Hi, can someone please help me with this question?
When each of three different numbers is added to the average of two other numbers, the results are 105, 106 and 125. What is the average of the three original numbers?
I'm not exactly sure how to go about this problem... I'm pretty sure I'll need variables, but I'm just having a really hard time setting it up. I'd really appreciate some hlep.
sorry, nevermund, I figured it out. :)
Of course, I'd be happy to help you with this problem!
To solve this problem, let's first set up some variables. Let's call the three different numbers as A, B, and C. We are given that when each of these three numbers is added to the average of two other numbers, the results are 105, 106, and 125.
Now, let's write down the equations for these statements:
A + (B + C)/2 = 105
B + (A + C)/2 = 106
C + (A + B)/2 = 125
To simplify these equations, let's multiply each equation by 2 to eliminate the fractions:
2A + B + C = 210
A + 2B + C = 212
A + B + 2C = 250
Now, we have a system of three equations with three variables. We can solve this system to find the values of A, B, and C. There are several methods to solve a system of equations, such as substitution, elimination, or using matrices. Let's use the substitution method in this case.
From the first equation, we can express A in terms of B and C:
A = 210 - B - C
Substituting this expression for A in the second and third equations, we get:
(210 - B - C) + 2B + C = 212
(210 - B - C) + B + 2C = 250
Simplifying these equations:
-B + C = 2
-B + 3C = 40
Now, we can solve for B and C in this new system of equations. Adding the first equation to two times the second equation, we get:
(-B + C) + 2(-B + 3C) = 2 + 2(40)
-B + C - 2B + 6C = 2 + 80
-3B + 7C = 82
Next, multiplying the first equation by 3 and adding it to the second equation, we get:
3(-B + C) + (-B + 3C) = 3(2) + 40
-3B + 3C - B + 3C = 6 + 40
-4B + 6C = 46
Now, we have a system of two equations with two variables, -3B + 7C = 82 and -4B + 6C = 46. Solving this system, we find B = 20 and C = 14.
Substituting these values back into the first equation, we can find A:
A = 210 - B - C = 210 - 20 - 14 = 176
Therefore, the three original numbers are A = 176, B = 20, and C = 14.
To find the average of the three original numbers, simply add them up and divide by 3:
Average = (A + B + C)/3 = (176 + 20 + 14)/3 = 210/3 = 70
So, the average of the three original numbers is 70.