How to use nested for-loops to find the x and y of a linear equation

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I'm trying to find how many 10 and 50 dollar bills go into $1760 if there are only 160 bills. I figured, with the help of a friend, that using a nested for-loop is the best way to go but I'm having issues with implementation. My idea is to iterate every x one-by-one until 160 and if then the equation != 1760, I increment y by 1 and restart.

These are the equations: 10x + 50y = 1760, x + y = 160

My desired output is: 10(156) + 50(4) = 1760. But so far, all I've been getting is this:

10(1) + 50(0) = 10
10(2) + 50(1) = 70
10(3) + 50(2) = 130
10(4) + 50(3) = 190
10(5) + 50(4) = 250
...
...
...
10(156) + 50(30) = 3060
10(157) + 50(30) = 3070
10(158) + 50(30) = 3080
10(159) + 50(30) = 3090
10(160) + 50(30) = 3100

And for some reason, y stuck at 30 starting when x = 31. I don't know why. Anyways, this is my code:

for i in range(1, 161):
    print("10" + "(" + str(i) + ") + 50" + "(" + str(y) + ")" + " = " + str((10*i)+(50*y)))
    if ((10*i)+(50*y) < 1760):
        y += 1

I don't want the exact answer. I just want to know where I'm going wrong.

algebra for-loop python
3 Answers

Welcome to StackOverflow!

My idea is to iterate every x one-by-one until 160 and if then the equation != 1760, I increment y by 1 and restart.

The problem here is that you're not actually writing a nested for loop. A nested for loop is of the form:

for(...):
   ...
   for(...):

Instead, you're incrementing y even when you're going through the xs the first time, since you have that if statement. It's getting "stuck" because 30*10+29*50 = 1750, but once you run the loop again, 31*10+30*50 = 1810 which is greater than 1760, and so the if statement evaluates to false and y never increments again. So instead, you'll want to create two for-loops nested like I demonstrated above, and after the inner one finishes, you'll want to increment y by 1 (which is what the for loop does automatically).

Edit from the comments:

Your code will want to look something like this:

for y in range(0,161):
   ...
   for x in range(0,161):
      ... #go through and check if the equations are satisfied

Note that the above is equivalent to:

x = 0
y = 0
while(y <= 160):
   ...
   while(x <= 160):
      ... #go through and check if the equations are satisfied
      x+=1 #add 1 to x
   y+=1 #when the second loop is done iterating, add 1 to y and loop again

If you examine what's happening here, when y=0, the computer is going to see the inner for loop and start running through every number for x between 0 and 160, and then after that point, if no match is found, the loop will run again (and since it's a for loop rather than a while loop, the incrementing of y by 1 already happens automatically).

user_38134110

Why your y gets 'stuck' at 30:

Once the sum exceeds 1760, y never gets incremented again as (10*x + 50*y) < 1760 is always False. This happens at x=31 and y=30, which is why your y is 'stuck' at 30.

Your approach is almost complete but there is a minor bug in the logic:

Instead of running the 10-bills, x, up to 160 and increment the 50-bills, y, by one only if the amount exceeds the desired sum, you can run the 10-bills up to 160 and directly set the amount of 50-bills in each iteration to be y = 160 - x. This is because you enforce this condition (160 = x + y) so only configurations that fulfill this requirement need to be considered.

So, you were right in that you only need a single for loop, but you can set y in every iteration directly as 160 = x + y, and then you only need to check if the amount is correct.

So the Ansatz for this solution looks something like:

for x in range(0, 161):
    y = 160 - x
    # calculate the sum
    # if the sum is correct, break
user_16229370

Can't comment yet so sorry. rb612 has rightly explained the issue.

replace your code after the second line as

for y in range(1, 160-i):
     if((10*i)+(50*y) <= 1760):
          y += 1
     else: break
user_108352610
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