Generating fraction of a decimal number

Calculate the generating fraction of these terminating decimals. Do not forget to simplify the fraction to its simplest form.

1. a) $0.8$
2. b) $1.25$
3. c) $0.064$
4. d) $12.5$

Find the generating fraction of the following pure recurring decimal numbers:

1. a) $0,\widehat{4}$
2. b) $0,\widehat{15}$
3. c) $2,\widehat{3}$
4. d) $1,\widehat{72}$

Express the following mixed recurring decimals as irreducible fractions:

1. a) $0.1\widehat{6}$
2. b) $1.2\widehat{3}$
3. c) $0.0\widehat{5}$
4. d) $2.1\widehat{45}$

Calculate the exact value of the following expression. To do this, you must first convert each decimal number into its respective irreducible generating fraction:

$$1.25 - 0,\hat{3} \times \left( \frac{1}{2} + 0.1\hat{6} \right)$$

A messaging app's storage currently occupies $3.875$ GB of a phone's memory. The user decides to delete files and frees up space equivalent to $1,\hat{45}$ GB.

1. Calculate the generating fraction of the space initially occupied.

2. Calculate the generating fraction of the space freed up.

3. What fraction of a GB remains occupied on the phone? Express the result as an irreducible fraction.

Calculate the exact value of the following mixed expression. To avoid approximation errors, convert each decimal number into its respective generating fraction before calculating and simplify the result to an irreducible fraction:

$$L = \frac{1.2\widehat{6} \times \sqrt{0.25}}{(-2)^{-2} + 0.8\widehat{3}}$$

In a Physical Education class, three students timed how long it takes them to run 100 metres. The times obtained are:

• Aoife: $12.5$ seconds.

• Cian: $12,\hat{5}$ seconds.

• Niamh: $12.0\hat{5}$ seconds.

Express the time of each student as an irreducible generating fraction and order them from fastest to slowest.

A young programmer is designing a video game and needs to set the probability of a rare item appearing. The code requires the probability to be entered as an irreducible fraction, but the graphics engine returns the value as a mixed recurring decimal: $P = 0.02\widehat{45}$.

Find the generating fraction that the programmer must enter and justify whether this probability is greater than or less than $\frac{1}{40}$.

Order the following real numbers from smallest to largest. To compare them accurately, convert all decimals into their generating fractions and find a common denominator:

$$\frac{13}{20} \quad ; \quad 0.6\widehat{5} \quad ; \quad 0,\widehat{65} \quad ; \quad \frac{2}{3} \quad ; \quad 0.65$$