Question Video: Finding Two Numbers given Their Arithmetic Mean and Product Mathematics

Video Transcript

Find the two numbers whose arithmetic mean is 308.5 and product is 616.

We recall that in order to calculate the arithmetic mean of any data set, we divide the sum of the values by the number of values. If we let the two numbers in this question be 𝑥 and 𝑦, their arithmetic mean will be equal to 𝑥 plus 𝑦 divided by two. We are told that this is equal to 308.5. Multiplying both sides of this equation by two gives us 𝑥 plus 𝑦 is equal to 617. We will call this equation one. We are also told that the product of the two numbers is 616. This means that 𝑥 multiplied by 𝑦 is equal to 616. If we call this equation two, we have a pair of simultaneous equations.

We can solve this pair of simultaneous equations by substitution. However, in this case, we might spot the answer by inspection. One multiplied by 616 is 616, and one plus 616 is equal to 617. The pair of numbers that have a sum of 617 and a product of 616 are one and 616. This means that these are the two numbers whose arithmetic mean is 308.5 and product is 616.

Let’s imagine that we hadn’t noticed this and solve the equations by substitution. Rearranging equation one by subtracting 𝑥 from both sides gives us 𝑦 is equal to 617 minus 𝑥. We can substitute this into equation two such that 𝑥 multiplied by 617 minus 𝑥 is equal to 616. Distributing our parentheses or expanding our brackets gives us 617𝑥 minus 𝑥 squared is equal to 616.

This is a quadratic equation that we can solve by factoring when it is equal to zero. Adding 𝑥 squared and subtracting 617𝑥 from both sides gives us zero is equal to 𝑥 squared minus 617𝑥 plus 616. In order to factor or factorize this equation, we need to find a pair of numbers with a product of 616 and a sum of negative 617. These values are negative one and negative 616 such that our two sets of brackets or parentheses are 𝑥 minus one and 𝑥 minus 616.

For the product of these two parentheses to equal zero, either 𝑥 equals one or 𝑥 equals 616. The corresponding values of 𝑦 are the same values. When 𝑥 equals one, 𝑦 is 616. And when 𝑥 is 616, 𝑦 is equal to one. This confirms that the two numbers whose arithmetic mean is 308.5 and product is 616 are one and 616.