Find the two numbers whose arithmetic mean is 126, given the ratio between them is one-half.
In order to calculate the arithmetic mean of a set of numbers, we divide the sum of the values by the number of values. When dealing with any two numbers 𝑥 and 𝑦, this can be written as 𝑥 plus 𝑦 divided by two. We are told in this question that the arithmetic mean is 126. Therefore, 𝑥 plus 𝑦 divided by two is 126. Multiplying both sides of this equation by two gives us 𝑥 plus 𝑦 is equal to 252.
We are also told in the question that the ratio between our two numbers is one-half. This means that one number is double the other one. If we assume that 𝑦 is the larger number, we can say that 𝑦 is equal to two 𝑥 or two multiplied by 𝑥.
We now have a pair of simultaneous equations, which we can solve to calculate the values of 𝑥 and 𝑦. If we substitute equation two into equation one, we can replace the 𝑦 with two 𝑥. This leaves us with 𝑥 plus two 𝑥 is equal to 252. Simplifying the left-hand side by grouping like terms gives us three 𝑥. Dividing both sides of this equation by three gives us 𝑥 is equal to 84. 252 divided by three is 84. As 𝑦 is equal to two 𝑥, 𝑦 must be equal to two multiplied by 84. This is equal to 168.
The two numbers whose arithmetic mean is 126, given that the ratio between them is one-half, are 84 and 168. We could check this answer by looking at a number line. 126 is the midpoint or halfway between 84 and 168. It is 42 above 84 and 42 below 168.