3.4 Solving simple problems by forming and solving equations

Mathematics

Grade 9

Number patterns, graphs, equations,

Statistics and probability

Module 16

Solving simple problems by forming and solving equations

ACTIVITY 1

To solve simple problems by forming and solving equations

[LO 2.2, 2.4]

How do we solve this problem?

Jamie’s dad is four times as old as Jamie. His father is 40. How old is Jamie?

a) We could think really hard, and try a few guesses. For instance: If Jamie is 1 year old, then his father must be 4. Not correct. What about 2 years old? And so on.

b) Make a table: Complete the empty spaces. Does this help?

c) Draw a flow diagram: Is this useful?

• As you can see, there is not much difference between the three methods – you can use any one that suits you. But they are not really very useful, because you have to do a lot of guessing. If the problems become very difficult, these methods become impossible to use.
• So we will use a better way.
• The way to do this problem is to make an equation from the information, and then to solve the equation and use the solution to write down an answer.
• Making equations is not very easy to start with, but it gets a lot easier with practice.
• An equation needs an equal sign as well as a variable (we usually use x , but it can be any letter).

Jamie’s dad is four times as old as Jamie. If his father is 40, how old is Jamie?

• This is the general form of the equation:
• Four times Jamie’s age = father’s age, which is 40.
• The question asks Jamie’s age; so we let Jamie’s age be x .
• Now the equation becomes: 4 x = 40

• and this asks us to find a number which gives 40 when multiplied by 4. If we divide 40 by 4, we get 10. So x must be 10, and therefore Jamie is ten years old. It is easy to see that this answer is correct.

A problem for you to do: Leon has 48 marbles. Amy has only a third as many as Leon; how many does she have? Use the last method.

The way you should set out your answer is:

State what the variable represents.

• Write an equation using the variable.
• Solve the equation.
• Write down the solution, namely what the value of the variable is.
• Write down the answer to the problem in words.

Please note that the first and last steps are in ordinary words, and the middle step(s) in algebra.

E x ercise:

Find the answers to the following problems:

1. Mr Jacobs has R295,45 in his pocket. Mrs Jacobs has R55,30 less than her husband in her purse. How much money does she have in her purse?

2. I think of a number. I multiply it by 7 and divide the answer by three. I get 49. What was the number I first thought of? Remember to check your answer.

3. In America Joanie buys an item that is marked $5,75. She works out that it would be R41,69 when she converts the dollars to rand. What is the rand/dollar exchange rate she used?

ACTIVITY 2

To develop effective methods for solving more complicated equations

[LO 2.3, 2.4]

• We will be doing more word problems, but we have to concentrate more on the algebra in our solutions. Make sure that you know e x actly what each of the steps means and why it is done.
• Some of the following problems are followed by the solutions. But try to find the answer without looking ahead. Then compare your answer and the way you set it out with the answer.