Practice Makes Perfect Algebra - Carolyn Wheater [7]
Many of the problems you encounter in algebra fall into common types, and if you recognize the type and have a strategy for dealing with it, the problems are easier to solve. Often these problems are about mixing things. A merchant wants to mix different types of coffees or nuts. An amount of money is a mixture of different denominations of coins or bills. A laboratory is mixing chemicals in a solution, or a theater is offering different admission prices for adults and children. Even the famous “Two trains leave Chicago…” problems are mixing the distances traveled by the trains(or cars).
When you’re faced with a problem from this mixture family, you may find it easier to solve if you organize the information in a chart before you try to write and solve an equation. If a merchant is blending 10 lb of coffee, for example, you want a row for each of the individual coffees and the blend, and columns for pounds of coffee, price per pound, and total value. If you call the amount of one coffee x and the other 10 − x and you enter the per-pound price of each, you can multiply across to get the total value. The total value of the individual coffees should add up to the total value for the mixture, so your equation comes from adding down the last column.
Mixing chemicals instead of coffee? The approach is the same, but the column headings change. Your equation still comes from adding down the last column.
For coin problems, the column headings are number of coins and value of the coins, and the equation is formed the same way.
0.1x + 0.25(12 − x) = 1.95
0.1x + 3 − 0.25x = 1.95
−0.15x + 3 = 1.95
−0.15x = −1.05
x = 7
You end up with 7 dimes and 5 quarters.
The cars, trains, and planes take a little more analysis, but the setup of the problem is very much the same. You want a row in the table for each vehicle, and your column headings come from a familiar formula: rate(of speed) times time equals distance. Suppose two trains do leave Chicago, traveling in opposite directions. One travels at 80 mph and the other at 75 mph. When will the trains be 1085 mi apart? Let x be the time it takes for this to happen.
Here’s where the little bit of extra thinking comes in. What do you do with the distances? Since the trains are going in opposite directions, you add the distances. If they were traveling in the same direction at different speeds and you wanted to know how far ahead the faster one had gotten, you’d subtract the distances they’d traveled. Draw a picture to help you imagine what’s happening in a particular problem.
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EXERCISE 2.7
Solve each problem by writing and solving an equation.
1. Jake had 12 coins in his pocket, totaling 95 cents. If the coins were all dimes and nickels, how many nickels did Jake have?
2. Two cars leave Omaha at the same time. One travels east at 55 mph and the other travels west at 65 mph. When are the cars 500 mi apart?
3. You decide to make 10 lb of a peanut-and-raisin mixture to sell at the class snack sale. You can buy peanuts for $2.50 per pound and raisins for $1.75 per pound. If you want to sell the mixture for $2 per pound, how many pounds of peanuts and how many pounds of raisins should you use?
4. A vaccine is available in full strength(100%) and 50% solutions, but latest research shows that the safe and effective concentration for children is 65%. How much full-strength vaccine and how much 50% solution should be mixed to produce 100 mg of a 65% solution?
5. Jessie and her friends pack a tailgate picnic and, at exactly 1 p.m., set out for the football game, driving at 35 mph. Half an hour after they leave the house, Jessie’s mom notices their picnic basket, fully packed, sitting on the driveway. She grabs the basket, jumps in her car, and drives at 40 mph, the legal limit. When will Mom catch up with Jessie?
6. Bert and Harriet collect quarters and pennies. When they wrapped coins to take to the bank, they had $42.50. If they wrapped a total of 410 coins, how many were pennies?
7. Admission to the school fair is $2.50 for students