Question

(a)A fraction $\frac{a}{b}$ can be interpreted as $a÷b$. Use a calculator to convert each unit fraction to a decimal. Do some of the decimals look different than the others? Explain.

$$\begin{gathered} \frac{1}{2}\frac{1}{3}\frac{1}{4}\frac{1}{5}\frac{1}{6}\frac{1}{7} \\ \frac{1}{8}\frac{1}{9}\frac{1}{10}\frac{1}{11}\frac{1}{12} \end{gathered}$$

(b) Compare & contrast the fractions in part (a) with the fractions you wrote in Exploration 1. What conclusions can you make?

(c) Does every fraction have a decimal form that either terminates or repeats? Explain your reasoning

Answer 1

Step 1: (a) to convert given unit fractions into decimal form using calculator:

$$\begin{gathered} \frac{1}{2}=0.5\frac{1}{3}=0.\underline{3}\frac{1}{4}=0.25 \\ \frac{1}{5}=0.2\frac{1}{6}=0.1\underline{6}\frac{1}{7}=0.\underline{142857} \\ \frac{1}{8}=0.125\frac{1}{9}=0.\underline{1}\frac{1}{10}=0.1 \\ \frac{1}{11}=0.\underline{09}\frac{1}{12}=0.\underline{83} \end{gathered}$$

Yes, some of the decimals look different from others.

Some decimals are terminating like 0.5, 0.2 and some are repeating decimal form like $0.\underline{3},0.\underline{142857}$.

Step 2:

(a) Fractions in exploration 1	Fractions in part (a)
Denominator of these fractions are in power of 10. Some fractions can be converted to mixed form.	Denominator of these fractions are different from power of 10, are different in natural numbers. Fractions can't be converted to mixed form.

Conclusion: Every fraction can’t be converted to mixed form & denominator can be any natural number.

Step 3: (c) Yes every fraction have decimal form that either terminates or repeats. The denominator with prime factorisation as powers of 2 & 5 terminates otherwise repeats.

$\frac{1}{2}=0.5$(terminates) $\frac{1}{7}=0.\underline{142857}$(repeats)