A word problem is usually a description of a situation, followed by a question. As much as anything, it is a problem in reading comprehension. You are to figure out what the situation is exactly and what relationships are being described. You are to use those relationships (along with your knowledge of the world) to develop an answer to the question.
The question usually tells you very specifically what you are to find. It often works well to assign variables to the things you are to find. (Sometimes, there are intermediate values that must be figured first. Not in this problem.)
The "words" usually tell you the relationships between the variables. Here, we're talking about revenue from ticket sales. You are expected to use your real-world knowledge of ticket sales to know that for each kind of ticket, the price is multiplied by the number of tickets, and those products are added to get the total sales for the day.
Given
- 4 senior tickets and 5 student tickets are sold for $102
- 7 senior tickets and 5 student tickets are sold for $126
Find
- the selling price of a senior ticket
- the selling price of a student ticket
Solution
It is always a good idea to start by looking at what you are given. Here, the number of student tickets sold is the same each day, so the difference in revenue is entirely from the difference in senior tickets sold.
Selling 3 more senior tickets increased ticket sales by $126 -102 = $24. This means each senior ticket is worth $24/3 = $8. Already, you're half done with the problem.
If a senior ticket is $8, then 4 of them is $32. The other $102 -32 = $70 of sales the first day must be made from 5 student tickets. So, a student ticket is
... $70/5 = $14.
Now, we can answer the question.
- The price of a senior ticket is $8.
- The price of a student ticket is $14.
_____
If you want to define variables and write equations, you can let x represent the price of a senior citizen ticket, and y represent the price of a student ticket. The given information tells us
... 4x +5y = 102
... 7x +5y = 126
In the above, we solved these equations by elimination (subtracting the first equation from the second one). There are other methods available. One is graphing. The attached graph shows the lines described by these equations intersect at the point (x, y) = (8, 14). This is the set of values of x and y that will satisfy both equations, hence is the solution. You have to translate those variable values to a statement that answers the question posed by the problem. (See the bold text above for an example.)
