Question

It costs dylan $x$ dollars to buy $3$ tickets to the zoo. he discovers that if he were to buy $5$ tickets, he would receive a total discount of $12$ dollars. how much cheaper is the cost of a single ticket when dylan buys $5$ total tickets instead of $3$?

Answer 1

Answer:

$2.40

Step-by-step explanation:

cost originally=x/3

5 tickets=5x/3-12

so

(x/3)-(1/5)((5x/3)-12)=x/3-x/3+12/5

We get 12/5 which equals 2.40

Answer 2

So, if Dylan has x dollars and he bought 3 tickets with them, the tickets were priced at k dollars per ticket. If he bought 5 tickets with the x dollars and saved 12 total dollars, it would be the same as buying the tickets with x-12 dollars, so we have:

$\frac{x}{3}=k\\ \frac{x-12}{5}=q$

So, with this we have:

$x=3k\\ x=5q+12$

If we're looking for a number that satisfies these constraints, we can work with modular arithmetic. We have:

$x\equiv 0 \pmod{3}\\ x\equiv 12 \pmod{5}$

So, we can use the chinese remainder theorem here. So, we clearly have x=3k, which means:

$3k\equiv12 \pmod{5} \implies\\ k \equiv 4 \pmod{5} \implies\\ k=5j+4$

So, since we have x=3k, we also have x=3(5j+4)=15j+12.

So, clearly j=0 won't work so we should have j=1. That means our money per ticket for the five tickets is:

$\frac{27-12}{5}=3$

And our money per three tickets is:

$\frac{27}{3}=9$

This is easily verifiable. Three tickets needs 27 dollars and 5 tickets needs 15 dollars, which is 12 less than 27 dollars. So we have our money per three dollar ticket at 6 more than money per five dollar.