Generating Fraction of a Decimal Number
A video game programmer in Guadalajara gets from the graphics engine the probability that a rare item appears: $0.02\overline{45}$. To program it, he needs an exact fraction, not a decimal that never ends. How does he get it? That is what you will learn here.
From fraction to decimal
First the way there: every fraction is a division. $\frac{3}{8}$ means $3 \div 8$.
When you divide the numerator by the denominator, one of two things happens:
-
The division ends (the remainder reaches $0$): the decimal is terminating. Example: $\frac{3}{8} = 0.375$.
-
The remainders repeat and so do the digits: the decimal is repeating. Examples: $\frac{5}{11} = 0.\overline{45}$ and $\frac{5}{6} = 0.8\overline{3}$.
Amina and Marcos are comparing how much progress they have made reading a book. Amina says she has read $\frac{3}{4}$ of the book, and Marcos says he has read $0.75$…
Amina and Marcos are comparing how much progress they have made reading a book. Amina says she has read $\frac{3}{4}$ of the book, and Marcos says he has read $0.75$ parts.
Calculate the generating fraction of Marcos's number and check if they have actually read the same amount.
Now the way back, which is the topic of this page: from a decimal to its generating fraction.
Types of decimals
Before converting to a fraction, you have to recognize what type of decimal you have.
-
Terminating: it ends, like $2.75$.
-
Purely repeating: after the decimal point only the repeating block appears, like $1.\overline{6}$ or $0.\overline{15}$.
-
Mixed repeating: between the decimal point and the repeating block there are digits that do not repeat (the non-repeating part), like $0.1\overline{25}$.
The bar goes over the digits that repeat. The whole-number part is the part before the decimal point.
Classify the following numbers as exact decimals, pure recurring decimals or mixed recurring decimals. Then, for each of them, state its integer part and, if it has…
Classify the following numbers as exact decimals, pure recurring decimals or mixed recurring decimals. Then, for each of them, state its integer part and, if it has them, its period and its non-repeating part (anteperiod).
- a) $2.75$
- b) $1,\widehat{6}$
- c) $0.1\widehat{25}$
- d) $34,\widehat{09}$
Terminating decimals
A terminating decimal is already a fraction in disguise: $0.75$ is read "seventy-five hundredths," that is, $\frac{75}{100}$. Then you simplify.
Calculate the generating fraction of these terminating decimals. Do not forget to simplify the fraction to its simplest form. a) $0.8$ b) $1.25$ c) $0.064$ d) $12.5$
Calculate the generating fraction of these terminating decimals. Do not forget to simplify the fraction to its simplest form.
- a) $0.8$
- b) $1.25$
- c) $0.064$
- d) $12.5$
A smartwatch shows that you have completed $0.65$ of your daily step goal, as shown by the ring in the figure. What fraction of the total number of steps does this…
A smartwatch shows that you have completed $0.65$ of your daily step goal, as shown by the ring in the figure. What fraction of the total number of steps does this amount represent? Express the result as an irreducible fraction.
Purely repeating decimals
With $0.\overline{3}$ we can't write "one third" just by looking at the digits, but an algebraic trick solves it. Let $x = 0.\overline{3}$. Multiply by $10$: $10x = 3.\overline{3}$. When you subtract, the infinite digits cancel out:
$$10x - x = 3 \quad\Rightarrow\quad 9x = 3 \quad\Rightarrow\quad x = \frac{3}{9} = \frac{1}{3}$$If the repeating block has two digits, you multiply by $100$, and the subtraction gives $99x$. That is where the rule comes from.
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) $1,\widehat{72}$
It rained a lot yesterday and the school rain gauge collected a total of $3,\widehat{3}$ liters per square meter. How would you express that exact amount of water using…
It rained a lot yesterday and the school rain gauge collected a total of $3,\widehat{3}$ liters per square meter. How would you express that exact amount of water using a fraction?
Find the generating fraction of the pure recurring decimal $12,\widehat{3}$. To do this, follow the algebraic method by calling the number $x$ and multiplying by the…
Find the generating fraction of the pure recurring decimal $12,\widehat{3}$. To do this, follow the algebraic method by calling the number $x$ and multiplying by the appropriate power of 10 to subtract the expressions and eliminate the decimal part. Do not forget to simplify the result.
Mixed repeating decimals
In $0.1\overline{6}$ the $1$ does not repeat. You do two multiplications so the repeating block lines up, and then subtract: with $x = 0.1\overline{6}$, $100x = 16.\overline{6}$ and $10x = 1.\overline{6}$. Subtracting: $90x = 15$, so $x = \frac{15}{90} = \frac{1}{6}$.
Express the following mixed recurring decimals as irreducible fractions: 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 irreducible fractions:
- 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:
-
Identify what type of decimal number it is.
-
Calculate its generating fraction following the steps of the powers of 10 method.
-
Simplify the fraction until it is irreducible.
Match each decimal number with its corresponding irreducible generating fraction in your notebook. Be careful, there is one fraction that does not correspond to any of…
Match each decimal number with its corresponding irreducible generating fraction in your notebook. Be careful, there is one fraction that does not correspond to any of the numbers:
| Decimal Number | Generating Fraction |
|---|---|
| $0,\widehat{45}$ | $\frac{9}{20}$ |
| $0.45$ | $\frac{41}{90}$ |
| $0.4\widehat{5}$ | $\frac{5}{11}$ |
| $\frac{1}{2}$ |
Operating without rounding
If you add $0.333 + 0.666$ you get $0.999$, but the exact result of $\frac13 + \frac23$ is $1$. Operating with truncated digits builds up error. The solution: convert all the decimals to fractions, operate, and simplify.
Calculate the result of the following operation by first converting the decimal numbers into generating fractions and then performing the operation: $$\frac{1}{3} +…
Calculate the result of the following operation by first converting the decimal numbers into generating fractions and then performing the operation:
$$\frac{1}{3} + 0,\widehat{6}$$In a popular strategy video game, a hero's attack speed is increased by a factor of $1.75$ and their defence is multiplied by $0,\hat{6}$. To make precise calculations…
In a popular strategy video game, a hero's attack speed is increased by a factor of $1.75$ and their defence is multiplied by $0,\hat{6}$. To make precise calculations in the game engine, the programmers need to work with fractions. Find the irreducible generating fractions of both numbers to help them.
Calculate the exact value of the following expression. To do this, you must first convert each decimal number into its respective irreducible generating fraction: $$1.25…
Calculate the exact value of the following expression. To do this, you must first convert each decimal number into its respective irreducible generating fraction:
$$1.25 - 0,\hat{3} \times \left( \frac{1}{2} + 0.1\hat{6} \right)$$Calculate the exact result of the following mixed operation. To do this, you must first find the generating fractions of each decimal number and simplify the…
Calculate the exact result of the following mixed operation. To do this, you must first find the generating fractions of each decimal number and simplify the intermediate results:
$$1.2 + 0,\widehat{3} - 0.1\widehat{6}$$Calculate the exact value of the following mixed expression. To avoid approximation errors, convert each decimal number into its respective generating fraction before…
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:
$$L = \frac{1.2\widehat{6} \times \sqrt{0.25}}{(-2)^{-2} + 0.8\widehat{3}}$$Comparing and solving problems
With the generating fraction you can also compare numbers that are very close together or calculate exact areas, times, and speeds.
Look at the following number line where two points, $A$ and $B$, are marked. Point $A$ represents the decimal number $0.8\hat{3}$ and point $B$ represents the number…
Look at the following number line where two points, $A$ and $B$, are 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 timed how long it takes them to run 100 meters. The times obtained are: Sara: $12.5$ seconds. Iván: $12,\hat{5}$ seconds.…
In a Physical Education class, three students timed how long it takes them to run 100 meters. The times obtained are:
-
Sara: $12.5$ seconds.
-
Iván: $12,\hat{5}$ seconds.
-
Lucía: $12.0\hat{5}$ seconds.
Express the time of each student as an irreducible generating fraction and order them from fastest to slowest.
A tailor needs to buy fabric to make some costumes. The design requires exactly $2.0\widehat{4}$ meters of blue fabric. In the store, pieces of fabric are sold only in…
A tailor needs to buy fabric to make some costumes. The design requires exactly $2.0\widehat{4}$ meters of blue fabric. In the store, pieces of fabric are sold only in fractional measurements. Which of the following pieces should the tailor choose to get the exact measurement?
- a) $\frac{202}{99}$ meters.
- b) $\frac{37}{18}$ meters.
- c) $\frac{92}{45}$ meters.
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…
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…
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:
$$\frac{13}{20} \quad ; \quad 0.6\widehat{5} \quad ; \quad 0,\widehat{65} \quad ; \quad \frac{2}{3} \quad ; \quad 0.65$$Back to the beginning: the programmer must enter $\frac{27}{1100}$, the generating fraction of $0.02\overline{45}$.
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 deduce which terminating decimal the mixed repeating decimal $4.2\widehat{9}$ is equivalent to, and verify your answer by performing the addition $4.2 + 0.0\widehat{9}$ using fractions.