Mason has some number cards that each have a whole number on them.

If he chooses a card at random, the probability that the number on it is even is [tex]\frac{2}{9}[/tex].

Work out the ratio of even cards to odd cards that Mason has. Give your answer in its simplest form.



Answer :

Let's start by defining what we know from the problem:

1. The probability of selecting an even card is given as [tex]\(\frac{2}{9}\)[/tex].
2. We need to determine the ratio of even cards to odd cards.

### Step-by-step breakdown:

#### Step 1: Define Variables
- Let [tex]\( E \)[/tex] be the number of even cards.
- Let [tex]\( O \)[/tex] be the number of odd cards.
- The total number of cards is then [tex]\( E + O \)[/tex].

#### Step 2: Set Up the Probability
The probability of picking an even card is:
[tex]\[ \text{Probability(Even)} = \frac{E}{E + O} \][/tex]
Given that this probability is [tex]\(\frac{2}{9}\)[/tex], we can set up the equation:
[tex]\[ \frac{E}{E + O} = \frac{2}{9} \][/tex]

#### Step 3: Solve for the Ratio [tex]\( \frac{E}{O} \)[/tex]
We can manipulate the equation to find the relationship between [tex]\( E \)[/tex] and [tex]\( O \)[/tex].

First, cross-multiply to clear the fraction:
[tex]\[ 9E = 2(E + O) \][/tex]

Next, distribute and solve for [tex]\( O \)[/tex]:
[tex]\[ 9E = 2E + 2O \][/tex]
[tex]\[ 9E - 2E = 2O \][/tex]
[tex]\[ 7E = 2O \][/tex]

Now, isolate [tex]\( O \)[/tex]:
[tex]\[ O = \frac{7E}{2} \][/tex]

#### Step 4: Determine the Ratio [tex]\( \frac{E}{O} \)[/tex]
From the above equation, we can express [tex]\( O \)[/tex] in terms of [tex]\( E \)[/tex]:
[tex]\[ E = E \][/tex]
[tex]\( O = \frac{7E}{2} \)[/tex]

Therefore, the ratio of even cards to odd cards [tex]\( \frac{E}{O} \)[/tex] is:
[tex]\[ \frac{E}{\frac{7E}{2}} = \frac{E \cdot 2}{7E} = \frac{2}{7} \][/tex]

Thus, the ratio of even cards to odd cards in its simplest form is:
[tex]\[ \frac{2}{7} \][/tex]