Suppose Alvin’s savings is x at first; Bernard’s is y at first. They spent ‘z’ respectively at the book fair.

(1-1/10) x : (y+1/10 x)=3:11

(9/10 x-z) : (y+1/10 x -z)=1:9

9/10 x – z +264= y+1/10 x -z

Solve for x,y,z.

x=110 which is the amount Alvin had at first.

a)
After Alvin gave 10% to Bernard’s the ratio becomes,
3 : 11
9 : 33
That means 9u is 90% of what Alvin had at first. At first Alvin would have 100%, which is 10u. So Alvin gave 1u to Bernard
At first the ratio of A : B is as follows,
10 : 32 => 5: 16 ###

b)
Constant difference Strategy
Before spending, A : B => 3 : 11
After spending, A : B => 1 : 9
Both have a difference of 2u for spending. Therefore constant difference makes the above ratio share common units.

9u-1u = 264
u = 264/8 =33

Before spending Alvin has 3u = $99
Before Alvin gave 10% to Bernard = 99/0.9 = $110 ###

(a)
9 – 1 = 8
264/8 = 33 (Alvin’s savings in the end)
33 x 9 = 297 (Bernard’s savings in the end)

Alvin’s savings : Bernard’s savings
3u : 11u

Assuming both spent 1p each,
3u – 1p ——- 33
11u – 1p ——- 297
11u – 3u = 8u ——- 297 – 33 = 264
1u ——- 264/8 = 33

After Alvin gave 1/10 of his saving to Bernard.
Alvin : 3u ——- 3 x 33 = 99
Bernard : 11u ——- 11 x 33 = 363

1 – (1/10) = 9/10 ——– 99
1/10 ——- 99/9 = 11
10/10 = 10 x 11 = 110 (Alvin’s savings at first) (b)
363 – 11 = 352 (Bernard’s savings at first)

The ratio of Alvin’s savings to Bernard’s savings at first is 110 : 352 = 5 : 16.

Ans : (a) 5 : 16; (b) $110.