The ratio of two numbers x and y is 3 : 4. If a is added to x and b is added to y, the ratio becomes 11 : 15. Instead if a is subtracted from x and b is subtracted from y, the ratio becomes 7 : 9. Find the smallest value of b, if a, b, x and y are positive integers.

Sure, let's solve this step by step.

Step 1: We know that the ratio of two numbers x and y is 3:4. So, we can write this as: x/y = 3/4

Step 2: We are given that if a is added to x and b is added to y, the ratio becomes 11:15. So, we can write this as: (x+a)/(y+b) = 11/15

Step 3: We are also given that if a is subtracted from x and b is subtracted from y, the ratio becomes 7:9. So, we can write this as: (x-a)/(y-b) = 7/9

Step 4: Now, we have three equations and four variables. To solve for the variables, we can use the method of substitution or elimination.

Step 5: Let's multiply the first equation by 4, the second equation by 15, and the third equation by 9 to make the denominators the same. We get: 4x = 3y 15x + 15a = 11y + 11b 9x - 9a = 7y - 7b

Step 6: Now, let's subtract the third equation from the second to eliminate a and b. We get: 6x = 4y This implies that x = 2/3 * y

Step 7: Substitute x = 2/3 * y into the first equation, we get: 4 * 2/3 * y = 3y This simplifies to y = 6.

Step 8: Substitute y = 6 into the first equation, we get: 4x = 3 * 6 This simplifies to x = 4.5. But since x must be a positive integer, the smallest possible value for x is 5.

Step 9: Substitute x = 5 and y = 6 into the second and third equations, we get: 15 * 5 + 15a = 11 * 6 + 11b 9 * 5 - 9a = 7 * 6 - 7b

Step 10: Solve these two equations simultaneously to find the values of a and b. The smallest possible value for b is 1.