## Fractions are Hard!

### 3.4 Decimal arithmetic

[For an introduction to decimals, see lesson 2.1 of Exploding Dots.]

Here are some potentially annoying questions:

What is $$0.05+0.006$$ ?

What is $$0.05-0.006$$?

What is $$0.05\times 0.006$$?

What is $$0.05 \div 0.006$$ ?

If in doubt about how to handle a decimal arithmetic problem, it never hurts – and in fact can often be helpful to – think of the decimals as fractions.

COMPUTING $$0.05+0.006$$:

We have

$$0.05+0.006 = \dfrac{5}{100}+\dfrac{6}{1000}=\dfrac{50}{1000}+\dfrac{6}{1000}=\dfrac{56}{1000}=0.056$$.

This problem is perhaps too straightforward. Consider, instead, $$13.276 + 5.94$$.

In the work of fractions we have

$$13.276 + 5.94= 13+\dfrac{276}{1000}+5+\dfrac{94}{100}$$

$$=18+\dfrac{276}{1000}+\dfrac{940}{1000}$$

$$=18+\dfrac{1216}{1000}$$

$$=18+1+\dfrac{216}{1000}$$

$$=19.216$$.

(In the work of Exploding Dots we see the answer as $$18.11|11|6$$. Explosions then convert this to $$19.216$$.)

 Question: a)    Agatha says that computing  $$0.0348+0.0057$$ is essentially a matter of adding 348 and 57. What does she mean by this? Is she right? b)    Percy says that computing $$0.0852+0.037$$  is essentially a matter of adding 852 and 37. He is not right. What is wrong with Percy’s thinking?

COMPUTING $$0.05-0.006$$:

We have

$$0.05-0.006 =\dfrac{50}{1000}-\dfrac{6}{1000}=\dfrac{44}{1000}=0.044$$.

(Exploding dots gives $$0.050 – 0.006 = 0.0|5|-6 = 0.044.$$

COMPUTING $$0.05 \times 0.006$$:

We have

$$0.05 \times 0.006 =\dfrac{5}{100} \times \dfrac{6}{1000}=\dfrac{30}{100\times 1000}$$

$$=30 \times \dfrac{1}{10}\times \dfrac{1}{10}\times \dfrac{1}{10}\times \dfrac{1}{10}\times \dfrac{1}{10}$$

$$=3 \times \dfrac{1}{10}\times \dfrac{1}{10}\times \dfrac{1}{10}\times \dfrac{1}{10}$$

$$=0.0003$$

(I personally tend to get lost in lots of zeros and prefer to count how many times I am dividing by ten.)

 Question: In computing $$0.04\times 0.5$$ is it helpful to write our fractions in “simplified” form $$0.04 \times 0.5 = \dfrac{4}{100} \times \dfrac{5}{10} = \dfrac{1}{25}\times \dfrac{1}{2}$$ or is there advantage to keeping fractions in terms of powers of ten?

 Question: a)    Compute $$10 \times 0.4$$  and explain carefully why the answer is $$4$$.   b)    Compute $$10 \times 0.069$$  and explain why the answer is $$0.69$$.   c)    Compute $$100 \times 0.0987$$  and explain why the answer is $$9.87$$.   d)    Compute $$28354.31 \div 1000$$ and explain why the answer is $$28.35431$$.   e)    Is there any reason to memorize a rule about shifting the decimal point left or right when multiplying or dividing by powers of ten?

COMPUTING $$0.05 \div 0.006$$:

We have

$$\dfrac{0.05}{0.006} = \dfrac{\dfrac{5}{100}}{\dfrac{6}{1000}}=\dfrac{\dfrac{5}{100}\times 1000}{\dfrac{6}{1000}\times 1000} =\dfrac{50}{6} = \dfrac{25}{3}=8\dfrac{1}{3}$$.

Thus

$$0.05 \div 0.006=8\dfrac{1}{3}=8.3333\cdots$$.

Let’s also compute $$0.9 \div 100$$.

$$\dfrac{0.9}{100}=\dfrac{0.9\times 10}{100\times 10}=\dfrac{9}{1000}=0.009$$.

 Question: Consider the “abstract” decimal $$0.abc$$.   a)    Compute $$0.abc \div 10$$  and show that it equals $$0.0abc$$. b)    Compute $$0.abc \div 100$$ and show that it equals $$0.00abc$$. c)    Compute $$0.abc \div 0.1$$ and show that it equals $$a.bc$$. d)    Compute  $$0.abc \div 0.01$$. e)    Compute $$0.abc \times 100$$. f)    Compute $$0.abc \times 0.001$$.

Let’s put it all together!

 Question: Compute   a) $$0.3\times \left(5.37-2.07\right)+\dfrac{0.75}{2.5}$$.   b)$$\dfrac{0.1+\left(1.01-0.1\right)}{0.11+0.09}$$.   c) $$\dfrac{\left(0.002+0.2\times 2.02\right)\left(2.2 – 0.22\right)}{2.22-0.22}$$.

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