1. **State the problem:** A class of students wants to buy a birthday present. If each pays 8.5, they are short 12. If each pays 10, they have 30 extra. We need to find: (a) The number of students. (b) The price of the present. 2. **Define variables:** Let $n$ = number of students. Let $P$ = price of the present. 3. **Form equations:** If each pays 8.5, total collected is $8.5n$, but this is 12 less than $P$: $$8.5n = P - 12$$ If each pays 10, total collected is $10n$, which is 30 more than $P$: $$10n = P + 30$$ 4. **Rewrite equations:** From first: $$P = 8.5n + 12$$ From second: $$P = 10n - 30$$ 5. **Set equal and solve for $n$:** $$8.5n + 12 = 10n - 30$$ Bring terms to one side: $$8.5n + 12 - 10n + 30 = 0$$ $$-1.5n + 42 = 0$$ $$-1.5n = -42$$ $$n = \frac{-42}{-1.5}$$ Show cancellation: $$n = \frac{\cancel{-42}}{\cancel{-1.5}} = 28$$ 6. **Find price $P$ using $n=28$:** $$P = 8.5 \times 28 + 12$$ Calculate: $$P = 238 + 12 = 250$$ 7. **Answer:** (a) Number of students is $28$. (b) Price of the present is $250$.