Math tips & tricks for your daily life

Introduction

Math is easy. Or incomprehensible; depending on whom you ask.

For example, if one memorizes how to do long division without understanding where the procedure comes from, or how division and fractions are different aspects of the same thing, it becomes really hard to come up with different “algorithms” for certain calculations.

In this article I’ll compile a few tricks that can be used to resolve some practical math problems and, at the same time, help understand the reasoning behind them. Most of these can be found with a Google search; I hope this article helps by putting all in one place.

Enjoy!

Calculating percentages

This is a nice one; it relies on the commutative property of multiplication. UK-based copywriter Ben Stephens tweeted it.

Let’s say someone asks you to calculate 14% of 25. Ugh, 14% is 14/100 or 0.14; now I have to do 0.14 x 25 or 14 x 25 and divide by 100. Man, that’s hard.

What you fail to realize, young padwan, is that 0.14 x 25 is the same as 0.25 x 14 (light bulb); in other words, 14% of 25 is 25% of 14.

Wait a second, I know 0.25 is the same as 1/4 (because 25 cents is a quarter).

So now I just have to calculate 14/4, or 7/2 = 3.5. Easy peasy lemon squeezy.

Now you have a tool to resolve some percentages; it obviously won't help a lot for stuff like 12.35% of 23.768 but hey, it’s a start.

Calculating the tip

The history of tipping in the US is a little troubling. After the Civil War, tipping was used by restaurant owners to avoid paying wages to emancipated slaves, justifying these employees were already being paid by the customers in tips.

I learned this neat trick from a colleague I worked with some 15 years back.

In the US the typical tip is 15%. And calculating 15% of some number is annoying. Or is it?

Here are the steps:

1. To make the math easier, round the bill to the nearest even number (if you’ll round up or down is your choice; I won’t judge) and then go to step 2
2. Divide the result from step 1 by 2 and add to that result
3. Divide the final value by 10. Voilá, this is the 15% tip

Don’t believe me? This is what we just did:

Quick numeric example:

1. Bill is $43.78 => make it $44.00
2. Divide $44.00 by 2 = $22.00 and add to $44.00 = $66.00
3. Divide $66.00 by 10 = $6.60; this is the tip (you can even round it up to $7.00 if you’re feeling generous)

Converting Celsius to Fahrenheit

The history behind these 2 temperature scales is interesting.

Andres Celsius proposed, in 1742, a scale from 0 to 100 degrees; the interesting part is that originally 0 degree was the temperature for boiling water and 100 degrees was the temperature for melting ice.

Daniel Gabriel Fahrenheit proposed his scale in 1724 — before Celsius — and used a different approach; he defined 0 degree as the temperature of a solution of ice, water and salt (also used to quickly chill beer for a party).

Then he postulated the healthy human temperature to be 96 degrees (his reasons are unknown but some believe it was because 96 is divisible by 2, 3, 4, 8 and 12, making it easier to define the degrees of the scale).

Then he postulated the water melting point to be 32 degrees, making the temperature difference between the human body and ice to be 64 degrees; he then determined the scale divisions bisecting the intervals between both temperatures 32 times. Clever.

Finally, he measured the water’s boiling temperature using his scale and found 212; the difference between boiling water and melting ice was 180 degrees; in a fortuitous coincidence his scale fits in the angle degrees of a semi-circle. Cool.

The world moved on and now some folks use Fahrenheit and some use Celsius; how do we convert temperatures between these scales?

The ratio between the height of the column of mercury and the total height of each thermometer must be same for both so we can write:

And that’s the exact formula. Not very convenient. So how can we make this a little more practical?

Well, 9/5 is close to 10/5 = 2; the difference between the two values is 0.2 = 2/10. So we can follow these steps to convert Celsius to Fahrenheit:

1. Multiply the temperature in Celsius by 2
2. Divide the result from step 1 by 10 and subtract from it (you can ignore this step if you just want a ballpark; the result will be 20% higher than the actual temperature)
3. Add 32 to the result from step 2

Quick numeric example:

TC = 32

Step 1: 32 x 2 = 64

Step 2: 64 / 10 = 6.4 ~ 6; then 64–6 = 58

Step 3: 58 + 32 = 90 (the exact value is 89.6, which we would get if we used 6.4 from step 2)

Roots of a quadratic equation

This one was recently discovered by Po-Shen Loh, a mathematician at Carnegie Mellon University in Pittsburgh. We all learned Bhaskara’s formula to calculate the roots for the quadratic equation:

We accept it, we memorize it and we move on.

But check out what Po-Shen Loh did:

1. Every quadratic equation is the product of 2 binomials

where R and S are the 2 roots of the quadratic equation.

2. B is the sum of the roots R and S (minus the signal) and C is the product of the roots

3. Now here is Po-Shen’s really clever insight; re-writing the equation for B he got:

From that we can express R and S as a function of z:

Equations 1 and 2

4. Now that we have expressions for R and S we replace them in the equation for C and calculate z

5. With z we easily determine R and S using equations 1 and 2

So now we can define the steps to resolve a generic quadratic equation of the form

1. Divide all terms by A and obtain

2. Calculate z by resolving

3. Calculate the roots R and S

Isn’t that cool? Not only it is easier to calculate the roots this way, you can derive the traditional formula from this result (go ahead, give it a shot).