Question

What are the 3 numbers in Marlene's puzzle if the quotient of a & c is 8, the quotient of a & b is -16, the quotient of b & c is -0.5, the product of a & c is 2, the product of a & b is -1,?

Answer 1

A = 4, b = -0.25, c = 0.5

or a = -4, b = 0.25, c = -0.5



Write down what you know in the form of equations:

a / c = 8

a / b = -16

b / c = -0.5

a * c = 2

a * b = -1



Now see if you can express one variable in terms of another. The a * c = 2 expression looks like a nice place to begin.

a * c = 2



Divide both sides by c

a = 2/c



Now look for another expression that uses a and c. The expression a/c = 8 looks like a good possibility.



a/c = 8



Substitute the expression defining a in terms of c into the expression

(2/c)/c = 8



Multiply both sides by c

2/c = 8c



Multiply both sides by c again

2 = 8c^2



divide both sides by 8

2/8 = c^2

1/4 = c^2



Take the square root of both sides

1/2 = c

0.5 = c



Now that we know the value for c, Calculate a using the a = 2/c expression we made.

a = 2/c = 2/.5 = 4



So we now know that a = 4, and c = 0.5

Look for an expression using b that we can solve. a / b = -16 looks like a good place to try

a / b = -16



Substitute the known value of a

4 / b = -16



Multiply both sides by b

4 = -16 b



Divide both sides by -16

-4/16 = b



Simplify

-0.25 = b



So a = 4, b = -0.25, c = 0.5

Verify using the equations given

a / c = 8: 4 / 0.5 = 8 Correct.

a / b = -16: 4 / (-0.25) = -16 Correct

b / c = -0.5: -0.25 / 0.5 = -0.5 Correct

a * c = 2: 4 * 0.5 = 2 Correct

a * b = -1: 4 * -0.25 = -1 Correct



All of the quotients and products come out as they should, so the values determined for a,b,c are correct.



Note: Since the square root was taken for the equation

0.25 = c^2

0.5 = c

The value of either 0.5, or -0.5 is correct for c. So this problem actually has 2 solutions.

They are a = 4, b = -0.25, c = 0.5

or a = -4, b = 0.25, c = -0.5