Dhvanil has a large number of cards. On the back of each card there is either a honeycomb pattern (hexagons) or a brick pattern (rectangles). On the front of each card there is either a \(1\) or a \(2\). As Dhvanil went through all the cards, he found that \(30\%\) of the cards have a honeycomb pattern on the back. Of the cards with a brick pattern on the back, \(80\%\) have a \(1\) on the front.
Determine the percentage of all the cards that have a brick pattern on the back and a \(2\) on the front.
Solution 1
Let’s suppose that Dhvanil has \(100\) cards.
If \(30\%\) of the cards have a honeycomb pattern on the back, that means that \(0.3\times 100 = 30\) cards have a honeycomb pattern on the back. Therefore, \(100-30 = 70\) cards have a brick pattern on the back.
Of the cards with a brick pattern on the back, \(80\%\) have a \(1\) on the front. Therefore, there are \(0.8\times 70 = 56\) cards with a brick pattern on the back and \(1\) on the front. Therefore, \(70-56 = 14\) cards have a brick pattern on the back and a \(2\) on the front.
Therefore, the percentage of all cards with a brick pattern on the back and a \(2\) on the front is \(\dfrac{14}{100}\times 100\% = 14\%\).
Solution 2
If \(30\%\) of the cards have a honeycomb pattern on the back, that means that \(70\%\) of the cards have a brick pattern on the back.
Of the cards with a brick pattern on the back, \(80\%\) have a \(1\) on the front. Therefore, of the cards with a brick pattern on the back, \(20\%\) have a \(2\) on the front. Since \(0.2\times 0.70 = 0.14\), \(14\%\) of all of the cards have a brick pattern on the back and a \(2\) on the front.