 bills, and the number of $5 bills. The total in dollars is 750. So the equation should be . Simplify on the left to . Gregor has 10 $20 bills, 20 $10 bills, 100 $1 bills, and 50 $5 bills.
15. Stella has 100 coins in nickels, dimes, and quarters. She has 18 more nickels than dimes and a total of $7.40. How many of each coin does she have? 40 dimes, 58 nickels, and 2 quarters.
Let x = the number of dimes. Then the number of nickels, and the number of quarters is . These coins total $7.40 or 740 cents:
So Stella has 40 dimes, 58 nickels, and 2 quarters.
16. Betty invested $10,000 in two different funds. She invested part at 2% and the rest at 3%. She earned $240 in simple interest. How much did she invest at each rate? (Hint: Use the simple interest formula: 1=Prt.) Betty has $6,000 invested at 2% and the other $4,000 invested at 3%.
Let x = the amount invested at 2%. Then the amount invested at 3% is 10,000 – x. Betty earns interest of 2% on x dollars and 3% on 10,000 – x dollars. The total interest is $240, so
Chapter 21
Getting a Handle on Graphing
In This Chapter
Plotting points and lines
Computing distances and slopes
Intercepting and intersecting
Writing equations of lines
Graphing basic conics
Graphs are as important to algebra as pictures are to books and magazines. A graph can represent data that you’ve collected, or it can represent a pattern or model of an occurrence. A graph illustrates what you’re trying to demonstrate or understand.
The standard system for graphing in algebra is to use the Cartesian coordinate system, where points are represented by ordered pairs of numbers; connected points can be lines, curves, or disjointed pieces of graphs.
This chapter can help you sort out much of the graphing mystery and even perfect your graphing skills; just watch out! The slope may be slippery.
Thickening the Plot with Points
Graphing on the Cartesian coordinate system begins by constructing two perpendicular axes, the x-axis (horizontal) and the y-axis (vertical). The Cartesian coordinate system identifies a point by an ordered pair, (x, y). The order in which the coordinates are written matters. The first coordinate, the x, represents how far to the left or right the point is from the origin, or where the axes intersect. A positive x is to the right; a negative x is to the left. The second coordinate, the y, represents how far up or down from the origin the point is.
Cartesian coordinates designate where a point is in reference to the two perpendicular axes. To the right and up is positive, to the left and down is negative. Any point that lies on one of the axes has a 0 for one of the coordinates, such as (0, 2) or (–3, 0). The coordinates for the origin, the intersection of the axes, are (0, 0).
Q. Use the following figure to graph the points (2, 6), (8, 0), (5, –3), (0, –7), (–4, –1), and (–3, 4).
A. Notice that the points that lie on an axis have a 0 in their coordinate.
1. Graph the points (1, 2), (–3, 4), (2, –3), and (–4, –1).
Solve It
2. Graph the points (0, 3), (–2, 0), (5, 0), and (0, –4).
Solve It