In my last post I talked about the importance of doing math in your head and using estimates to make it easier. My example focused on flying because it is something that I enjoy doing and math plays a key role in it.

I was thinking that I needed a better example, an everyday example. I found it when I went shopping for shoes last week.

With winter on its way out, several stores are having their annual “end of winter sale.” I live in Chicago and I like to be outdoors. This means that I need a good pair of boots that can withstand the cold, keep the slush out, and be comfortable to walk in.

In one shoe store, I found a pair of high quality boots that originally cost $84.99, and were on sale for 35% off. Using a few simple math tricks, I was able to calculate how much the shoes would cost in my head.

Why bother doing the math? You will find that out in the end.

How did I do the math? Let me share this with you now.

Simplify the problem!

$84.99, is a hard number to use. When I was younger, I thought that the $0.99 at the end of every price was put there to scare me away from doing math in my head. This is a lame excuse. I quickly learned to round the prices up to make the number easier to handle. In this case we will use $85.

Percentages are also a intimidating, especially for a number like 35%. The key to figuring out percentages is to break it down into simpler numbers. When calculating a sale price I find that the easiest percentages to calculate are 50%, 10%, and 5%.

I selected these numbers because…

50% is half, so divide the price by two.

10% is 1/10, so divide the price by 10.

5% is half of 10%, so determine the 10% price and then divide it in half.

For numbers like 35%, use a combination of the numbers above. For example, the number 35 breaks down into 10 + 10 + 10 + 5, or (3 x 10) + 5.

Since the boots cost $85, 10% is $8.50 and 5% is half of the 10% price, or $4.25. To figure out what 35% is, multiply (3 x $8.50) and then add $4.25.

Ok, I know you are thinking that 3 x $8.50 is not simple math.

Here’s how I see it. I know that 3 x $8 is $24 and 3 x $0.50 is $1.50. Add them together to get $25.50. Doing math this way makes the problem easier to solve.

To figure out the final discount, add $4.25 and $25.50. This gives $29.75 for the 35% price. That’s nearly $30 off!

This means I should pay about $55 before taxes. The actual price is $55.24, so our estimate is very good.

So why bother doing the math?

When I asked people about how they solve this kind of math problem, most said that they trust the register so they do not worry about doing math in their head.

I disagree with them because I have had many experiences like the one below.

When I took the boots to the register, I was expecting the price to be around $55. The clerk scanned the shoes and the computer said the shoes cost $63.74.

There had to be a mistake somewhere.

I quickly estimated that $63.74 reflected 25% off, not the 35% that was advertised. How did I do this?

I figured this out by first rounding $63.74 up to $64 because it is easier to deal with.

The difference between $64 and $55 is $9. This is close to $8.50, or about 10% (we figured this out earlier). Since the price rang up higher, that means the discount was 10% less than 35%, giving 25%.

I told the sales clerk that that the shoes were on sale for 35%, not 25%. As it turns out, the same pair of shoes were on sale last week for 25%, and this week they were on sale for 35%.

The register was not updated with the new price.

By doing math in my head I was able to catch the mistake. If I had trusted the register, I would have paid $9 more than I should have.

With $9 I could buy lunch or about 3 gallons of gas.

What could you do with $9?

-Dr. Dave

When I was young, I wanted to earn my pilot’s license. All my teachers knew this, and they always told me that pilots were good at doing math without calculators.

During my flight training, I encountered all sorts of math problems in the form of how to properly loading the plane with people and fuel (weight & balance) and navigation. I am very glad I followed my teachers’ advice.

Let’s take a closer look at the issues pilots have to face and examples of the problems they have to solve. Towards the end, I will share how I do math in my head.

Weight and Balance

Flying an airplane is a balancing act. Think of an airplane as a seesaw.

Pilots try to achieve the a condition similar to when a seesaw is in balance. This is possible by distributing the weight evenly. If an airplane is out of balance, it will fly poorly or not at all. Gasp!

Navigation

Pilots have to know how to get to their destination using tools like radios and GPS. Modern airplanes have a lot of fancy computers onboard that make navigation easier, but mistakes can occur. I was trained to always double check the computer because the best onboard computer is the pilot’s brain.

Always Ask, “Does the Answer Make Sense?”

When I was in elementary school, I had a friend who used calculators a lot. On the day of a big test, his calculator broke and so he borrowed a calculator that was different from the one he was used to; it had twice as many buttons. He did very poorly on this test, and he blamed the calculator for giving him the wrong answers. It turns out he was at fault. Why?

Calculators and computers are just machines that do exactly what you tell it to do. For example, if your problem is 44 x 5, but you accidentally type 44 + 5, the math will be correct, but the answer will be wrong.

Similarly, onboard flight computers are very useful tools for flying, but it is up to the pilot to make sure the computer is correct. How do we do it?

The Power of Estimation

When flying an airplane, I have to deal with numbers that are not as easy to use. Let’s say that my cruising speed is 167 miles-per-hour (mph), and my destination is 302 miles away.

The onboard flight computer says it will take 1 hour 49 minutes.

Is the flight computer right? Does this flight time make sense?

If I used the exact numbers in my head, I would probably be more focused on doing math than flying the plane. This is dangerous. Instead, I will round numbers up or down to make the math easier.

The Easy Way

In this example I would round the distance down to 300 miles (from 302 miles) and use 150 mph (instead of 167 mph).

300 miles divided by 150 mph is easy to solve.

The answer is 2 hours, which is close to what the onboard computer is saying.

Also, since I know I am going faster than 150 mph, I know I will reach my destination sooner than 2 hours. My logic agrees with the onboard computer and so the flight time “makes sense.”

It Works!

I have been in a situation where my flight examiner intentionally programmed an error into the onboard flight computer. I noticed that the error early on because I always asked myself “does the flight computer make sense?” I was able to catch the error before it became a problem later in the flight. The examiner was impressed.

Asking, “does the answer make sense” when solving any sort of problem helps to check your work. I can remember many times where I checked my work without a calculator, using estimation and simple math, and I uncovered a calculation error.

My teachers were right; doing math in your head is an important skill. Thanks to their advice, my dream came true and I earned my pilot’s license.

-Dr. Dave

P.S. There is another way that I use to solve the flight time problem shown above. It is more advanced so I saved it at the end for those who are interested.

A Little More Advanced

I know that 10% of 150 mph is 15 mph. If I add 150 mph with 15 mph, I get 165 mph, which is very close to 167 mph.

If I am going 10% faster than 150 mph, I know I will arrive 10% sooner than 2 hours (120 minutes). 10% of 120 minutes is 12 minutes.

Arriving 10% sooner means I will arrive in 2 hours – 12 minutes, or 1 hour and 48 minutes.

This agrees with the flight computer!