Two numbers have a sum of 56. If one number is one-third of the other, what are the numbers?
To answer this question, we’re going to need to form some equations. We’ll let the two numbers be represented by the letters 𝑎 and 𝑏. The first piece of information we’re given in the question is that the two numbers have a sum of 56. We can express this using algebra as 𝑎 plus 𝑏 is equal to 56. The other piece of information we’re given is that one number is one-third of the other number. Let’s assume then that 𝑎 is the smaller of the two numbers. And we can express this fact as 𝑎 is equal to 𝑏 over three. 𝑎 is one-third of 𝑏.
It’s usually easier to work with integers than fractions though. So an equivalent way of expressing this is that if 𝑎 is equal to 𝑏 over three, then three 𝑎 is equal to 𝑏. We can form this equation by multiplying both sides of our original equation by three.
What we now have is a pair of simultaneous equations in the variables 𝑎 and 𝑏. And we need to solve these equations to find the values of 𝑎 and 𝑏 that satisfy both. Our second equation gives an explicit expression for 𝑏 in terms of the other variable 𝑎. 𝑏 is equal to three 𝑎. We can take this expression for 𝑏 and substitute it into our first equation in place of the 𝑏 here. When we do, we obtain the equation 𝑎 plus three 𝑎 is equal to 56, which is an equation in one variable only.
What we’ve done then is used the method of substitution to swap one variable for an expression in terms of the other. We can now solve this equation to determine the value of 𝑎. Firstly, grouping like terms, we have four 𝑎 is equal to 56. And then dividing both sides of the equation by four, we find that 𝑎 is equal to 14. So we’ve determined the value of one of the two numbers.
To determine the value of the other number, we need to take this value of 𝑎 and substitute it into the equation for 𝑏. 𝑏 is equal to three 𝑎. So that’s three multiplied by 14, which is equal to 42. We found the two numbers then. They are 14 and 42. With a quick check, we can confirm that the sum of these two numbers is indeed 56 and that one-third of 𝑏, that’s one-third of 42, is indeed equal to 𝑎. That’s 14.
So by forming a pair of linear simultaneous equations in these two variables and then solving them using the substitution method, we’ve found that the two numbers, which have a sum of 56 and such that one number is one-third of the other, are 14 and 42.