## 8.5 Problem solving with algebraic expressions

## Worked Example 8.21: Solving problems with algebraic expressions

Kabelo has \(k\) sweets. Precious has \(4\) more sweets than Kabelo. Dakalo has twice as many sweets as Precious.

- Give simplified expressions, in terms of \(k\), for:
- the number of sweets that Precious has
- the number of sweets that Dakalo has.

- Determine a simplified expression, in terms of \(k\), for the number of sweets that Kabelo, Precious, and Dakalo have in total.

### Write an expression for Precious’s sweets.

We don’t know how many sweets Kabelo has. We use the variable \(k\) to stand in the place of Kabelo’s sweets.

Precious has \(4\) more sweets than Kabelo. Kabelo has \(k\) sweets.

Therefore, Precious has \(k + 4\) sweets.

### Write an expression for Dakalo’s sweets.

Dakalo has twice as many sweets as Precious. Precious has \(k + 4\) sweets.

\[2(k + 4) = 2k + 8\]Therefore, Dakalo has \(2k + 8\) sweets.

### Add the three expressions together, and simplify.

- Kabelo has \(k\) sweets.
- Precious has \(k + 4\) sweets.
- Dakalo has \(2k + 8\) sweets.

We must add the expressions together to get the total.

\[\begin{align} & k + (k + 4) + (2k + 8) \\ & = k + k + 4 + 2k + 8 \\ & = 4k + 12 \end{align}\]Therefore, they have \(4k + 12\) sweets in total.

Once we know the number of sweets Kabelo has (which is \(k\)), we will also be able to tell how many sweets Precious and Dakalo each have. If Kabelo has \(8\) sweets, then:

- Precious has \(k + 4 = 8 + 4 = 12\) sweets
- Dakalo has \(2k + 8 = 2\left(8 \right) + 8 = 24\) sweets
- and in total they have \(8 + 12 + 24 = 44\) or \(4k + 12 = 4\left( \mathbf{8} \right) + 12 = 32 + 12 = 44\) sweets

## Worked Example 8.22: Solving critical thinking problems with algebraic expressions

Raya has simplified an expression using the distributive law. Her working out is shown below. But, there is one term missing from within the brackets of the expression that she had to simplify.

\[(5f^{2} + \mathbf{?})( - 1) = - 5f^{2} - 4f\]What is the missing term?

### Figure out what will help you get the answer.

This question is like doing the distributive law in reverse: you know the expanded expression and have to work out something about the “original” expression. In this case, you need to find the second term in the brackets.

Start by breaking the equation down into matching parts. You need to work out how the terms on the right are related to the expression on the left.

\[(5f^{2} + \mathbf{?})( - 1) = - 5f^{2} - 4f\]You can see that \(−1\) times \(5f^{2}\) leads to the \(- 5f^{2}\) in the answer. That means that the \(- 4f\) in the answer must come from multiplying \(−1\) with the missing term. Focus on those terms:

\[( - 1)(?) = - 4f\]### Work out the answer.

What number(s) should you put in place of the question mark to make the expression true? (Since this is about multiplication, you can use the signs to figure out the sign of the missing term.)

\[( - 1)(?) = - 4f\]When you think that you have an answer, it is very important that you test it by substituting it into the expression above. If your answer does not work out correctly when you substitute it into the expression, you need to try again!

The correct value for the missing term is: \(4f\), because \(- 1 \times 4f = - 4f\).