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Alex and Sam want to buy tickets for a concert. Alex has 2/3 of the money needed for one ticket, while Sam has 3/5 of the money needed for one ticket. They discover a promotional code that reduces the price of each ticket by $5. With this discount, they find that together they have exactly enough money to buy two tickets.
What is the original price of one concert ticket?
Let’s solve this step-by-step:
1. Let’s say the original price of one ticket is $x.
2. Alex has 2/3 of x, which is (2/3)x
Sam has 3/5 of x, which is (3/5)x
3. Together, they have: (2/3)x + (3/5)x = (10/15)x + (9/15)x = (19/15)x
4. With the $5 discount, each ticket now costs (x – 5)
5. For two discounted tickets, they need: 2(x – 5) = 2x – 10
6. We know that the money they have together is exactly enough for two discounted tickets:
(19/15)x = 2x – 10
7. Now let’s solve this equation:
19x = 30x – 150
150 = 11x
x = 150/11 = 13.64 (rounded to two decimal places)
Therefore, the original price of one concert ticket is $13.64 (or $13.65 if we round up to the nearest cent).
We can verify:
* Alex has: 2/3 × $13.65 = $9.10
* Sam has: 3/5 × $13.65 = $8.19
* Together they have: $9.10 + $8.19 = $17.29
* Two discounted tickets cost: 2 × ($13.65 – $5) = 2 × $8.65 = $17.30
The slight difference ($0.01) is due to rounding.