Generating Fraction of a Decimal Number
In a video game, a mission's progress bar shows $0.\overline{6}$ complete. What exact fraction of the mission is that? Every terminating or repeating decimal comes from a fraction: it is its generating fraction.
Generating fraction of a terminating decimal
A terminating decimal has a limited number of digits after the point. You read it just as you say it: $0.75$ is "75 hundredths."
Before calculating, it helps to know what type of decimal you have. Look at how many digits repeat:
| Type | Example | What it looks like |
|---|---|---|
| Terminating | $3.15$ | it ends |
| Purely repeating | $5.232323\ldots = 5.\overline{23}$ | the repeating block starts right after the point |
| Mixed repeating | $12.4777\ldots = 12.4\overline{7}$ | there are digits before the repeating block |
In a repeating decimal, the whole-number part comes before the point, the non-repeating part is the decimal digits that do not repeat, and the repeating block is the digits that repeat.
To start working with generating fractions, it is essential to know what type of number we are dealing with. Classify the following decimal numbers into terminating…
To start working with generating fractions, it is essential to know what type of number we are dealing with. Classify the following decimal numbers into terminating decimal, pure recurring or mixed recurring:
- a) $3.15$
- b) $0.666...$
- c) $12.4777...$
- d) $5.232323...$
- e) $8.02333...$
Calculate the generating fraction of the following terminating decimal numbers. Remember to simplify the result wherever possible: a) $0.5$ b) $1.2$ c) $0.25$ d) $0.08$
Calculate the generating fraction of the following terminating decimal numbers. Remember to simplify the result wherever possible:
- a) $0.5$
- b) $1.2$
- c) $0.25$
- d) $0.08$
In the Natural Sciences laboratory, you have weighed a sample of sand and the scale reads $0.75$ grams. Write this number as a decimal fraction (with a power of 10 in…
In the Natural Sciences laboratory, you have weighed a sample of sand and the scale reads $0.75$ grams. Write this number as a decimal fraction (with a power of 10 in the denominator) and then simplify it to obtain its irreducible generating fraction.
Generating fraction of a purely repeating decimal
A video game indicates that you have completed $0,\overline{6}$ of a mission. To find out the exact part of the map you have covered, convert that pure recurring decimal…
A video game indicates that you have completed $0,\overline{6}$ of a mission. To find out the exact part of the map you have covered, convert that pure recurring decimal number into its generating fraction.
Find the generating fraction of the following pure recurring decimal numbers with a single digit in the recurring part. Simplify if you can: a) $0,\overline{3}$ b)…
Find the generating fraction of the following pure recurring decimal numbers with a single digit in the recurring part. Simplify if you can:
- a) $0,\overline{3}$
- b) $0,\overline{7}$
- c) $1,\overline{1}$
- d) $0,\overline{5}$
Identify the elements of the following repeating decimal numbers. For each one, state its integer part, its non-repeating part (if it has one) and its repeating part…
Identify the elements of the following repeating decimal numbers. For each one, state its integer part, its non-repeating part (if it has one) and its repeating part (period):
- a) $5.888...$
- b) $0.1232323...$
- c) $14.5262626...$
Going deeper: generating fraction of a mixed repeating decimal
When there are digits between the point and the repeating block (the non-repeating part), the rule expands a little.
In a strategy video game, a loading bar indicates that $0.85$ of a special mission has been completed. What irreducible fraction of the mission has already been…
In a strategy video game, a loading bar indicates that $0.85$ of a special mission has been completed. What irreducible fraction of the mission has already been completed? Calculate the generating fraction of that decimal number and simplify it.
Classify the following decimal numbers into terminating, pure recurring or mixed recurring. Once classified, calculate the generating fraction of each of them and…
Classify the following decimal numbers into terminating, pure recurring or mixed recurring. Once classified, calculate the generating fraction of each of them and simplify the result to obtain the irreducible fraction:
- a) $3.15$
- b) $0,\widehat{12}$
- c) $2.1\widehat{6}$
Back to the beginning: $0.\overline{6}$ is purely repeating with a one-digit block, so its generating fraction is $\frac{6}{9} = \frac{2}{3}$. You have completed exactly two thirds of the mission.