Answer :
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.
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.
