A man’s age is nine more than two times his son’s age. Given that the sum of their ages is 57, find each of their ages.
It may be tempting to try and answer this question using trial and error. But instead let’s take an algebraic approach. We don’t know the ages of either of these two people. So we’ll use some letters to represent them. We’ll let the man’s age be represented by the letter 𝑚 and the son’s age be represented by the letter 𝑠. We can then form some equations using the information given in the question.
Firstly, we were told that the man’s age is nine more than twice his son’s age. Well, if his son’s age is 𝑠, then twice his son’s age is two 𝑠. And nine more than this is two 𝑠 plus nine. We can therefore form the equation 𝑚 is equal to two 𝑠 plus nine. The other piece of information we’re given is that the sum of their ages is 57. So we can form a second equation: 𝑚 plus 𝑠 equals 57.
What we now have is a pair of linear simultaneous equations in the unknowns 𝑚 and 𝑠. To solve these equations, we should observe that our first equation gives an explicit expression for the variable 𝑚 in terms of the other variable. This means that we can substitute this expression for 𝑚 into the second equation. And this will give an equation in 𝑠 only.
Let’s see what this looks like then. We take our second equation, and where we had 𝑚, we replace this with two 𝑠 plus nine, which gives the equation two 𝑠 plus nine plus 𝑠 is equal to 57. That’s an equation in one variable only. We can now solve this equation to determine the value of 𝑠. Grouping the like terms on the left-hand side gives three 𝑠 plus nine equals 57. We can then subtract nine from each side to give three 𝑠 is equal to 48. Finally, we can divide both sides of the equation by three, giving 𝑠 is equal to 48 over three, which is 16. This tells us then that the son is 16 years old.
All that remains is to find the age of his father. We can do this by taking the value we found for 𝑠 and substituting it into either of the two equations. Let’s use equation one. This gives 𝑚 is equal to two multiplied by 16 plus nine. That’s 32 plus nine, which is equal to 41. Finally, we should check our answer by confirming that the sum of the ages of the man and his son is equal to 57, which it is.
So, by forming a pair of linear simultaneous equations, which we then solved using the substitution method, we found that the son is 16 years old and the man is 41 years old.