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Station  8C
Multiplication and Division 

Even though the addition and subtraction of decimals is straightforward in a 1101 \leftarrow 10 machine, performing multiplication and division on them can be awkward. The trouble is that the 1101 \leftarrow 10 machine is based on whole dots, not parts of dots, and so it is tricky to make easy sense of pictures. But there are ways to play with dots-and-boxes nonetheless.

Question 1


We can certainly do some basic multiplication.

For example, a picture of dots and boxes shows that 2.615×72.615 \times 7equals 14.4273514.42|7|35.

I8S8C - Image139

With explosions, this becomes 18.30518.305. (Check this!)

We can also explain a rule that students are taught to memorize:

If you multiply a decimal by 1010, just shift the decimal point to the right.

For example, 31.42×1031.42 \times 10 gives the answer 3010.402030|10.40|20. With explosions, this becomes 314.20314.20 and the answer looks as though we just shifted the decimal point. (Most people leave off the final zero.)
I8S8C - Image145

By the way, if you multiply a number a little larger than 3131 by ten, you should get and answer a little larger than 310310. Memorizing the direction of travel of a decimal point is unnecessary.

Explain why multiplying a decimal by 100100 has the same effect as shifting the decimal point two places?
(Do you need to memorize the direction of the shift?)

Question 2

We can also multiply decimals by decimals, but now the work is getting less fun. We need to follow the same approach as we did when we explored long multiplication in Island 3.

Let’s do an example. Consider 2.15×0.32.15 \times 0.3.

Here’s what 2.152.15 looks like.
I8S8C - Image151

Here’s what 0.30.3 looks like: it’s three dots to the right of where one dot should be.
I8S8C - Image153

So to compute 2.15×0.32.15 \times 0.3 we need to replace each dot that appears in the picture of 2.152.15 with three dots to the right of where that dot should be. Doing so gives this picture.

I8S8C - Image155
We see the answer 0.63150.6|3|15, or 0.6450.645 after an explosion.

So multiplication with decimals is do-able, but awkward. It is actually easier, in practice, to go back to basic arithmetic and think of multiplication of 0.3, say, as multiplication by 3 followed by division by 10, or work with decimals as fractions. For example,

0.2×0.4=210×410=8100=0.080.2 \times 0.4 = \dfrac {2}{10} \times \dfrac {4}{10} = \dfrac {8}{100} = 0.08


0.05×0.006=5100×61000=30100000=310000=0.00030.05 \times 0.006 = \dfrac {5}{100} \times \dfrac {6}{1000} = \dfrac {30}{100000} = \dfrac {3}{10000} =0.0003


2.15×0.3=(2+15100)×310=610+451000=0.6+0.045=0.6452.15 \times 0.3 = \left(2+ \dfrac {15}{100} \right) \times \dfrac {3}{10} = \dfrac {6}{10}+\dfrac {45}{1000} = 0.6+0.045 = 0.645.

We also see the effect of multiplying a decimal by ten this way. For example

10×3.142=10×(3+110+4100+21000)=30+1+410+2100=31.4210 \times 3.142 = 10 \times \left(3+\dfrac{1}{10}+\dfrac{4}{100}+\dfrac{2}{1000}\right)=30+1+\dfrac{4}{10}+\dfrac{2}{100}=31.42.

Compute 0.04×0.50.04 \times 0.5.

Question 3

Compute 1000×0.03851000 \times 0.0385.

Question 4


Consider 0.08÷0.0050.08 \div 0.005.

To do this problem I could try drawing a picture of 0.080.08 in a 1101 \leftarrow 10 machine and attempt to make sense of finding groups of 0.0050.005 in that picture, that is groups of five dots three places to the right of where they should be. Although possible to do, it seems hard to keep straight in my head and it makes my brain hurt!

Here’s a piece of advice from a mathematician: Avoid hard work! Mathematicians will, in fact, work very hard to avoid hard work!

Since converting decimals to fractions makes the multiplication of fractions easier, let’s do the same thing for division.

Here are some examples.

Example: Examine 0.08÷0.0050.08 \div 0.005.

A fraction is a number that is an answer to a division problem. And, conversely, we can think of a division problem as a fraction. For example, the quantity 0.08÷0.0050.08 \div 0.005 is really this “fraction” 0.080.005\dfrac {0.08}{0.005}.

(It is okay to have non-whole numbers as numerators and denominators of fractions.)

To make this fraction look friendlier, let’s multiply the top and bottom by factors of ten.

0.08×10×10×100.005×10×10×10=805\dfrac {0.08 \times 10 \times 10 \times 10}{0.005 \times 10 \times 10 \times 10} = \dfrac{80}{5}

The division problem 805\dfrac{80}{5} is much friendlier. It has the answer 1616.
(Use a 1101 \leftarrow 10 machine to compute it if you like!)

Example: Examine 8.5100\dfrac{8.5}{100}.

Dividing by 100100 is the same as multiplying by 1100\dfrac{1}{100}. So, 8.5100\dfrac{8.5}{100}, for example, can be thought of as

1100×8.5=1100(8+510)=8100+51000=0.085\dfrac{1}{100} \times 8.5 = \dfrac{1}{100} \left(8+\dfrac{5}{10}\right)=\dfrac{8}{100} + \dfrac{5}{1000} = 0.085

Example: Examine 1.51÷0.071.51 \div 0.07.

If I was asked to compute 1.51÷0.071.51 \div 0.07, I would personally do this problem instead:

1.51×10×100.07×10×10=1517\dfrac{1.51 \times 10 \times 10}{0.07 \times 10 \times 10}= \dfrac{151}{7}.

I see the answer 214721 \dfrac{4}{7}. If that same person asked me to give the answer as a decimal, then I would have to convert the fraction 47\dfrac{4}{7} into a decimal. And that is possible. The next lesson explains how!

Explain why the answer to 0.9÷100.9 \div 10 must be 0.090.09.

Question 5

Explain why the answer to 2.34÷10002.34 \div 1000 must be 0.002340.00234.

Question 6

Explain why the answer to 40.04×0.0140.04 \times 0.01 must be .4004.4004.

Question 7

Compute 0.7525\dfrac {0.75} {25}.

Question 8

Let’s put it all together. Here are some unpleasant computations. Do you want to try evaluating them?

0.3×(5.372.07)0.3 \times \left( 5.37 - 2.07 \right)

Question 9

0.1+(1.010.1)0.11+0.09\dfrac {0.1 + \left( 1.01 - 0.1 \right)} {0.11 + 0.09}

Question 10

(0.002+0.2×2.02)(2.20.22)2.220.22\dfrac {\left( 0.002 + 0.2 \times 2.02 \right) \left( 2.2 - 0.22 \right)} {2.22 - 0.22}

You can either play with some of the optional stations below or go to the next island!

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