To solve this problem, let's denote the two numbers as \(2x\) and \(5x\), where \(x\) is a common multiplier.
According to the problem, when 4 is added to each number, the new numbers are in the ratio \(4 : 9\). Therefore, we can set up the following relationship:
[tex]\[
\frac{2x + 4}{5x + 4} = \frac{4}{9}
\][/tex]
Next, we cross-multiply to remove the fractions:
[tex]\[
9(2x + 4) = 4(5x + 4)
\][/tex]
Expanding both sides, we get:
[tex]\[
18x + 36 = 20x + 16
\][/tex]
Now, we need to collect like terms to solve for \(x\). Subtract \(18x\) and 16 from both sides:
[tex]\[
18x + 36 - 18x = 20x + 16 - 18x - 16
\][/tex]
Simplifying, we have:
[tex]\[
20 = 2x
\][/tex]
To find \(x\), we divide both sides by 2:
[tex]\[
x = 10
\][/tex]
Now that we have the value for \(x\), we can determine the original numbers:
[tex]\[
2x = 2 \cdot 10 = 20
\][/tex]
[tex]\[
5x = 5 \cdot 10 = 50
\][/tex]
Thus, the two numbers are 20 and 50.
