## Geometric Series to the Rescue!

On a recent road trip, the driver asked me to compute the car’s mileage per gallon. She handed me a little notebook with odometer readings and gasoline gallons at each filler-up. I guess when you run a math club for girls, people think you’re a human calculator.

Looking over the gallons at each filler-up I couldn’t help wishing that the column was filled with tens. I felt too lazy to pull out a pen and do long division and I am decidedly not a “human calculator.” What could be done?

### Geometric series to the rescue!

Instead of doing several long divisions by numbers like 10.07, 10.37, and 9.22, let’s think about dividing by $10 + x$.

Using the formula for an infinite geometric series, notice that

$\frac{1}{10+x} = \frac{1}{10}\frac{1}{1+x/10} = \frac{1}{10}(1 - (\frac{x}{10}) + (\frac{x}{10})^2 - (\frac{x}{10})^3 + (\frac{x}{10})^4 - \dots)$,

provided that $\vert x \vert < 10$.

This shows us how to turn division by a number close to 10 into a computation that only involves divisions by 10, and division by 10 might be more appealing than dividing directly by numbers like 10.07 or 9.22.

Let’s try the first computation: 306 divided by 10.07.  Here $x = 0.07$ so $x/10 = 0.007$.  I’ll first compute $306(1 - (\frac{x}{10}) + (\frac{x}{10})^2 - (\frac{x}{10})^3 + (\frac{x}{10})^4 - \dots)$ and then divide this result by 10 to get the mileage per gallon.

The first term is 306.
The next term is $306 \times -0.007$, which is -2.142.
The next term is the prior term multiplied by -0.007, which is about 0.015.
The next term is the prior term multiplied by -0.007, which is really small.

So, to the nearest tenth, the geometric series adds up to 303.9, and so, to the nearest tenth, the mileage per gallon for the first tank full of gas comes out to 30.4 mpg.

Let’s do the next computation, which is 300 divided by 10.75. Here $x = 0.75$, so the common ratio in the geometric series is 0.075.

We compute in our head: 300 – 22.5 + 1.6875, which is about 279.2, so, to the nearest tenth, we get 27.9 mpg.

If you actually do this, it gets easier and easier to do because of pattern repetition:

308 divided by 10.16: Common ratio is 0.016. So, 308 – 4.928 + (something small) is about 303.072. To the nearest tenth: 30.3 mpg.

330 divided by 10.37: Common ratio is 0.037. So, 330 – 12.21 + 0.04…, or about 317.83. To the nearest tenth: 31.8 mpg.

273 divided by 9.24: Common ratio is -0.076. So, 273 + 20.748 + 1.6…, or about 295.3. To the nearest tenth: 29.5 mpg.

This technique makes division by 10.1 or 9.9 particularly easy.

For example, what is 80 divided by 9.9? The common ratio in the geometric series is 0.01.  So we just compute 80 + 0.8 + 0.008 + 0.00008 + …, and then divide by 10 to get 8.080808080808… (Perhaps you’re thinking that 1/99 = 0.01010101… so this computation can be done by multiplying 80 by 0.101010101… This is true. The above analysis effectively demonstrates that 1/99 is 0.010101…)

Or how about this: What is 700 divided by 95 to the nearest ten thousandth? With practice, one can think: 700 + 35 + 1.75 + 0.0875 + 0.003… + …, so the answer is 7.3684.  (You might like to try doing 700 divided by 95 using long division in your head. It’s not so bad.)

If you enjoy this sort of thing, try these entirely in your head:

1. What is 400 divided by 10.5 to the nearest thousandth?

2. What is 25 divided by 17 to the nearest ten thousandth?

3. What is 300 divided by 37 (exactly)?