Every week Roger pays for a movie ticket and a soda out of his allowan

Let the value of A = 100, the cost of his movie ticket = T and the cost of his soda = S.

Difference between A and cost of the soda = 100 – S

Difference between A and cost of the movie ticket = 100 – T

As per the information given about the cost of the movie ticket,

T = 20% (100 – S) = \(\frac{1}{5}\) (100 – S) = 20 –\(\frac{ 1}{5}\) * S

And the cost of the soda, S = 5% (100 – T) = \(\frac{1}{20}\) (100 – T) = 5 – \(\frac{1}{20}\) * T

Therefore, T = 20 – \(\frac{1}{5}\) * S and S = 5 – \(\frac{1}{20}\) * T

Substituting the value of S in the first equation,

T = 20 – \(\frac{1}{5}\) [5 – \(\frac{1}{20}\) * T]

T = 20 – 1 + \(\frac{1}{100}\) * T

T = 19 + \(\frac{1}{100}\) * T
.
T – \(\frac{1}{100}\) * T = 19

\(\frac{99 }{ 100}\) * T = 19

T = \(\frac{19 }{ 99}\) * 100 ≅ 19% of A, since A = 100

Substituting the value of T in the equation,

S = 5 – \(\frac{1}{20}\) [\(\frac{19}{99}\) * 100]

S = 5 – \(\frac{19}{99}\) * 5

S = 5 [1 – \(\frac{19}{99}\)]

S = 5 [\(\frac{80}{99}\)]

S = \(\frac{400 }{ 99}\)

Therefore, S = \(\frac{4}{99}\) * 100 ≅ 4% of A, since A = 100.

So, Roger paid approximately 23% of his allowance A, to pay for his movie ticket and soda.
The correct answer option is D.