A family bought a total of 16 adult and child tickets to a magic show. Adult tickets are $10.50 each and child tickets are $7.50 each. The family paid a total of $141. How many of adult tickets, a, and child tickets, c, did they buy?

9514 1404 393

Step-by-step explanation:

Using the variables defined in the problem statement, we can write the equations ...

  a + c = 16 . . . . . . . . . . . . number of tickets bought

  10.50a +7.50c = 141 . . . cost of tickets bought

Substituting for c in the second equation, we get ...

  10.50a +7.50(16 -a) = 141

  3.00a +120 = 141 . . . . . . . . simplify

  3a = 21 . . . . . . . . . . . subtract 120

  a = 7 . . . . . . . . divide by 3

  16 -a = c = 9 . . . . . . . find the number of child tickets

They bought 7 adult tickets and 9 child tickets.

If you examine the solution, you see that the number of adult tickets (the higher priced item) is the quotient of the difference between the actual expenditure (141) and the amount that would have been spent on all child tickets (16×7.50=120), and the difference in ticket prices (10.50-7.50=3.00). Knowing this, you can often work problems of this kind in your head.

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step-by-step explanation:

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because linear equations are a lot simpler than quadratics equations, and linear equations formula is a lot easier to memorize. that's why i think they're harder.