7.5 Solve problems
The first step in translating a word problem into a mathematical equation is to use a variable as the name for the unknown quantity. For example, if the unknown is the number of chocolates that someone has, then we can call that number \(a\). If you have double the number of chocolates compared to that someone, then you have \(2 \times a\) or \(2a\) number of chocolates.
Now, if together you have \(10\) chocolates, then the equation will look like this:
\[a + 2a = 10\]Worked Example 7.20: Setting up and solving equations from word problems
Leen and Themba attend school together. Leen has three times as many chocolates as Themba. Let the number of chocolates that Themba has be \(a\), and use this to answer the questions that follow.
- Write an expression for the number of chocolates that Leen has, in terms of \(a\).
- Altogether, Leen and Themba have \(28\) chocolates. How many chocolates does Themba have?
Write Leen’s number of chocolates in terms of \(a\).
We are told that Leen has three times as many chocolates as Themba has.
To get an expression for Leen’s chocolates we must multiply \(a\) by \(3\).
Leen’s chocolates \(= 3 \times a = 3a\)
Create an equation.
We know that (Themba’s chocolates) \(+\) (Leen’s chocolates) \(= 28\)
Now we can substitute the expressions for Leen and Themba’s chocolates in terms of \(a\):
\[(3a) + (a) = 28\]Solve the equation.
We have an equation that we can solve, using techniques that we know already.
\[\begin{align} 3a + a &= 28 \\ 4a &= 28 \end{align}\]Since \(4 \times \mathbf{7} = 28\), that means \(a = \mathbf{7}\).
Therefore, Themba has \(7\) chocolates.