Poll results

The following technique is used in chemistry to derive an empirical formula based on given percentages.

First, divide each of the percentages by the smallest percentage. With some luck, we may recognize simple-number ratios. $\begin{array}{rll} 25.79 \div 13.21 & = 1.95231 & \\ 22.64 \div 13.21 & = 1.71385 & \approx 1\tfrac57 \\ 19.81 \div 13.21 & = 1.49962 & \approx 1\tfrac12 \\ 18.55 \div 13.21 & = 1.40424 \\ 13.21 \div 13.21 & = 1.00000 & = 1 \end{array}$ If the number of votes is not too large, the fractions on the right should be exact. This means that the number of people who voted for "7-8" must be a multiple of 14. We multiply all ratios by 14 and see what happens: $\begin{array}{rll} 14 \times 1.95231 & = 27.3323 & \approx 27\tfrac13 \\ 14 \times 1.71385 & = 23.9939 & \approx 24 \\ 14 \times 1.49962 & = 20.9947 & \approx 21 \\ 14 \times 1.40424 & = 19.6593 & \approx 19\tfrac23 \\ 14 \times 1 & = 14.0000 & = 14 \end{array}$ Again, we multiply by the common denominator 3 and find the numbers $82, 72, 63, 59, 42$ for a total of $\boxed{318}$ . It is not difficult to check that the fractions $82/318$ , $72/318$ , etc. indeed correspond to the given percentages.

Brute force method

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perc = [0.2579,0.2264,0.1981,0.1855,0.1321]

N = 1
found = False
while N < 10000 and not found:
    found = True
    for p in perc:
        X = int(round(p*N))
        pp = int(round(X/N*10000) - round(p*10000))
        found = found and (pp == 0)

    if not found:
        N += 1

if found:
    print(N)

Output:

1

318