The bat and ball puzzle is almost everywhere and you may well have seen it:
Together, a bat and ball cost $1.10. The bat costs $1.00 more than the ball. How much does the ball cost?
Please answer this question before you read on.
Your first guess
This puzzle has been given to many university students and the majority answer: “The ball costs 10 cents.” Unfortunately, this is incorrect!
This is an illustration of what Nobel Prize psychology professor, Daniel Kahneman, calls “System 1 Thinking”, that is arriving at a quick answer using our intuition. As the title of this post suggests, this is a question for our more deliberative “System 2”.
Where our intuition goes wrong
We hear “The total is $1.10”. We also hear “$1.00 more for the bat”. We understandably rush to subtract and think the answer is “10 cents for the ball”.
If the ball’s price were 10 cents, then the bat would cost $1.10. Together, they would cost us $1.20.
The answer illustrated
Figure 1 below illustrates the following:
- The price of the bat is the same as the price of a ball plus $1.00
- The contribution of the two pink boxes, together, is 10 cents.
It follows that the value of each pink box, of the price of a ball, is 5 cents.
Let’s verify: The ball costs $0.05, the bat costs one dollar more, so $1.05. Together, the costs add up to $1.10
Over to you
Again, this is the maths teacher in me, assigning you a problem to check that you were paying attention!
Together, a basic violin with a bow cost $360. The violin costs $300 more than the bow. What is the price of the bow?
Write your answer in the comments below. If you get it right, I will send you a virtual chocolate. I hear that they are virtually delicious!
More on counterintuitive mathematics
This is part 4 of a series about the pitfalls of over-relying on our intuition. Here are the previous three:
- Part 1: How many lineups, which asks in how many possible orders can you send 11 players to bat?
- Part 2: How to win a car on a game show? This post is about updating the odds of winning as new information is presented to us.
- Part 3: Exponential growth. This is a type of growth, such as in the case of compound interest, that we’re not so good at grasping with our intuition
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