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]
