9.5 Solving problems
The first step in solving problems that involve algebraic equations is usually to translate the problem into an equation. If the equation is correct and you found the right answer, it should make sense when you put the answer back into the initial problem.
For example, if you are solving a problem about ages in a family, you should never end up with child’s age bigger than the parent’s age. Always check your answer when solving a problem. If it is correct, it should make sense.
Worked example 9.15: Finding the sum of consecutive integers
The sum of five consecutive integers is \(-65\). What is the value of the largest integer?
Write expressions for the consecutive integers.
Consecutive means that the numbers go up by one each time, such as \(5; 6; 7; 8; 9\) or \(- 11; - 10; - 9; - 8; - 7\).
- First, we name one of the numbers as a variable: let the smallest integer be \(z\).
- Then the next integer will be \(z + 1\).
- The third integer will be one more than this: \((z + 1) + 1 = z + 2\).
- In the same way, the fourth integer will be \(z + 3\).
- The fifth integer will be \(z + 4\).
So, the five consecutive integers are \(z;z + 1;z + 2;z + 3;z + 4\).
Set up the equation.
We were told that the sum of the consecutive integers is \(-65\). This means that if we add the numbers together we will get \(-65\).
\[z + (z + 1) + (z + 2) + (z + 3) + (z + 4) = - 65\]Solve the equation.
Now we can solve this equation:
\[\begin{align} 5z + 10 &= - 65 \\ 5z + 10 - 10 &= - 65 - 10 \\ 5z &= - 75 \\ \frac{5z}{5} &= - \frac{75}{5} \\ \therefore z &= - 15 \end{align}\]Answer the question.
The question asks us to determine the largest integer. The largest integer is \(z + 4\), so:
\[z + 4 = - 15 + 4 = - 11\]The largest integer is \(-11\).
We can check that the answer is correct by adding up the integers:
\[( - 15) + ( - 14) + ( - 13) + ( - 12) + ( - 11) = - 65\]
Worked example 9.16: Solving the age-old problem
Jamie’s father is \(36\) years older than he is. In \(7\) years’ time, his father will be double Jamie’s age. How old is Jamie’s father now?
Write expressions for their ages now.
Let Jamie’s age be 𝑘. We are told that his father is \(36\) years older than he is.
Present ages:
Jamie \(= k\)
Father \(= k + 36\)
Write expressions for their ages in \(7\) years’ time.
In \(7\) years’ time, both Jamie and his father will be \(7\) years older. This means that we can add \(7\) to both of their present ages to get an expression for their ages in \(7\) years’ time.
Future ages:
Jamie \(= k + 7\)
Father \(= k + 36 + 7 = k + 43\)
Create an equation and solve it.
We were also told that in \(7\) years’ time, Jamie’s father will be double Jamie’s age. This means that if we multiply the expression for Jamie’s future age by \(2\), we will get the value of his father’s future age.
\[\begin{align} 2 \times \text{(Jamie’s future age)} &= \text{(father’s future age)} \\ \therefore 2(k + 7) &= k + 43 \\ 2k + 14 &= k + 43 \\ 2k - k + 14 &= k - k + 43 \\ k + 14 &= 43 \\ k + 14 - 14 &= 43 - 14 \\ \therefore k &= 29 \end{align}\]Jamie is \(29\) years old today.
This question has asked us to calculate the age of Jamie’s father today. We know that Jamie’s father is \(36\) years older than Jamie, so Jamie’s father is \(29 + 36 = 65\) years old today.
You can check your answer by calculating their ages at both points in time. Jamie’s father is \(65\) years old today. In \(7\) years’ time, Jamie will be \(36\) years old, and his father will be \(72\) years old. This is double Jamie’s age in \(7\) years’ time.