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Video: Puzzle 0001

Tim Burnham

In this video, we present a short puzzle about the price of a bow and arrow. There's a seemingly obvious (but wrong) solution, so we explain how to use some simple algebra to get to the right answer.


Video Transcript

Welcome to puzzle time. Here’s puzzle number one. This puzzle concerns a bow and arrow salesman with a slightly mismatching jumper and a pair of trousers. Here he is now, jumping up and down saying get your bow and arrow here. Twenty one dollars for the pair! But when his first customer comes along, he says but what if I just want an arrow. How much? Now the bow and arrow salesman doesn’t wanna make it too easy for the customer to just buy an arrow, so they say, “Well the bow is twenty dollars more than the arrow.”

And at this point in the animation, they realised that the bow and arrow salesman hasn’t been jumping up and down as we might have thought at first, but he’s just been floating in the air all the time. But that was soon fixed by the animator. And at that point, the customer neatly sums up the question by saying the bow and arrow are twenty one dollars for the pair. The bow is twenty dollars more than the arrow, how much is the arrow? Okay, you’ve got three seconds to think about it. Maybe you wanna pause the video while you work this out.

Well hopefully, you’ve got an answer for that now, but let’s do a little bit of algebra and see how we can work it out. So let’s define our variables. Let 𝑎 be the cost of the arrow and let 𝑏 be the cost of the bow. And let’s make sure we know what unit we’re specifying those variables to be and it’s dollars in this case. So we know that the arrow plus the bow cost twenty one dollars together, so 𝑎 plus 𝑏 is equal to twenty one.

And we also know that the bow cost twenty dollars more than the arrows, so the 𝑏 is the price of the arrow plus twenty. And what we’ve created is two simultaneous equations. We’ve got equation number one and we’ve got equation number two. Now in equation number two, we can see that the value 𝑏 is equal to 𝑎 plus twenty. So we can replace the 𝑏 in equation one with 𝑎 plus twenty like this. And now we’ve got an equation with just one thing that we don’t know, so we can do a bit of simplifying and rearranging and we can solve this.

Well without any modification going on, the brackets are fairly irrelevant, so we can ignore those for now. And we’ve got 𝑎 plus 𝑎 which is two 𝑎 and then another twenty on the left-hand side and we’ve got twenty-one on the right-hand side. Now if I subtract twenty from both sides of my equation then twenty minus twenty is nothing on the left-hand side, so I’m just left with two 𝑎. And on the right-hand side, twenty-one take away twenty is just one. So I know that two 𝑎 is equal to one. Well I want to know what one 𝑎 is, the cost of one arrow. So if I divide both sides by two, I get an answer of two 𝑎 divided by two is just 𝑎 and one divided by two is a half or nought point five.

Now remember that 𝑎 was in dollars, so that’s nought point five dollars, but we don’t just leave that as one decimal place. If we’re dealing with money, we’re gonna keep that in to two decimal places or we could just say 𝑎 is fifty cents. So the price of an arrow is nought point five dollars or fifty cents. And although the question didn’t ask for it, we can work out the price of a bow. So that’s 𝑎 nought point five plus twenty. So that’s twenty point five. And again, being money, we’ve got to have two decimal places. So the bow is twenty dollars and fifty cents. Hope you worked it out and hope you found our explanation useful.