In this instructional video, Paul Sloane argues that mathematical numeracy is essential for everyday life and especially for business decision‑making. He illustrates the point with four classic puzzles that many people answer incorrectly: (1) the average speed of a round‑trip journey, showing that the simple average of two speeds is wrong; (2) the bat‑and‑ball cost problem, solved with elementary algebra; (3) the "rope around the Earth" problem, revealing that adding a foot to a massive circumference only increases length by about 6.3 feet, a result that scales to planetary and cosmic sizes; and (4) a river‑width estimation trick that uses triangulation and right‑triangle similarity to measure a river with only a yardstick. After walking through each puzzle, Sloane offers concrete advice for improving mathematical thinking: calculate everyday totals, help children with homework, practice puzzles, draw graphs and diagrams, make estimates, and read mathematics books. He recommends his own book, Mathematical Lateral Thinking Puzzles (co‑authored with Des McHale), as a resource for learners at all levels. The overall message is that deliberate practice of simple quantitative reasoning sharpens the mind and builds confidence in numbers.
Chapters
As you go through life, you meet people who tell you that they never liked mathematics. They're frightened or wary of mathematical concepts. They're clumsy with numbers or struggle with means, probabilities, or statistics. You should feel sorry for these people. They will be at a severe disadvantage in many walks of life, but particularly in business, where it's very helpful to have a good grasp of numbers, arithmetic, and percentages. The mathematically innumerate are not only restricted in the thinking tools they can use, but also denied an appreciation of the power and beauty of mathematics. Most of our thinking is verbal thinking. We think in terms of words, verbal concepts, verbal reasoning. We apply some logic, but there are some problems for which words and logic are poor tools. We need the accuracy of mathematics. So, let's start with this little problem. You drive from A to B at an average speed of 20 miles per hour. There's very heavy traffic, a lot of congestion, takes you a long time. You drive back late at night, the roads are all clear, you come back at 60 miles per hour, an average speed. What is your average speed for the whole journey? Now, many people would say, well, the average of 20 and 60 is 40, so it's going to be 40 miles per hour. But this would be wrong, and not just a little bit wrong, but very wrong. So we can solve this problem with some elementary arithmetic. Let's say that A to B is 60 miles. It could be any distance, but 60 works out well for us. At 20 miles per hour, it will take us 3 hours to get there, and at 60 miles per hour, it just takes 1 hour to get back. So the total journey there and back is 120 miles. And the total time spent is 4 hours. So the average speed is 30 miles per hour, not 40 miles per hour. So relying on your intuition or guesswork here doesn't give you a good answer at all. You need a little mathematics. Try this puzzle, which stumps many people. A bat and a ball together cost $11. The bat costs $10 more than the ball. How much does the ball cost? Write down your answer straight away, please. What did you say? Most people say $1 for the ball, but this isn't right. If the ball was $1, then the bat would be $10, and it would be $9 more than the ball. So, you can solve this puzzle quite easily with guesswork or estimation, but a good way to solve it is with some elementary algebra. Let the cost of the bat be x and the cost of the ball be y. Then x plus y equals 11, and x minus y, the difference between the bat and the ball, is $10. So we have 2 equations with 2 unknowns, and that's easy to solve. We can solve it by rearrangement, or in this case, I've added the 2 equations together. I've added the 2 left-hand sides and the 2 right-hand sides. And we get 2x equals 21, or x equals 10.5. The bat costs $10.50, and the ball costs 50 cents. Try this famous problem, the rope at the equator. So, the diameter of the Earth at the equator is about 8,000 miles. Imagine a rope running around the Earth At the equator, which is the circumference of the globe running all the way around, and imagine a second rope that's exactly 1 foot higher than the first rope all the way around. So it's 1 foot above the ground all the way around, or 1 foot above the water. How much longer is the second rope than the first? Well, I can tell you the first rope is almost 26,000 miles long. It's The circumference of the Earth. So how much longer do you think the second rope will be? It's just 1 foot off the ground all the way around, but all the way around that 26,000 miles. So what is your guess? And many people get this estimation completely wrong. So let's do a little geometry. Circumference is diameter times pi, the diameter of the Earth is 8,000 miles. Therefore, the circumference is 8,000 pi miles, which is just under 26,000, I think. So, the length of the first rope is 8,000 miles times pi. The length of the second rope is 8,000 miles plus 2 feet. There's 1 foot extra on either side, times pi. Now, if you subtract one from the other, The difference between those two is two pi feet. That's all. Six point three feet. Much much less than most people would estimate. Most people estimate miles. But the interesting thing about this, and it's very counterintuitive. It's not what you'd expect. The interesting thing I think about this is that if you put a rope around the solar system just outside Pluto. All around our huge solar system and another rope exactly 1 foot further out, the second rope would only be 6.3 feet longer than the first. Or a rope around the galaxy, the Milky Way, or a rope around the entire known universe and a second rope just 1 foot outside it, the second rope would only be 6.3 feet longer than the first rope, even though the first rope is unimaginably long. So, it's very counterintuitive, and we need mathematics to give us the accurate answer. Let me try this on you, the river problem. This is a very clever problem with a very elegant solution. How can you accurately estimate the width of a river with just a simple yardstick? You're standing on one bank of a river looking across it, and you want to know how wide the river is. There is no bridge. There's no boat. You don't have GPS. You don't have a map. You don't have laser beams or anything else. All you've got is a yardstick. Trust me, there's no trick in this question at all. But there's a very elegant solution. So I suggest what you do here is you pause the video, go and get yourself a cup of coffee and a pencil and paper, and just sketch this out and try and figure out how you could estimate accurately the width of the river. Well, Well, you can solve the problem with triangulation, and what you do is you choose an object on the other side of the river, a visible object. It might be a tree or a bush or some unusual feature, and you stand exactly opposite that. So that's— the object is at point A, you're at point B. Now, you then turn 90 degrees, right angles to AB, and you walk along the side of the river Any distance, say BC, and you measure that distance. You measure your steps, and then you do ex— and you put a stick in the ground at point C. You put the stick in as clearly as you can. Then you walk on the same distance again, that CD equals BC. So if you walked 50 steps, you now walk another 50 steps exactly. When you get to point D, you then turn at right angles through 90 degrees. And you walk along a line, that line, looking, and then when you can line up C and A in an exact line, that's the tree and the stick, you've reached point E and you stop there. And as it happens, the distance DE is exactly equal to the distance AB. Why? Well, you've constructed 2 identical triangles. They're both right-angle triangles. They've got this opposite angle in common, and BC equals CD. We know that. Therefore, the 2 triangles are identical, and therefore DE equals AB. Very, very elegant way to use geometry to solve this problem. So, here are some tips for you to help you improve your mathematics. I think everyone can improve their mathematics, no matter what level you are, whether you're Really starting out or intermediate or advanced. So the first thing I would say is calculate. And when you get— before you get your bill at the coffee shop, you've seen the price of the coffee, you've seen the price of the pastries, you've worked out what it should be. And when the barista or the assistant says that's $13.25, you know whether that's right or wrong. So calculate things where you can. If you've got children who are doing maths homework, Help them with it. Some of the concepts they use for teaching mathematics now are very, very clever, and you'll learn stuff as you look over your children's shoulder, and you might be able to help them, they might be able to help you. Try puzzles and brain teasers. Very often newspapers have these simple mathematical puzzles, logic puzzles. These are all good for the brain and will help you. Draw Graphs and diagrams. Think visually. And if you've got some complex problem to solve, try drawing it as a graph. And it's a very, very powerful way of seeing a problem is to draw a graph. Estimate things. So when you do your grocery shopping, estimate what you think it will come to as you go around. Do a little estimate. That's nearly $1, that's nearly $5. And as you go around, add up things. Estimate how many people there are at a football match. Estimate how many tiles might be needed to cover a floor, something like that, and it just uses your brain in a mathematical sense. And then finally, I would say read some maths books, whatever level you are. Choose a mathematics book that suits your level. It can be very elementary, it can be intermediate level, it can be advanced level, and you might find it's fun, you might enjoy it. And while I'm on that subject, I would recommend this book, Mathematical Lateral Thinking Puzzles by me, Paul Sloane, and my co-author Des McHale. And these are puzzles we've chosen because in each puzzle there's a routine way to solve the puzzle and there's a clever lateral way to solve it, just like the width of the river. So, if you like that sort of puzzle where you have to use your brain and look for a clever way to do things, I would recommend this book. It's at various levels. I would say junior school, a bright kid at junior school could start it, Somebody, a keen maths student at secondary school would really enjoy it, right up to undergraduate level. But it's got the different levels, and I think you will enjoy this book. So, there it is. I recommend that whatever kind of thinker you are, you develop your mathematical thinking skills. Make it a little bit better, make it a little bit sharper, and it will make you more confident with maths, more confident with numbers, more confident with variables, and it will make you a better thinker.
