SOLUTION: -Adrian and Betty have been teaching for a total of 36 years. -Charlie and Betty have been teaching for a total of 22 years. -Charlie and Adrian h

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Question 980579: -Adrian and Betty have been teaching for a total of 36 years.
-Charlie and Betty have been teaching for a total of 22 years.
-Charlie and Adrian have been teaching for a total of 28 years.
How long has each been teaching?
Found 3 solutions by ikleyn, Alan3354, MathTherapy:
Answer by ikleyn(53763)   (Show Source):
You can put this solution on YOUR website!

Let a be the number of years Adrian has been teaching.
Let b be the number of years Betty has been teaching.
Let c be the number of years Charlie has been teaching.

Then we have the system of three equations in three unknowns

a + b       = 36,         (1)
      b + c = 22,         (2)
a +       c = 28.         (3)

It is very special kind of system, and we will apply special method to solve it.
Add all three equations (left sides and right sides separately).
You will get

2(a + b + c) = 86,     or

a + b + c = 43.         (4)

Now, distract the equation (1) from the equation (4). You will get

c = 43 - 36 = 7.

Hence, Charlie has been teaching for 7 years.
Next, distract the equation (2) from the equation (4). You will get

a = 43 - 22 = 21.

Hence, Adrian has been teaching for 21 years.

Next, distract the equation (3) from the equation (4). You will get

b = 43 - 28 = 15.

Hence, Betty has been teaching for 15 years.

For more problems similar to this one see the lesson The tricks to solve some word problems with three and more unknowns using mental math in this site.

Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website!
-Adrian and Betty have been teaching for a total of 36 years.
-Charlie and Betty have been teaching for a total of 22 years.
-Charlie and Adrian have been teaching for a total of 28 years.
How long has each been teaching?
==================
a + b + 0 = 36
0 + b + c = 22
a + 0 + c = 28
---------------
---
a + b + 0 = 36
0 + b + c = 22
------------------- Subtract
a - c = 14
a + c = 28 3rd equation
---------------------------- Add
2a = 42
a = 21
Can you do the rest?

Answer by MathTherapy(10809)   (Show Source):
You can put this solution on YOUR website!

-Adrian and Betty have been teaching for a total of 36 years.
-Charlie and Betty have been teaching for a total of 22 years.
-Charlie and Adrian have been teaching for a total of 28 years.
How long has each been teaching?

Let number of years Adrian, Betty, and Charlie have been teaching be A, B, and C, respectively
We then get:
A + B = 36______B = 36 - A -------- eq (i)
C + B = 22 -------- eq (ii)
C + A = 28 -------- eq (iii)
C + 36 - A = 22 ----- Substituting 36 - A for B in eq (ii)
C - A = 22 - 36
C - A = - 14 -------- eq (iv)
2C = 14 ------------- Adding eqs (iii) & (iv)
C, or amount of time Charlie has been teaching = , or years
7 + B = 22 ---------- Substituting 7 for C in eq (ii)
B = 22 - 7
B, or amount of time Betty has been teaching = years
7 + A = 28 ---------- Substituting 7 for C in eq (iii)
A = 28 - 7
A, or amount of time Adrian has been teaching = years