Two numbers are such that their difference, their sum, and their product are in the ratio 1 : 4 : 15. What are the two numbers?

(a-b):(a+b):ab = 1:4:15

(a-b)/(a+b) = 1/4
4a-4b = a+b
3a = 5b

(a-b)/ab = 1/15
15a-15b = ab
15a-9a = a(3a/5)
30a = 3a^2
a = √10
b = 3/5 √10

15a+15b = 4ab
15a + 9a = 4a(3a/5)
120a = 12a^2
a = √10

a-b = 2/5 √10
a+b = 8/5 √10
ab = 6 = 30 * 2/5 √10

Dang! Doesn't work out at the end.
15(a-b) = 6√10, not just 6

Where's my mistake?

To find the two numbers, let's assume the first number is x and the second number is y.

Given that their difference, sum, and product are in the ratio 1:4:15, we can write the following equations:

1. Difference: y - x = k (where k is a constant)
2. Sum: x + y = 4k
3. Product: x * y = 15k

To solve these equations, we can use substitution or elimination method.

Let's solve using substitution method:
From equation 2, we can write x = 4k - y

Substituting this value of x in equation 3:
(4k - y) * y = 15k
4ky - y^2 = 15k
y^2 - 4ky + 15k = 0

Now we have a quadratic equation in terms of y. We can solve it using the quadratic formula:
y = [-b ± √(b^2 - 4ac)] / 2a

For this equation, a = 1, b = -4k, and c = 15k. Substituting the values, we get:
y = [4k ± √((-4k)^2 - 4*1*15k)] / 2*1
y = [4k ± √(16k^2 - 60k)] / 2
y = [2k ± √(4k^2 - 15k)] / 1

To simplify this equation further, we need to find the discriminant of the quadratic equation. The discriminant is given by b^2 - 4ac. Substituting the values, we get:
Discriminant = (-4k)^2 - 4*1*15k
Discriminant = 16k^2 - 60k

If the discriminant is positive, we will have two real and distinct values of y. If the discriminant is zero, we will have one real and repeated value of y. If the discriminant is negative, we will have no real values of y.

So, set the discriminant equal to zero and solve for k:
16k^2 - 60k = 0
k(16k - 60) = 0

Either k = 0 or 16k - 60 = 0

If k = 0, then we won't have any values for y.
If 16k - 60 = 0, then k = 3.75

Substituting k = 3.75 in the expression for y, we get:
y = [2(3.75) ± √(4(3.75)^2 - 15(3.75))] / 1
y = [7.5 ± √(56.25 - 56.25)] / 1
y = [7.5 ± √0] / 1
y = 7.5 ± 0
y = 7.5

Therefore, the two numbers are x = 4(3.75) - 7.5 = 7.5
And y = 7.5

So, the two numbers are 7.5 and 7.5.