What does x equal?

Question

I have 4 numbers -- a, b, c, and d -- that I can pair up in 6 different ways to multiply -- ab, ac, ad, bc, bd, cd .when I do, I get products in the set {5, 6, 10, 12, 18, x}. what does x equal ?

My solution was as follows:

$cd = \frac{(ac)(bd)}{ab} = \frac{6 * 18}{5} = \frac{108}{5}$

or

$cd = \frac{(ad)(bc)}{ab} = \frac{10 * 12}{5} = 24$

I solved some of the equations together to get these values of (a, b, c) $(\frac{\sqrt{10}}{2}, \sqrt{10}, \frac{6 \sqrt{10}}{5})$

Now if I use these values to get d, I get two different values:

if I used the equation $ad = 10$ thus $d = 2 \sqrt{10}$ and $cd = \frac{6 \sqrt{10}}{5} * 2 \sqrt{10} = 24$

if I used the equation $bd = 18$ thus $d = \frac{9 \sqrt{10}}{5}$ and $cd = \frac{6 \sqrt{10}}{5} * \frac{9 \sqrt{10}}{5} = \frac{108}{5}$

Is this solution correct? I got confused that there could be more than one value for x!

Answer 1

To answer if this solution was correct before the edit, as pointed out in the comments, the mistake made by OP is assuming that the products are given in order, which they need not be. Such an assumption led to an inconsistent system, which is why OP calculated differing values.

To answer if this solution was correct after the edit, the mistake made by OP is that they didn't verify that they had a consistent system (which they didn't).
EG For $ (a, b, c) = ( \frac{ \sqrt{10}}{2} , \sqrt{10}, \frac{ 6\sqrt{10}}{5})$, if $ ad = 10$, then $ d= 2\sqrt{10}$. But then, we do not have 2 terms whose product is 18, so this doesn't satisfy the conditions of the problem.

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To answer the problem that was posed, the hint I provided, was to consider what the value of $abcd$ is. The solution sketch follows, fill in any gaps as needed.

1. $abcd = ab \times cd = ac \times bd = ad \times bc$.

2. Out of the 5 given products, we must find two pairs of products which multiply to the same value. This will be a possible value of $abcd$.

3. Hence, conclude that the only possible value of $abcd$ is $ 5 \times 12 = 10 \times 6 = 18 \times x$.

4. Thus, the missing term is $ x = \frac{10}{3}$.

   • Because there is only 1 way to get $abcd$, this missing term is uniquely determined. See below for more details.

5. For completeness, let's establish that this system is consistent (else the answer is "no solution exists"). Solving $ ab = 5, ac = 10, ad = 18, abcd = 60$ gives us $ a = \sqrt{15}, b = \sqrt{5/3}, c = \sqrt{20/3} d = \sqrt{ 108/5}$. (Unique up to sign)

   • There are more solution sets that satisfy the original conditions, like taking $ ab = 5, ac = 10, ad = \frac{10}{3}, abcd = 60$.

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To answer OP's question of "could there be multiple answers", then this would require multiple sets of "two pairs which have the some product", which give multiple possible values of $abcd$, which hopefully might give multiple possible values of $x$.

As an explicit example, if the products in the set were $\{1, 2, 4, 8, 16, x \}$, then by following the steps in the above solution, we could have:

• $1 \times 16 = 2 \times 8 = 4 \times x$, so the missing term is $x = 4$. This can be achieved by $ a = \frac{1}{\sqrt{2}}, b = \sqrt{2}, c = 2\sqrt{2}, d = 4 \sqrt{2}$.
• $ 1 \times 8 = 2 \times 4 = 16 \times x $, so the missing term is $x = \frac{1}{2}$. This can be achieved by $ a = \frac{1}{2\sqrt{2} }, b = \sqrt{2} , c = 2\sqrt{2} , d = 4\sqrt{2} $.