Video Transcript
Find the two numbers whose arithmetic mean is 77 and the difference between them is 64.
We recall that in order to calculate the arithmetic mean of a set of numbers, 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, then the arithmetic mean is equal to 𝑥 plus 𝑦 divided by two. We are told that this is 77. Multiplying both sides of this equation by two gives us 𝑥 plus 𝑦 is equal to 154. The sum of our two numbers is 154. We will call this equation one.
We are also told that the difference between our numbers is 64. If 𝑥 is the larger number, this means that 𝑥 minus 𝑦 is equal to 64. We will call this equation two, and we now have a pair of simultaneous equations. We could solve these by substitution or elimination. In this case, we’ll use the elimination method.
We begin by adding equation one and equation two, as this will eliminate the 𝑦-terms. Remember, when the signs are the same, we should subtract. And when the signs are different, we add. 𝑥 plus 𝑥 is equal to two 𝑥. 𝑦 plus negative 𝑦 is equal to zero. So, the 𝑦’s cancel. 154 plus 64 is equal to 218. Dividing both sides of this equation by two gives us 𝑥 is equal to 109 as a half of 218 is 109.
We can now substitute this value of 𝑥 back in to equation one or two. In this case, we will choose equation one. This gives us the equation 109 plus 𝑦 is equal to 154. Subtracting 109 from both sides of this equation gives us 𝑦 is equal to 45.
The two numbers 45 and 109 have a difference of 64 and a sum of 154. This means that their arithmetic mean is 77. We could also have shown this on a number line as 45 and 109 are in equal distance away from 77. 77 is 32 below 109 and 32 above 45.
