Math
Problem Solving

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What is a problem?

For the present, we can say that a problem consists of three types of information, namely:

1. givens

2. operations

3. goals

Suggested approach for problem solving:

A pencil and paper is recommended and please use them extensively!! The basic and most important step is to read the problem and represent the information in either symbolic or diagrammatic form. Then identify the three types of information. It is usual to start with the goals.

Decide on symbols

• single letters e.g., x, y, t
• multiple letters and subscripts e.g., t₀ , t_p for time initially and time after p seconds

Use expressions; e.g., Let x kg be Bill's weight and then (x+30) kg be John's weight.

Now represent the givens in symbolic form using the symbols defined for the goals. The operations may include:

• add, subtract, multiply, divide, exponents
• derivative, integral
• vector operations
• matrix operations

As you get extensive practice in problem solving techniques, you will become skilled so that the process becomes more automatic.

 

Worked example

Tom, Anne, and Paul mow lawns to earn money. They each have a lawn mower and they accept a contract to mow 5900 sq ft of lawn together, using all three machines. Tom mows 70 sq ft/minute, Anne 50 sq ft/minute and Paul 40 sq ft/minute. Anne and Paul start mowing the lawn at the same time, but Tom has trouble starting his mower and is delayed for 30 minutes. All three stop mowing at the same time, when the lawn is finished. How long does Tom mow?

Procedure:

• Read and re-read information.
• Define goal or goals.
• Represent goal/goals in symbolic form.
• Represent givens in symbolic form.
• Solve equations.

Define the goal and represent it in symbolic form.

Let Tom mow for x minutes.

Note: units are minutes

Now represent givens:

 

Note: Do not be afraid of writing down words.

Tom mows at 70 sq ft per minute.

in x minutes he will mow 70x sq ft

Tom was delayed in starting by 30 minutes.

Anne and Paul mowed for (x+30) minutes each.

Anne mows at 50 sq ft per minute

in (x+30) minutes, Anne will mow 50(x+30) sq ft

Paul mows at 40 sq ft per minute

in (x+30) minutes, Paul will mow 40(x+30) sq ft.

Together they mow 5900 sq ft.

70x + 50(x + 30) + 40(x + 30) = 5900

70x + 50x + 1500 + 40x + 1200 = 5900

160x = 5900 – 2700 = 3200

x = 3200 / 160 = 20

Tom mows for 20 minutes.

Check:

Tom mows for 20 minutes and will mow	1400 sq ft.
Anne mows for 50 minutes and will mow	2500 sq ft.
Paul mows for 50 minutes and will mow	2000 sq ft.
Total	5900 sq ft.