The average of two numbers 'A' and 'B' is 20; that of 'B' and 'C' is 19; and that of 'C' and 'A' is 21. What is the value of 'A'?

Steps to Solve

1. Set up equations from the averages

From the problem, we have the following averages:

• The average of (A) and (B) is 20:

  $$ \frac{A + B}{2} = 20 $$ Multiplying both sides by 2 gives:

  $$ A + B = 40 , , , (1) $$

• The average of (B) and (C) is 19:

  $$ \frac{B + C}{2} = 19 $$ Multiplying both sides by 2 gives:

  $$ B + C = 38 , , , (2) $$

• The average of (C) and (A) is 21:

  $$ \frac{C + A}{2} = 21 $$ Multiplying both sides by 2 gives:

  $$ C + A = 42 , , , (3) $$

2. Solve the equations

Now we have a system of equations:

1. ( A + B = 40 )
2. ( B + C = 38 )
3. ( C + A = 42 )

We can solve these equations step by step.

3. Express one variable in terms of another

From equation (1):

$$ B = 40 - A , , , (4) $$

Substituting equation (4) into equation (2):

$$ (40 - A) + C = 38 $$

Simplifying:

$$ C = 38 - 40 + A $$ $$ C = A - 2 , , , (5) $$

4. Substituting C into equation (3)

Substituting equation (5) into equation (3):

$$ (A - 2) + A = 42 $$ This simplifies to:

$$ 2A - 2 = 42 $$

5. Solve for A

Adding 2 to both sides gives:

$$ 2A = 44 $$

Dividing both sides by 2:

$$ A = 22 $$