Mr Bogacz arranged a maths race. First prize would receive half of the prize money, second place would receive one fifth of the money, third place would receive one seventh of the money. One fourteenth of the money was to be used for administration costs. The remainder of the money was pocketed by Mr Bogacz himself as he thought no one would notice. a) What fraction did Mr Bogacz take? b) If Mr Bogacz took \( \$ 84 \), what is the prize money that third place would have received?

To find the fraction that Mr. Bogacz took, we first determine the total fractions allocated for the prizes and administration. The fractions are: - First prize: \( \frac{1}{2} \) - Second prize: \( \frac{1}{5} \) - Third prize: \( \frac{1}{7} \) - Administration costs: \( \frac{1}{14} \) Finding a common denominator for these fractions, which is 70, gives: - First prize: \( \frac{35}{70} \) - Second prize: \( \frac{14}{70} \) - Third prize: \( \frac{10}{70} \) - Administration costs: \( \frac{5}{70} \) Adding these fractions together: \( \frac{35+14+10+5}{70} = \frac{64}{70} \) Mr. Bogacz takes the remainder: \( 1 - \frac{64}{70} = \frac{6}{70} = \frac{3}{35} \) So, Mr. Bogacz took \( \frac{3}{35} \) of the total prize money. For part b, if Mr. Bogacz took $84, we set up the equation based on his fraction: \( \frac{3}{35} \times \text{Total Prize Money} = 84 \) Solving for the total prize money: Total Prize Money = \( 84 \div \frac{3}{35} = 84 \times \frac{35}{3} = 980 \) Now we can find what third place would have received: Third place rate \( = \frac{1}{7} \) of the total prize money. Third place prize = \( \frac{1}{7} \times 980 = 140 \). Thus, third place would have received $140.