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,?

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