I am trying to add up integer multiples of 4 specific numbers and getting them as close as possible if not exact to a specific number

Question

I want to know how to solve this:

5587.93728 = (101.4*a) + (208.8*b) + (429.8*c) + (884.1*d)

but all the variables a,b,c,d have to be integers . it is ok if its not exact but I want the right side of equation to be as close as possible to the left(less than 1 difference or at least 10)

Thank you!

Answer 1

Another solution: a=4, b=8, c=2, d=3

Answer 2

I'll agree that there is probably some algebra with an optimum solution for the problem.

Being a pragmatic type, I just set up a simple simulator which put in values for each of the coefficients from 0 to a specific max (8,10,5, 4) for each of the variables. {a,b,c,d}. This I ran quite a few times because within just 500 iterations of my simulation, even when I tried changing the randomization parameters, I kept coming up to the same minimum error for the solution (error = difference between right side and left side) , the absolute value of which was .03728 which is equal to the un reducible remainder of the left side of the equation (as none of the values on the right have more than one decimal portion). The simulation converged fairly quickly each time. So, yea... Daniel, this looks like an exact answer. a = 4. b = 8. c = 2 and d = 3

Answer 3

Using the Euclidean algorithm, the least common multiple of 8841 and 4298 is 7.

The least common multiple of 8841 and 2088 is 3.

Thus, these three numbers will be sufficient to solve the problem.

We will let a = 0

Solve 8841/7 a + 4298/7 b = 1 and 8841/3 c and 2088/3 d = 1

1263 a + 614 b = 1

2947 c + 696 d = 1

I took the lazy way to solve and wrote a loop (we will flip the negative sign)

for(a in 1:613){for(b in 1:1262){if(a* 1263 - b * 614 == 1){

print(a)

print(b)

}}}

a = 193

b = 397

for(c in 1:695){for(d in 1:2946){if(c* 2947 - d * 696 == 1){

print(c)

print(d)

}}}

c = 427

d = 1808

Then, as 7 - 2*3 = 1, we have a solution to the triplet diophantine equation

Our solution for 8841 f + 4298 g + 2088 h = 1 will be

f = a - 2 * c

g = -b

h = 2d

(193 - 2 * 427)8841 - 397*4298 +(2 * 1808)2088 = 1

Now, let's revisit the original problem; as the left hand side is 5587.9, we will multiply our solutions by 55879

d = 55879(193 - 2 * 427)

c = 55879(-397)

b = 55879(2*1808)

a = 0

a = 0

b = 202058464

c = -22183963

d = -36936019