Section 2.6More about Problem Solving

Objectives

Solve investment problems Solve uniform motion problems Solve liquid mixture problems Solve dry mixture problems Solve number-value problems

Objective 1: Solve Investment Problems

To find the amount of simple interest I an investment earns, we use the formulaI = Prt.P is the principal (the amount invested), r is

the annual interest rate, and t is the time in years.

EXAMPLE 1 Paying Tuition

A college student invested the $12,000 inheritance he received and decided to use the annual interest earned to pay his tuition cost of $945. The highest rate offered by a bank at that time was 6% annual simple interest. At this rate, he could not earn the needed $945, so he invested some of the money in a riskier, but more profitable, investment offering a 9% return. How much did he invest at each rate?

Objective 2: Solve Uniform Motion Problems

If we know the rate at which we will be traveling and the time we will be traveling at that rate, we can find the distance d traveled by using the formula d = rt.

EXAMPLE 2 Rescues at Sea

A cargo ship, heading into port, radios the Coast Guard that it is experiencing engine trouble and that its speed has dropped to 3 knots (this is 3 sea miles per hour). Immediately, a Coast Guard cutter leaves port and speeds at a rate of 25 knots directly toward the disabled ship, which is 56 sea miles away. How long will it take the Coast Guard to reach the ship?

(Sea miles are also called nautical miles.)

Objective 3: Solve Liquid Mixture Problems

We now discuss how to solve mixture problems. In the first type, a liquid mixture of a desired

strength is made from two solutions with different concentrations.

EXAMPLE 4 Mixing Solutions

A chemistry experiment calls for a 30% sulfuric acid solution. If the lab supply room has only 50% and 20% sulfuric acid solutions, how much of each should be mixed to obtain 12 liters of a 30% acid solution?

Amount of solution x strength of the solution = amount of pure sulfuric acid

Objective 4: Solve Dry Mixture Problems

In another type of mixture problem, a dry mixture of a specified value is created from two differently priced ingredients.

EXAMPLE 5 Snack Foods Because cashews priced at $9 per pound were not selling, a

produce clerk decided to combine them with less expensive peanuts and sell the mixture for $7 per pound. How many pounds of peanuts, selling at $6 per pound, should be mixed with 50 pounds of cashews to obtain such a mixture?

Objective 5: Solve Number–Value Problems

When problems deal with collections of different items having different values, we must distinguish between the number of and the value of the items.

For these problems, we will use the fact that

Number x value = total value

EXAMPLE 6 Dining Area Improvements

A restaurant owner needs to purchase some tables, chairs, and dinner plates for the dining area of her establishment. She plans to buy four chairs and four plates for each new table. She also plans to buy 20 additional plates in case of breakage. If a table costs $100, a chair $50, and a plate $5, how many of each can she buy if she takes out a loan for $6,500 to pay for the new items?