Video: AQA GCSE Mathematics Higher Tier Pack 4 • Paper 2 • Question 8

Daniel was running a raffle. There were 300 green tickets numbered from 1 to 300 and 500 blue tickets numbered from 1 to 500. The tickets cost £1 each. Green tickets ending in zero zero or five zero win a prize each of which cost Daniel £25. Blue tickets ending in 3 win a prize each of which cost Daniel £2.50. At the end of the day, all of the prizes had been won, and there were still some losing tickets which hadn’t been sold. Given that Daniel made a profit of £92, calculate the number of losing tickets that Daniel sold.

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Video Transcript

Daniel was running a raffle. There were 300 green tickets numbered from one to 300 and 500 blue tickets numbered from one to 500. The tickets cost one pound each. Green tickets ending in zero zero or five zero win a prize each of which cost Daniel 25 pounds. Blue tickets ending in three win a prize each of which cost Daniel two pound 50. At the end of the day, all of the prizes had been won, and there were still some losing tickets which hadn’t been sold. Given that Daniel made a profit of 92 pounds, calculate the number of losing tickets that Daniel sold.

So first of all, to solve this problem, what we want to do is look at the number of winning green tickets and winning blue tickets there are. So I’m gonna start by looking at the winning green tickets. So we know that for a green ticket to win, it needs to end in zero zero or five zero. And we also know that there are 300 green tickets numbered from one to 300. So the winning green tickets are going to be 50, 100, 150, 200, 250, and 300. So there are going to be six winning tickets.

So now, we’re gonna take a look at the winning blue tickets. And a blue ticket is a winner if it ends in a three. And we know that there are 500 blue tickets numbered from one to 500. Well, the winning blue tickets, therefore, would be three, 13, 23. And then we carry on this pattern all the way up to 483, 493. So I could work this out by writing them all out. But is there a quicker way to do this? Well yes, there is, because what we can do is work out the number of winning tickets per 100 tickets, and then multiply this by five because we know there are 500 blue tickets.

So as we can see, for the first 100 tickets, we’re gonna have three, 13, 23, 33, 43, 53, 63, 73, 83, and 93. So that’ll be 10 winning tickets per 100 tickets. And this will carry on for the next 100 tickets. For instance, we’d have 103, 203, et cetera. So now, what we’re gonna do is multiply it by five because as we said, there are 500 tickets. So there are 5 hundreds. So when we do that, we get five multiplied by 10 which is equal to 50. So we’re gonna have 50 winning blue tickets.

So now, we’ve worked out how many winning tickets there’s gonna be for green and blue. What’s the next stage? So the next step is to work out the total money spent on prizes. And once again, we’re gonna start with the green tickets. So we can look at the prizes that will be received if you get a green winning ticket. Well, the amount of money that Daniel spent on the green prizes is six, because that’s the number of winning tickets, multiplied by 25 pounds, because that was the cost of a prize for one of the green ticket prizes. And then, we’re gonna add on the cost for the blue prizes. And that’s gonna be 50, because that’s the number of winning tickets, multiplied by two pound 50. And that’s because that was how much it cost Daniel to buy a prize for the blue tickets.

So when we add these together, we get 275 pounds. And we got that because if we do six multiplied by 25, well, we know four multiplied by 25 is 100. So then two more gives us 150 pounds. And then, if we’ve got 50 multiplied by two pound 50, well, what we can do is we can say it’s the same as five multiplied by 25 which would give us 125 pounds. Add these together, we get our 275 pounds. So that’s our total money spent on prizes.

So the next thing we want to work out is how much money he made from ticket sales. Well, we don’t know how many tickets were sold. But we do know how to calculate this because we’re told that he made a profit of 92 pounds. Well, if he’d made 92 pounds profit, then he must have made enough money to cover the total that he spent on prizes. That’s 275 pounds plus the 92 pounds he made as a profit, which is gonna be equal to 367 pounds. So this is gonna be the total money from the ticket sales. So therefore, we know the number of tickets sold. And that’s gonna be 367. And that’s because the tickets each cost one pound. So therefore, all we do is 367 divided by one which gives us 367. So we know that Daniel sold 367 tickets.

Well, does that mean that we’ve completed the question? Well, no, because the question asks us how many losing tickets did Daniel sell. Well, how can we calculate that? So therefore, to work out the number of losing tickets sold, it’s gonna be equal to 367, which is the total number of tickets sold, minus 50, because that was the number of blue winning tickets, minus six, because that was the number of green winning tickets. And we know we can subtract 50 and six because we’re told that all of the prizes have been won. So we’re gonna get a total of 311. So therefore, we can say that the number of losing tickets that Daniel sold was 311.

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