Problem of the Week Problem C and Solution Two Bricks

Problem

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

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.