#
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.