SOLUTION: the average of four numbers is 40. the third is 12 more than the second. the second is five times the first. the fourth is 8 less than the first. find the numbers.

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Question 629795: the average of four numbers is 40. the third is 12 more than the second. the second is five times the first. the fourth is 8 less than the first. find the numbers.
Answer by Charles3475(23) (Show Source):
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Let the four numbers be a, b, c, d

(a+b+c+d)/4=40 (Definition of average)

a+b+c+d = 160 (multiply by 4)

c = b + 12 (Given)

b = 5*a (Given)

d = a - 8 (Given)

c = (5*a) + 12 (Substitution)

Each number has been expressed in terms of "a", we may now solve for "a" by substituting each expression back into the original equation.

a+ 5*a + (5*a + 12) + (a-8) = 160 (Substitution)

a + 5*a + 5*a + 12 + a -8 = 160 (Expansion of terms)

12*a +4 = 160 (add like terms)

12*a = 156 (subtract 4 from both sides)

a = 13 (divide both sides by 4)

With "a" we may solve for each of the other numbers.

b = 5*a = 5*13 = 65 (Substitution and simplify)

c = 5*a + 12 = 5*13 + 12 = 77 (Substitution and simplify)

d = a - 8 = 13 - 8 = 5 (Substitution and simplify)

13 + 65 + 77 + 5 = 160 (Check)