A = B × 2 = C × 3 = D × 4

So let's look at them one by one.

A = B × 2
So ½ × A = B

A = C × 3
So ⅓ × A = C

A = D × 4
So ¼ × A = D


Therefore,


A + B + C + D

= A + ½ × A + ⅓ × A + ¼ × A

= A × (1 + ½ + ⅓ + ¼)

= A × (12/12 + 6/12 + 4/12 + 3/12)

= A × 25/12


Since A + B + C + D = 450,

A × 25/12 = 450

A = 450 ÷ 25/12

= 450 × 12/25

= 18 × 12

= 216

B = 216 × ½ = 216 ÷ 2 = 108

C = 216 × ⅓ = 216 ÷ 3 = 72

D = 216 × ¼ = 216 ÷ 4 = 54


Check :

A+B+C+D

= 216 + 108 + 72 + 54
= 450

Alternative method (Ratio)

A = B × 2 = C × 3 = D × 4


A = B × 2
So A : B = 2 : 1

A = C × 3
So A : C = 3 : 1

A = D × 4
So A : D = 4 : 1


LCM of 2, 3 and 4 is 12 (2² × 3¹ = 12)

Change all the ratios such that A is 12 units


A : B = 2 : 1 = 12 : 6
A : C = 3 : 1 = 12 : 4
A : D = 4 : 1 = 12 : 3


So we can say A = 12 units, B = 6 units, C = 4 units, D = 3 units

A + B + C + D
= (12 + 6 + 4 + 3) units
= 25 units


Since A + B + C + D = 450,

25 units = 450
1 unit = 450 ÷ 25 = 18


A = 12 × 18 = 216
B = 6 × 18 = 108
C = 4 × 18 = 72
D = 3 × 18 = 54

Model way :


⬜⬜⬜⬜⬜⬜⬜⬜⬜⬜⬜⬜
|---------------------A---------------------|

⬜⬜⬜⬜⬜⬜⬜⬜⬜⬜⬜⬜
|---------B----------|---------B----------|

⬜⬜⬜⬜⬜⬜⬜⬜⬜⬜⬜⬜
|-----C------|------C------|------C-----|

⬜⬜⬜⬜⬜⬜⬜⬜⬜⬜⬜⬜
|---D----|----D---|----D----|----D----|

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