Three cards are on a table. Each card has a letter on one side and a positive number on the other side. One card has an R on it, one card has a G on it, and one card has a B on it. The number side of each card is face down on the table. The following facts are known about the three concealed numbers:
The product of the number on the card with an R and the number on the card with a G is equal to the number on the card with a B.
The product of the number on the card with a G and the number on the card with a B is \(180\).
Five times the number on the card with a B is equal to the number on the card with a G.
Determine the product of the numbers on the three cards.
Solution 1
Let the three numbers be represented by \(r,\) \(g\), and \(b\).
Since the product of the number on the card with an R and the number on the card with a G is equal to the number on the card with a B, \(r \times g=b\). We are looking for \(r \times g \times b = (r \times g) \times b=(b) \times b= b^2\). So when we find \(b^2\) we have found the required product \(r \times g \times b\).
We are also given that \(g \times b=180\) and \(g =5 \times b\). Substituting \(g=5\times b\) into \(g \times b=180\), we have \((5 \times b)\times b=180\) or \(5 \times b^2=180\). Dividing by \(5\), we obtain \(b^2=36\). This is exactly what we are looking for since \(r\times g\times b=b^2\).
Therefore, the product of the three numbers is \(36\).
For those who would like to know the values of the three numbers, we can continue to solve. We know \(b^2=36\), so \(b=6\), since \(b\) is a positive number. Therefore, \(g=5 \times b=5 \times 6=30\). And finally, \(r \times g=b\) so \(r \times (30)=6\). Dividing by \(30\), we get \(r=\frac{6}{30}=\frac{1}{5}\).
We can verify the product \(r \times g \times b=(\frac{1}{5})\times(30)\times(6)=6\times 6=36\).
Solution 2
In this solution we try to determine the numbers by working with the factors of \(180\).
The product of the number on the card with a G and the number on the card with a B is \(180\), and the number on the card with a G is five times the number on the card with a B. The number \(180\) can be written as \(2\times 2\times 3\times 3\times 5\). If G and B are integers, then G and B must share the same factors, along with G having one more factor, that being \(5\). By playing with the factors of \(180\), we find a possibility is that the number on the card with a G is \(5\times 2\times 3\) and the number on the card with a B is \(2\times 3\). That is, the number on the card with a G could be \(30\) and the number on the card with a B could be \(6\).
Now using the fact that the number on the card with an R multiplied by the number on the card with a G is equal to the number on the card with a B, we determine that some number multiplied by \(30\) is equal to \(6\). It follows that the number on the card with an R would be \(6\div 30=\frac{1}{5}\).
The product of the three numbers is \(\frac{1}{5}\times 30 \times 6=6\times 6=36\).
This solution works because the number on the card with a G and the number on the card with a B happen to be integers.