The numbers are in the ratio 2:3.Let the common ratio be x.Therefore the numbers are 2x and 3x.If 5 is added to both the numbers the ratio become 5:7.Therefore, (2x + 5):(3x + 5)= 5 : 7Therefore, 14x + 35 = 15x + 25Therefore, x = 10Thus, the numbers are 2x = 2 x 10 = 20 and 3x = 3 x 10 = 30.

Problem: Two numbers are in the ratio 2:3. If 5 is added to each number, the ratio becomes 5:7. Find the numbers.

Solution:

Let's assume the two numbers in the ratio 2:3 are x and y, respectively.

Step 1: Set up the equations

We can set up the following equations based on the given information:

Equation 1: x/y = 2/3 (since the two numbers are in the ratio 2:3)
Equation 2: (x+5)/(y+5) = 5/7 (since 5 is added to each number and the new ratio is 5:7)

Step 2: Solve the equations

To solve these equations, we can use the method of cross-multiplication.

Equation 1: x/y = 2/3

Cross-multiplying, we get:
3x = 2y

Equation 2: (x+5)/(y+5) = 5/7

Cross-multiplying, we get:
7(x+5) = 5(y+5)
7x + 35 = 5y + 25
7x - 5y = -10 (rearranging the equation)

Step 3: Solve the system of equations

We now have a system of two equations:
3x = 2y
7x - 5y = -10

We can solve this system of equations through substitution or elimination method.

Step 3.1: Solving by substitution

From Equation 1, we can express x in terms of y:
x = (2/3)y

Substituting this value into Equation 2:
7((2/3)y) - 5y = -10
(14/3)y - 5y = -10
(14y - 15y)/3 = -10
-y/3 = -10
y = 30

Substituting the value of y back into Equation 1:
x = (2/3)(30)
x = 20

Therefore, the two numbers are 20 and 30.

Step 3.2: Solving by elimination

Multiplying Equation 1 by 7:
21x = 14y

Multiplying Equation 2 by 3:
21x - 15y = -30

Adding the two equations:
21x + 21x - 15y + 14y = -30
42x - y = -30

Simplifying further:
y = 42x + 30

Substituting this value of y into Equation 1:
x/(42x + 30) = 2/3
3x = 2(42x + 30)
3x = 84x + 60
-81x = 60
x = -60/81
x = -20/27

Substituting the value of x back into Equation 1:
y = (2/3)(-20/27)
y = -40/81

Therefore, the two numbers are -20/27 and -40/81.

Step 4: Conclusion

After solving the equations

20 and 30