If the arithmetic mean between 𝑎 and 𝑏 is nine and the arithmetic mean between seven 𝑎 and five 𝑏 is 15, then 𝑏 minus 𝑎 equals blank.
Let’s start out by defining what we mean by arithmetic mean. The arithmetic mean is the sum of the values divided by the number of values. And that means we can set up an equation for the arithmetic mean of 𝑎 and 𝑏, which would be 𝑎 plus 𝑏 divided by two. We need the sum of the values of 𝑎 and 𝑏, and there are two values, so we divide that by two. But we already know that this mean is equal to nine. We can set up a second equation for the arithmetic mean of seven 𝑎 and five 𝑏. This would be seven 𝑎 plus five 𝑏 divided by two.
Again, we’re only dealing with two terms, and that means we need to divide their sum by two. And we’ve been told that this is equal to 15. At this point, we’ll want to rearrange the equations to see if we can solve simultaneously. On the left, if we multiply both sides of the equation by two, we get 𝑎 plus 𝑏 equals 18. And if we do the same thing with our second equation, we’ll get seven 𝑎 plus five 𝑏 equals 30. Now that we have two equations with variables 𝑎 and 𝑏, we could solve equation one for one of our variables and use substitution.
If we wanted to solve simultaneously, we see that in our second equation our 𝑏-variable is five 𝑏. So if we take our first equation and multiply through by negative five, we get an equivalent equation that says negative five 𝑎 minus five 𝑏 equals negative 90. And now we can combine both of these equations and eliminate that 𝑏-variable. Since five 𝑏 minus five 𝑏 equals zero, seven 𝑎 minus five 𝑎 equals two 𝑎, and 30 minus 90 equals negative 60. If two 𝑎 equals negative 60, then 𝑎 equals negative 30. And we know from our first equation that 𝑎 plus 𝑏 must equal 18. So we plug in negative 30 for 𝑎 and add 30 to both sides of the equation to give us 𝑏 equals 48.
We haven’t finished solving here since we’re trying to solve for 𝑏 minus 𝑎, which will be 48 minus negative 30. Make sure you recognize that 𝑎 is a negative value. We’re subtracting a negative. Negative 48 minus negative 30 is equal to 48 plus 30, which is 78. Therefore, 𝑏 minus 𝑎 equals 78.