Difference between revisions of "2001 AMC 12 Problems/Problem 7"

(Solution 2)
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Hence, <math>A+140</math> has to be a factor of 4002.
 
Hence, <math>A+140</math> has to be a factor of 4002.
  
By inspection, we see that the prime factorization of <math>4002=2\times3\times23\times29
+
By inspection, we see that the prime factorization of <math>4002=2\times3\times23\times29</math>
  
We see that </math>A=34<math> through inspection. We also find that </math>H=23<math>
+
We see that <math>A=34</math> through inspection. We also find that <math>H=23</math>
 +
 
 +
Hence, the price of full tickets out of <math>2001</math> is <math>23\times34=782</math>.
  
Hence, the price of full tickets out of </math>2001<math> is </math>23\times34=782$.
 
 
== See Also ==
 
== See Also ==
  

Revision as of 22:21, 19 May 2017

The following problem is from both the 2001 AMC 12 #7 and 2001 AMC 10A #14, so both problems redirect to this page.

Problem

A charity sells $140$ benefit tickets for a total of $2001$. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?

$\text{(A) } \textdollar 782 \qquad \text{(B) } \textdollar 986 \qquad \text{(C) } \textdollar 1158 \qquad \text{(D) } \textdollar 1219 \qquad \text{(E) }\ \textdollar 1449$

Solution

Let's multiply ticket costs by $2$, then the half price becomes an integer, and the charity sold $140$ tickets worth a total of $4002$ dollars.

Let $h$ be the number of half price tickets, we then have $140-h$ full price tickets. The cost of $140-h$ full price tickets is equal to the cost of $280-2h$ half price tickets.

Hence we know that $h+(280-2h) = 280-h$ half price tickets cost $4002$ dollars. Then a single half price ticket costs $\frac{4002}{280-h}$ dollars, and this must be an integer. Thus $280-h$ must be a divisor of $4002$. Keeping in mind that $0\leq h\leq 140$, we are looking for a divisor between $140$ and $280$, inclusive.

The prime factorization of $4002$ is $4002=2\cdot 3\cdot 23\cdot 29$. We can easily find out that the only divisor of $4002$ within the given range is $2\cdot 3\cdot 29 = 174$.

This gives us $280-h=174$, hence there were $h=106$ half price tickets and $140-h = 34$ full price tickets.

In our modified setting (with prices multiplied by $2$) the price of a half price ticket is $\frac{4002}{174} = 23$. In the original setting this is the price of a full price ticket. Hence $23\cdot 34 = \boxed{(\text{A})782}$ dollars are raised by the full price tickets.

Solution 2

Let the cost of the full price ticket be $x$, let the number of full price tickets be $A$ And half price tickets be $B$

Multiplying everything by two first to make cancel out fractions.


We have

$2Ax+Bx=4002$

And we have $A+B=140\implies B=140-A$

Plugging in, we get $\implies 2Ax+(140-A)(x)=4002$

Simplifying, we get $Ax+140x=4002$

Factoring out the $x$, we get $x(A+140)=4002\implies x=\frac{4002}{A+140}$

Obviously, we see that the fraction has to simplify to an integer.

Hence, $A+140$ has to be a factor of 4002.

By inspection, we see that the prime factorization of $4002=2\times3\times23\times29$

We see that $A=34$ through inspection. We also find that $H=23$

Hence, the price of full tickets out of $2001$ is $23\times34=782$.

See Also

2001 AMC 12 (ProblemsAnswer KeyResources)
Preceded by
Problem 6
Followed by
Problem 8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions
2001 AMC 10 (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions

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