Mo' Money- Making An "algorithm" To Solve Two Variable Algebra Problems

A cash drawer contains 160 bills, all 10s and 50s. The total value of the 10s and 50s is $1,760. How many of each type of bill are in the drawer? You can figure this out by tr

Solution 1:

Using numpy for the system:

x+y=160

10x+50y=1760

import numpy as np
a = np.array([[1, 1], [10, 50]])
b = np.array([160, 1760])
x = np.linalg.solve(a, b)
print(x)

Outputs:

[156.   4.]

Solution 2:

There are a total of two variables, the number of tens and the number of fifties. So you could do nested for-loops. A really blunt way of doing this would be:

for fifties in range(161):
      for tens in range(161-fifties):
            if (fifties+tens == 160) and (50*fifties + 10*tens == 1760):
                 break

We can improve that a bit by noting that each time we increase the number of fifties, we decrease the possible number of tens:

for fifties in range(161):
      for tens in range(161-fifties):
            if (fifties+tens == 160) and (50*fifties + 10*tens == 1760:
                 break

It can be improved even further by noting that although there are a total of two variables, we have the constraint that they add up to 160. Thus, we can use this constraint to get one given the other:

for fifties in range(161):
      tens =160 - fifties
      if50*fifties + 10*tens == 1760:
           break

Solution 3:

You can take a dynamic programming approach to this for a general solution:

Set up some edge conditions where there is no answer:

1. If the length of notes it less than 1
2. You have one note but the total is not divisible by the denomination there's no answer.

The take a note off and see if it works by recursively calling the function with the remaining notes and adjusted bill-count/total.

defcountNotes(notes, total, bills):
    iflen(notes) < 1:
        returniflen(notes) == 1:
        return [bills] if bills * notes[0] == total elseNonefor count inrange(bills+1):
        amount = notes[0] * count
        rest = countNotes(notes[1:], total - amount, bills - count)
        if rest:
            return [count, *rest]

countNotes([50, 10], 1760, 160)
#[4, 156]

countNotes([100, 20, 5], 173, 2)
# None -- not possible

countNotes([100, 20, 5, 2], 1255, 61)
#[1, 57, 3, 0]

This will return the counts as a list in the same order as the notes passed in. If you're dealing with large lists and totals if could be improved by memoizing it.

Solution 4:

d = {'x': 10, 'y': 50} #create dictionary to hold bill values
total_value = 1760
num_bills = 160

y = (total_value - num_bills *d['x']) / (d['y']-d['x']) #isolating y 

x = num_bills - y # isolating x with y value knownprint("the number of ten dollar bills is: " + str(x))
print("the number of fifty dollar bills is: " + str(y))