Writing Decimals as Fractions
A video game engine says the chance of a rare item dropping is $0.02454545\ldots$, a decimal that never ends. But the game's code needs an exact fraction, not a decimal that goes on forever. How do you turn a decimal that never stops into a fraction? By the end of this page you will know, and you will use it to calculate and compare numbers exactly.
Three kinds of decimals
Every decimal number is one of three kinds. The kind tells you which method to use.
Some problems draw a hat over the repeating digits, like $0.\widehat{6}$, and ask for the generating fraction. It is the same as the bar, and the task is the same: write the decimal as a fraction.
For any decimal, you can name three pieces: the whole-number part (left of the point), the non-repeating part, and the repeating block. In $34.\overline{09}$ the whole-number part is $34$, there is no non-repeating part, and the block is $09$.
Classify the following numbers according to whether they are terminating decimals, pure recurring decimals or mixed recurring decimals. Then, indicate for each of them…
Classify the following numbers according to whether they are terminating decimals, pure recurring decimals or mixed recurring decimals. Then, indicate for each of them what its integer part is and, if they have one, its recurring part and its non-recurring part.
- a) $2,75$
- b) $1,\widehat{6}$
- c) $0,1\widehat{25}$
- d) $34, \widehat{09}$
Terminating decimals to fractions
A terminating decimal is just a fraction written in place value. Read it out loud: $0.65$ is "sixty-five hundredths," so it is $\frac{65}{100}$. Then simplify.
The bar above shows $\frac{13}{20} = \frac{65}{100} = 0.65$. Same amount, different names.
Calculate the generating fraction of these terminating decimals. Do not forget to simplify the fraction until you reach the irreducible fraction. a) $0,8$ b) $1,25$ c)…
Calculate the generating fraction of these terminating decimals. Do not forget to simplify the fraction until you reach the irreducible fraction.
- a) $0,8$
- b) $1,25$
- c) $0,064$
- d) $12,5$
A smartwatch indicates that you have completed $0,65$ of your daily step goal. What fraction of the total steps does this amount represent? Express the result as an…
A smartwatch indicates that you have completed $0,65$ of your daily step goal. What fraction of the total steps does this amount represent? Express the result as an irreducible fraction.
Ashley and Mark are comparing how much they have progressed in reading a book. Ashley says she has read $\frac{3}{4}$ of the book, and Mark says he has read $0,75$…
Ashley and Mark are comparing how much they have progressed in reading a book. Ashley says she has read $\frac{3}{4}$ of the book, and Mark says he has read $0,75$ parts.
Calculate the generating fraction of Mark's number and check if they have really read the same amount.
Pure repeating decimals
You cannot write $0.\overline{7}$ as $\frac{7}{10}$: that would be $0.7$, which stops. We need a trick that makes the endless part disappear. The trick: subtract a shifted copy of the number from itself.
Let $x = 0.\overline{7} = 0.7777\ldots$ Multiply by $10$ to slide the decimal point one place: $10x = 7.7777\ldots$ The endless tails are identical, so subtracting cancels them.
The whole-number part just comes along: $2.\overline{3} = 2 + \frac{1}{3} = \frac{7}{3}$, and the algebra method gives the same result.
Find the generating fraction of the following pure recurring decimal numbers: a) $0,\widehat{4}$ b) $0,\widehat{15}$ c) $2,\widehat{3}$ d) $1,\widehat{72}$
Find the generating fraction of the following pure recurring decimal numbers:
- a) $0,\widehat{4}$
- b) $0,\widehat{15}$
- c) $2,\widehat{3}$
- d) $1,\widehat{72}$
Yesterday it rained a lot and the school's rain gauge collected a total of $3,\widehat{3}$ gallons per square foot. How would you express that exact amount of water…
Yesterday it rained a lot and the school's rain gauge collected a total of $3,\widehat{3}$ gallons per square foot. How would you express that exact amount of water using a fraction?
In a popular strategy video game, a hero's attack speed is increased by a factor of $1,75$ and their defense is multiplied by $0,\hat{6}$. To perform precise…
In a popular strategy video game, a hero's attack speed is increased by a factor of $1,75$ and their defense is multiplied by $0,\hat{6}$. To perform precise calculations in the game engine, the programmers need to work with fractions. Find the irreducible generating fractions of both numbers to help them.
Mixed repeating decimals
When digits come before the block, as in $0.1\overline{6}$, first multiply to put the block right after the decimal point, then multiply again to shift one full block.
Express the following mixed recurring decimals as an irreducible fraction: a) $0,1\widehat{6}$ b) $1,2\widehat{3}$ c) $0,0\widehat{5}$ d) $2,1\widehat{45}$
Express the following mixed recurring decimals as an irreducible fraction:
- a) $0,1\widehat{6}$
- b) $1,2\widehat{3}$
- c) $0,0\widehat{5}$
- d) $2,1\widehat{45}$
A Twitch streamer has calculated that the average duration of their subscriptions is $2,41\hat{6}$ months. If they want to express this time exactly for their quarterly…
A Twitch streamer has calculated that the average duration of their subscriptions is $2,41\hat{6}$ months. If they want to express this time exactly for their quarterly report:
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Identify what type of decimal number it is.
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Calculate its generating fraction following the steps of the powers of 10 method.
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Simplify the fraction until it is irreducible.
A number is said to be "curious" if by adding the digits of its numerator and its denominator in the irreducible generating fraction, we obtain the same value. Check if…
A number is said to be "curious" if by adding the digits of its numerator and its denominator in the irreducible generating fraction, we obtain the same value. Check if the numbers $0,4\hat{6}$ and $1,\hat{3}$ are "curious" numbers by calculating their generating fractions.
Computing exactly with fractions
If you add or multiply decimals that repeat, you have to cut them off somewhere, and the answer becomes a little wrong. Fractions avoid that: change every decimal to a fraction first, then compute.
The same idea works for multiplying: $0.\overline{6} \times 1.5 = \frac{2}{3} \times \frac{3}{2} = 1$. Follow the order of operations: parentheses first, then multiplication and division, then addition and subtraction.
Calculate the result of the following operation by first converting the decimal numbers to generating fractions and then operating: $$\frac{1}{3} + 0,\widehat{6}$$
Calculate the result of the following operation by first converting the decimal numbers to generating fractions and then operating:
$$\frac{1}{3} + 0,\widehat{6}$$Calculate the value of the following expression exactly. To do this, you must first transform each decimal number into its respective irreducible generating fraction:…
Calculate the value of the following expression exactly. To do this, you must first transform each decimal number into its respective irreducible generating fraction:
$$1,25 - 0,\hat{3} \cdot \left( \frac{1}{2} + 0,1\hat{6} \right)$$The storage of a messaging app 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.…
The storage of a messaging app 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.
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Calculate the generating fraction of the initially occupied space.
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Calculate the generating fraction of the freed space.
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What fraction of GB remains occupied on the phone? Express the result as an irreducible fraction.
Comparing and ordering
Two numbers can look almost the same when one is a decimal and the other is a fraction. Convert both to fractions with a common denominator, or write out enough decimal digits to see which is larger.
The number line shows $\frac{7}{10} = 0.7$ and $\frac{7}{9} = 0.\overline{7}$. They are close, but $0.\overline{7}$ is farther right, so it is greater.
When you have to order several decimals, line up the digits and write extra places: $12.5 = 12.500\ldots$, $12.\overline{5} = 12.555\ldots$, $12.0\overline{5} = 12.0555\ldots$ Compare place by place from the left.
Look at the following number line where two points, $A$ and $B$, have been marked. Point $A$ represents the decimal number $0,8\hat{3}$ and point $B$ represents the…
Look at the following number line where two points, $A$ and $B$, have been marked. Point $A$ represents the decimal number $0,8\hat{3}$ and point $B$ represents the number $1,\hat{2}$.
Calculate the generating fraction of each point and find the exact distance between $A$ and $B$ expressed as a single irreducible fraction.
In a Physical Education class, three students have timed how long it takes them to run the 100-meter dash. The times obtained are: Sarah: $12,5$ seconds. Ivan:…
In a Physical Education class, three students have timed how long it takes them to run the 100-meter dash. The times obtained are:
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Sarah: $12,5$ seconds.
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Ivan: $12,\hat{5}$ seconds.
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Lucy: $12,0\hat{5}$ seconds.
Express each student's time as an irreducible generating fraction and order them from fastest to slowest.
A young programmer is designing a video game and needs to define the probability of a rare item appearing. The code requires the probability to be entered as an…
A young programmer is designing a video game and needs to define 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 in mixed recurring decimal format: $P = 0,02\widehat{45}$.
Find the generating fraction that the programmer must enter and justify whether this probability is greater or less than $\frac{1}{40}$.
Back to the start: to put the drop chance into the game's code, you enter the fraction $\frac{27}{1100}$, which is exactly $0.02\overline{45}$.
From fractions back to decimals
To go the other way, divide the numerator by the denominator. The result either terminates or repeats. For instance, $\frac{7}{9} = 7 \div 9 = 0.\overline{7}$ and $\frac{5}{6} = 0.8\overline{3}$.
You can tell which one happens before dividing.
Examples: $\frac{3}{8}$ terminates ($8 = 2^3$), giving $0.375$. But $\frac{4}{15}$ repeats, since $15 = 3 \times 5$ has a $3$. And $\frac{1}{7}$ repeats with a long block: $\frac{1}{7} = 0.\overline{142857}$.
You can check that last one with the method from before. The block has 6 digits, so $0.\overline{142857} = \frac{142857}{999999}$. Since $999{,}999 = 7 \times 142{,}857$, this simplifies to $\frac{1}{7}$.
A young graphic designer is creating a logo based on concentric circles. The radius of the inner circle must be exactly $\frac{7}{9}$ of the radius of the outer circle.…
A young graphic designer is creating a logo based on concentric circles. The radius of the inner circle must be exactly $\frac{7}{9}$ of the radius of the outer circle. If the radius of the outer circle measures $4,5$ in:
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Express the radius of the inner circle as a recurring decimal number.
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If the area of the outer circle is calculated as $A = \pi \cdot r^2$, and we use the approximation $\pi \approx 3,14$, calculate the area of the small circle always using the generating fraction to obtain the maximum possible precision.
Two chemical tanks contain different mixtures. The first has $2,1\widehat{6}$ gallons of a reagent and the second has $1,8\widehat{3}$ gallons. If we mix both contents…
Two chemical tanks contain different mixtures. The first has $2,1\widehat{6}$ gallons of a reagent and the second has $1,8\widehat{3}$ gallons. If we mix both contents in a third container and, due to evaporation, a quarter of the total mixture is lost:
- a) What volume of liquid is left in the container? (Use generating fractions for the calculation).
- b) Is the result a terminating or recurring decimal?
A satellite sends data in bursts of microseconds. A burst lasts $0,\widehat{142857}$ seconds. The control center receives $2,1$ bursts per clock cycle. If a clock cycle…
A satellite sends data in bursts of microseconds. A burst lasts $0,\widehat{142857}$ seconds. The control center receives $2,1$ bursts per clock cycle.
If a clock cycle lasts exactly what the sum of these bursts indicates, calculate the duration of a cycle, expressing it as an irreducible fraction.
Hint: Look at the period $142857$, does it remind you of the division of a natural number by 7?
Special cases: $0.\overline{9}$ and digit puzzles
The method has a surprising result. Let $x = 0.\overline{9}$.
So $0.\overline{9} = 1$ exactly. It is not "almost" $1$: two different numbers always have another number between them, and there is no number between $0.\overline{9}$ and $1$.
The same logic shows that $0.0\overline{9} = \frac{1}{10}$, so $4.2\overline{9} = 4.2 + 0.1 = 4.3$. A decimal ending in a repeating $9$ is just a terminating decimal in disguise.
Find a digit $x$ so that the pure recurring decimal $0,\widehat{x7}$ has as its generating fraction an expression whose simplified denominator is $33$. Is there more…
Find a digit $x$ so that the pure recurring decimal $0,\widehat{x7}$ has as its generating fraction an expression whose simplified denominator is $33$. Is there more than one solution? Justify your answer by analyzing the sum of the digits of the numerator.
We know that in the decimal numeral system, the number $0,\widehat{9}$ is equal to $1$. Prove this using the generating fraction algorithm. Next, use that same logic to…
We know that in the decimal numeral system, the number $0,\widehat{9}$ is equal to $1$. Prove this using the generating fraction algorithm.
Next, use that same logic to reason which integer the mixed recurring decimal $4,2\widehat{9}$ is equivalent to, and check your answer by performing the sum $4,2 + 0,0\widehat{9}$ using fractions.
Multi-step problems with roots and powers
Longer problems mix repeating decimals with roots, powers, and division. The plan stays the same: turn every decimal into a fraction first, then use fraction rules.
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A square root of a fraction: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ (when $a$ and $b$ are perfect squares). For example, $\sqrt{0.25} = \sqrt{\frac{25}{100}} = \frac{5}{10} = \frac{1}{2}$.
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A negative exponent flips the base: $a^{-n} = \frac{1}{a^n}$.
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Dividing by a fraction means multiplying by its reciprocal.
Calculate the exact value of the following combined expression. To avoid approximation errors, transform each decimal number into its respective generating fraction…
Calculate the exact value of the following combined expression. To avoid approximation errors, transform each decimal number into its respective generating fraction before operating and simplify the result until you obtain an irreducible fraction:
$$L = \frac{1,2\widehat{6} \cdot \sqrt{0,25}}{(-2)^{-2} + 0,8\widehat{3}}$$In an archaeological excavation in ancient Egypt, a square tile has been found whose surface area measures exactly $5,4\widehat{4}$ square decimeters. The archaeologists…
In an archaeological excavation in ancient Egypt, a square tile has been found whose surface area measures exactly $5,4\widehat{4}$ square decimeters. The archaeologists need to know the measurement of the side of the tile to reconstruct the original mosaic.
Determine the length of the side as a fraction and, subsequently, calculate the perimeter of the tile, expressing it as a decimal number.
A surveillance drone travels a distance of $14,2\widehat{7}$ miles in a time of $0,3\widehat{8}$ hours. To calculate the average speed with total precision (without…
A surveillance drone travels a distance of $14,2\widehat{7}$ miles in a time of $0,3\widehat{8}$ hours. To calculate the average speed with total precision (without rounding in intermediate steps), the control system must operate with fractions.
Calculate the average speed of the drone in mph using the generating fractions. Express the final result as an integer or an irreducible fraction.