Seven ninths of the people present in a hall are sitting on nine thirteenths of the chairs available, and the rest of the people are standing. If there are 28 empty chairs, how many chairs would still be empty if everyone in the hall were sitting?

Introduction / Context:
This is a classic arithmetic reasoning problem based on fractions and word interpretation. It involves people, chairs, and the concept of empty seats. By carefully translating the statements into algebraic equations involving fractions, you can find both the total number of chairs and the total number of people, and then answer the question about how many chairs remain empty if every person sits down.

Given Data / Assumptions:
- Let the number of people in the hall be P.- Let the total number of chairs be C.- Seven ninths of the people are sitting.- These seated people occupy nine thirteenths of the chairs.- There are 28 empty chairs under this seating arrangement.- We assume all chairs are identical and each chair can hold one person.

Concept / Approach:
Translate the fractions into equations. If seven ninths of the people are sitting, then the number of seated people is (7 / 9) * P. The problem states that those seated people occupy nine thirteenths of the chairs, so the number of occupied chairs is (9 / 13) * C. Since each seated person uses exactly one chair, these two quantities must be equal. At the same time, we know the number of empty chairs is C minus the occupied chairs, and this is given as 28. Using these two relations, we can solve for C and then find P. Finally, if every person sits, the number of empty chairs will be C - P.

Step-by-Step Solution:
- Number of seated people = (7 / 9) * P.- Number of occupied chairs = (9 / 13) * C.- Equate seated people and occupied chairs: (7 / 9) * P = (9 / 13) * C.- The number of empty chairs is C - (9 / 13) * C = (4 / 13) * C.- We are told that empty chairs are 28, so (4 / 13) * C = 28.- Solve for C: C = 28 * 13 / 4.- Compute 28 * 13 = 364, and 364 / 4 = 91, so C = 91.- Now find P using (7 / 9) * P = (9 / 13) * 91.- Compute (9 / 13) * 91 = 9 * 7 = 63.- So (7 / 9) * P = 63, which gives P = 63 * 9 / 7 = 81.- If everyone sits, the number of empty chairs becomes C - P = 91 - 81 = 10.

Verification / Alternative check:
Check the original situation using C = 91 and P = 81. Seated people are (7 / 9) * 81 = 63. Chairs used are (9 / 13) * 91 = 63. Empty chairs are 91 - 63 = 28, which matches the given information. Under full seating, 81 chairs are used and 10 remain empty. This confirms the solution.

Why Other Options Are Wrong:
- Option 15: Would correspond to P = 76 given C = 91, which does not respect the initial fraction relations.- Option 12 and option 18: Both contradict either the fraction equalities or the given 28 empty chair condition.- Option 8: Similarly fails to satisfy the equations when you recompute the number of people and chairs.

Common Pitfalls:
Students often misinterpret which fraction applies to people and which to chairs, or they forget to equate the number of seated people with the number of occupied chairs. Another common error is to ignore the condition about empty chairs, which is essential to compute the total number of chairs. Work step by step: write precise equations, solve for one variable at a time, and then answer the final question about empty chairs when everyone sits.

Final Answer:
If everyone in the hall were sitting, there would still be 10 chairs empty.