Let A be the number of stamps Alberto has and K be the # stamps Kaye has.

K/A = 5/3
(K-10)/(A+10) = 7/5

Solve this equation for A and K and get A=90 and K=150

That's the number before the gift.
After the gift
A = A + 10 = 90 + 10 = 100
K = K - 10 = 150 - 10 = 140

How many more?
140-100 = 40

Let the number of stamp Kaye an Alberto have be x and y respectively.
At first, x : y = 5:3
or $$\frac{x}{y}$$ = $$\frac{5}{3}$$
5y = 3x or 3x-5y = 0 ....................................equation 1
As a rsult of the gift Kaye now has (x-10) stamps and Alberto have (y+10) stamps, and the ratio is x-10 : y+10 = 7:5
or
$$\frac{x-10}{y+10}$$ $$\frac{7}{5}$$
= 5(x-10) = 7(y+10)
5x-50 = 7y+70
or 5x-7y = 120.........equation 2
Solving the two equations simultaneously, we have:
3x-5y=0......(i) * 5
5x-7y=120.......(ii) * 3
Now we have 15x -25y=0
- 15x-21y=360
--------------------------------
-4y =-360
or y= $$\frac{360}{4}$$
=90
SInce y = 90, 3x-5(90) =0
3x=450, 450/3
x=150

As a result of the gift KAye has (150-10)stamps, which is 140, and Alberto now have (90+10 =100) stamps
Therefore, the ratio of their number of stamps will be
= 140:100
=14:10
=7;5
which means we are correct.
Therefore, as a result of the gift Kaye have
(1140-100) stamps more than Alberto
=40 stamps