1.4.7 Converting a/40 and b/80, etc… to Decimals

The following isn’t necessarily a trick but more of a procedure I like to follow when I am approached with converting $\frac{a}{40}$ and $\frac{b}{80}$ into decimals (usually on the first column of problems). So for $\frac{a}{40}$ I treat it as:

$$\frac{a}{40} = \frac{a}{40} \times \frac{1/4}{1/4} = \frac{a/4}{10}$$

So the technique is to divide the numerator by 4 then shift the decimal point over. Similarly, for $\frac{b}{80}$ you want to divide by 8 and shift the decimal point over. Let’s look at a couple of examples:

$$\frac{43}{40} = 1 + \frac{3}{40} = 1 + \frac{.75}{10} = 1.075$$ $$\frac{27}{80} \Rightarrow \frac{27}{8} = 3.375 \Rightarrow \frac{3.375}{10} = .3375$$

Problem Set 1.4.7