258 people are going to a theatre show on a Saturday evening. 103 are members of the theatre. Three-fifths of the nonmembers have bought their tickets online. Altogether, two-thirds of the audience have bought their tickets online. a) Complete the two-way frequency table.
There is also a second part to this question. We’ll look at that in a moment. We’ll begin for part a) by adding the numerical information we’ve been given in the question into our table. We know that 258 people are going to the theatre show. That’s the total. And of these 258 people, 103 are members. So we can say that the total number of members is 103.
Once we have all but one piece of information in a column or a row, we can complete that final piece of information. Since there are 258 people at the theatre show and 103 of them are members, we can find the number of nonmembers at the show by finding the difference between these two numbers. That’s 258 minus 103, which is 155.
Next, we’re told that three-fifths of the nonmembers have bought their tickets online. We said that, in total, there were 155 nonmembers at the show. So we need to find three-fifths of 155. Remember, in maths, the word “of” is commonly interchanged with the multiplication symbol. So we could use a calculator to work out three-fifths of 155 by typing three-fifths multiplied by 155. And that tells us that 93 of the nonmembers bought their tickets online.
Alternatively, we could’ve worked out the value of one-fifth by dividing 155 by five and then multiplying that number by three to get the value of three-fifths. Either method will give us an answer of 93. Remember, we said when we had all but one piece of information from a row or a column, we had enough to complete that final piece of information. This means we can find the total number of tickets bought at the ticket booth by nonmembers by subtracting 93 from 155. That’s 62. So 62 nonmembers bought their tickets at the ticket booth.
Finally, we’re told that, altogether, two-thirds of the audience have bought their tickets online. In total, there are 258 people at the show. So we need to work out two-thirds of 258. We could work out two-thirds multiplied by 258. Or we could find the value of one-third by dividing it by three and then multiply that number by two. Either way, we get 172. So we know that, in total, 172 people bought their tickets online.
At this stage, we can either work out the number of tickets bought online by members or the total number of tickets bought at the ticket booth. There’s no real order to this.
To find the number of tickets bought online by members, we’ll subtract the number of tickets bought online by nonmembers from the total. That’s 172 minus 93, which is 79. And to find the total number of tickets bought at the ticket booth, we’ll subtract the total number of tickets bought online from the total number of tickets sold. That’s 258 minus 172, which is 86.
Finally, we need to work out the number of tickets bought at the ticket booth by members. There are two ways we can do this. We could subtract the total number of tickets bought online by members from the total number of members. Or we could subtract the number of tickets bought at the ticket booth by nonmembers from the total number of tickets bought at the ticket booth. Either way, we’re hoping we get the same number for each calculation. That will show us that we’ve probably done the rest of the table correctly.
Either of these calculations does actually give us the number 24. So that’s a nice little way of checking the values in our table. And we can see that 24 members bought their tickets at the ticket booth. And we have a completed two-way frequency table.
Part b) To encourage people to prebook and pay for their tickets online, tickets bought at the ticket booth cost seven percent more than tickets bought online. A ticket bought at the ticket booth cost 26 pounds 75. How much does an online ticket cost?
A common mistake here is to think that we can find seven percent of the increased price and take it off that. That’s simply not true. Instead, we’ll look at two different methods you might have come across.
The first is to consider what we really mean by a seven-percent increase. The original number is 100 percent. So to find a seven-percent increase, that’s the same as finding 107 percent of that original number. We then recall that “percent” means out of 100. And we can find a relevant decimal multiplier that corresponds to this percentage by dividing it by 100. That gives us 1.07.
So if we knew the original price, we could multiply it by 1.07 to find the value of a seven-percent increase. We don’t. So we’re going to say that the online price of a ticket is 𝑥. We do know that increasing this price by seven percent tells us the price of a ticket bought at the booth. It’s 26.75. So we can say that the original online price, 𝑥, multiplied by 1.07 is 26.75.
We need to solve this equation to find the value of 𝑥. And to do that, we’ll perform the inverse operation. The inverse or the opposite to multiplying by 1.07 is to divide by 1.07. 26.75 divided by 1.07 is 25. And we can say that an online ticket cost 25 pounds.
An alternative method relies on the fact that we’re finding 107 percent of the online ticket price. And when we do, we get 26.75. So we can say that 107 percent of the ticket price is 26 pounds 75. We need to work out what one percent is equivalent to. To do that, we’ll divide through by 107. And that tells us that one percent is equal to 0.25 or zero pounds and 25 pence.
We’re trying to get back to the original, and that original is 100 percent. So we’re going to multiply through by 100. And we can see that 100 percent is equal to 25 pounds. And we’ve shown that the online ticket prices are 25 pounds.