You saw a shirt for $97. You didn’t have the cash, so you borrowed $50 from your mum and $50 from your dad = $100. You bought the shirt, and had $3 change. You gave your dad $1 and your mum $1 and kept the other $1 for yourself. Now you owe your mum $49 and your dad $49. 49+49 = 98 + your $1 = 99. Where is the missing $1?
[Image is a moving gif of someone making ‘my head is exploding’ type gestures overlaid with explosions and fireworks.]
So it seems like $what-you-owe-mum + $what-you-owe-dad + $the-money-you’ve-got-left should = $100, because $100 is what you borrowed from your parents originally. But it doesn’t. And that’s okay: it shouldn’t. It should = $99, as it does in the story above.
A mathematician could probably see the problem really quickly and explain it very simply, but I’m pretty bad at maths so here’s how I (after some heavy thinking) worked it through.
What’s confusing here is that we have two numbers in play that are very close to each other: $100 (the total you originally borrow from your parents) and £97 (the cost of the shirt). Make it less confusing by imagining that, in stead of borrowing $50 from each parent, you borrow $500 from each parent. Now you have $1,000. You buy the shirt and get $903 change. You give $451 to each parent and you keep $1 for yourself. So now you still owe your mum $500 - $451 = $49, and the same to your dad. $49 + $49 + your $1 = $99. So $what-you-owe-mum + $what-you-owe-dad + $the-money-you’ve-got-left = $99 again.
It’s the same if you originally borrow $54 from each parent, or $67, or $86,914,037. As long as you buy the shirt, keep $1 of the change for yourself, and then pay half of the rest back to each of your parents, you will still always end up owing them $98, which added to your $1 always comes to $99. It’s nothing to do with what you borrow from them: it’s all to do with the price of the shirt.
The shirt costs $97. You start the story with $0. You end the story with the shirt (worth $97) and $1 cash. So you’re up $98. No matter how much you originally borrow from your parents, if you repay all of it except $1, you end the story with $97 of shirt + $1 = $98. And that’s why you always owe your parents $98. Effectively it’s the same as if you just asked your parents to buy the shirt for you and also give you $1. That’s where the $98 comes from.
But where does the $99 come from? Well, this is where the riddle plays its trick. It asks you to add $what-you-owe-mum + $what-you-owe-dad + $the-money-you’ve-got-leftwithout actually telling you why you’d ever want to do that sum. You wouldn’t. It’s a silly sum to do. Because the $1 comes out of what you borrowed. It’s part of what you owe. So why would you add it to the $98, which is already the whole of what you owe? It isn’t a sum you’d never do. If you wanted to work out what you owe your parents, you’d just do $what-you-originally-borrowed-from-mum + $what-you-originally-borrowed-from-dad - $the-money-you’ve-already-paid-back, which is $50 + $50 - $2 = $98. The $1 is already included in that, you don’t have to add it separately. On the other hand, if you wanted to work out what’s happened to the original $100, you’d do a completely different sum. You’d do $what-you-spent-on-the-shirt + $the-money-you’ve-already-paid-back + $the-money-you’ve-got-left, which is $97 + $2 + $1 = $100. There’s no reason ever to do the sum in the riddle.
Does that make sense?