Equalising Units Concept (Identical Total Amount)

Given: 2 Relations of the Same Set
First
There are some red, blue and white beads in a box. 1/2 of the beads were red, 3/8 of the beads were blue and the remaining beads were white. If there were 12 more red than white beads, how many beads were there in the box.

Fractions Method
Red = 1/2 = 4/8 of total
Blue = 3/8 of total
White = 1 whole - 4/8 - 3/8 = 1/8

4u - 1u = 3u
3u = 12
Total = 8u = 32

Ratio Method
Red : Total
= 1 : 2
= 4 : 8

Blue : Total
= 3 : 8

White = 8 - 4 - 3= 1u
Red - White = 4 - 1 = 3u
3u = 12
Total = 8u = 32

Given: 2 Relations of Different Sets (Same Total Amount)

First

There was an equal number of Strawberry tarts and Blueberry pies in a shop.
After selling 4/9 of the Strawberry tarts and 2/7 of the Blueberry pies, there were 160 Strawberry tarts and Blueberry pies left.
How many Strawberry tarts were sold?

Fractions Method:

(Left)
Strawberry = 1 - 4/9
= 5/9 = 35/63

Blueberry = 1 - 2/7
= 5/7 = 45/63

Left = 35 units + 45 units = 80 units
80 units = 160
1 unit = 160 ÷ 80 = 2

(Sold)
Strawberry = 4/9 = 28/63
28 units = 2 × 28 = 56

Ratio Method:

(Strawberry)
Left : Sold : Original
= 5 : 4 : 9
= 35 : 28 : 63

(Blueberry)
Left : Sold : Original
= 5 : 2 : 7
= 45 : 18 : 63

Left = 35u + 45u = 80u
80u = 160
Strawberry (Sold) = 28u = 160 ÷ 80u × 28u = 2 × 28 = 56

Second

There were an equal number of apples and oranges. Apples were packed equally into 3 boxes and oranges were packed equally into 5 crates. There was a total of 128 fruits in 2 boxes of apples and 2 crates of oranges. Find the total number of apples and oranges at first. 

Third
Alice, Betty, Clara and Denise shared $168. Denise received 1/7 of the total amount of money received by Alice, Clara and Betty. Alice received 3/4 of the total amount of money received by Clara and Betty. Betty received 2/5 as much as Clara. How much did Betty receive?