If 3A = 4B = 5C = 6D and A + B + C + D = 1026, find the values of A, B

3A = 4B = 5C = 6D and all the choices available are positive integers. So,

A = 4B/3 = 5C/3 = 2D.
**A=2D --> choices A and C are out.

3A/4 = B = 5C/4 = 6D/4
** 3A/4=B. A must be divisible by 4. --> choice B is eliminated.

Left with choice D. Let's plug in number:
1) 3A = 4B = 5C = 6D (OK!!)
3(360) = 4(270) = 5(216) = 6(180)
2) A + B + C + D = 1026 (OK!!)
360+270+216+180=1026

Final answer is (D)

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The numbers must be in descending order as A is bigger than B etc. Only options B and D matches this.

Try with D:

3A = 4B
3*360 = 4*270

Answer is D.

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If 3A=4B=5C=6D=60, the following ratio is yielded:
A:B:C:D = 20:15:12:10

Sum of the ratio in blue = 20+15+12+10 = 57
Since the actual sum = 1026, the multiplier for the ratio = \(\frac{1026}{57} = \frac{342}{19} = \frac{9*38}{19} = 9*2 = 18\)

Thus:
A = 20*18 = 360
B = 15*18 = 270
C = 12:18 = 216
D = 10*18 = 180

Let’s express each of the variables in terms of D. We see that A = 6D/3 = 2D; B = 6D/4 = 3D/2; and C = 6D/5. Substituting these quantities in the second equation, we have:

2D + 3D/2 + 6D/5 + D = 1026

Multiplying the equation by 10, we have:

20E + 15D + 12D + 10D = 10,260

57D = 10,260

D = 180

So A = 2(180) = 360, B = 3(180)/2 = 270, and C = 6(180)/5 = 216.

Answer: D

1026
=A + B + C + D
since 3A = 4B, we substitute A with (4/3B)
=(4/3B) + B + C + D
same as A, substitute B with 5/4C
=4/3 * 5/4 C + 5/4C + C + D
eliminate the same number 4 with denominator and numerator on the first term, thus it becomes
=5/3C + 5/4C + C + D
then in the last replace C with 6/5D
=5/3 * 6/5D + 5/4 * 6/5D + 6/5D + D
to simplify, eliminate the same number on the fraction
=6/3D + 6/4D + 6/5D + D
=6/3D + 6/4D + 6/5D + 6/6D
the denominator are 3、4、5、6 respectively, now find the LCM for these four numbers
LCM(3, 4, 5, 6)= LCM(3, 2^2, 5, 2*3) = 2^2 * 3 * 5 =60
expand the fraction to make the denominator, which we set as 60, the same
=120/60D + 90/60D + 72/60D + 60/60D
=342/60D=1026
D= (60*1026)/342 =180 thus ans(D)

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