## 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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