Exercises: Generating fraction of a decimal number

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.

1. a) $2,75$
2. b) $1,\widehat{6}$
3. c) $0,1\widehat{25}$
4. d) $34, \widehat{09}$

Calculate the generating fraction of these terminating decimals. Do not forget to simplify the fraction until you reach the irreducible fraction.

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 an irreducible fraction:

1. a) $0,1\widehat{6}$
2. b) $1,2\widehat{3}$
3. c) $0,0\widehat{5}$
4. d) $2,1\widehat{45}$

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.

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?

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.

Calculate the result of the following operation by first converting the decimal numbers to generating fractions and then operating:

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.

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.

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:

1. Identify what type of decimal number it is.

2. Calculate its generating fraction following the steps of the powers of 10 method.

3. Simplify the fraction until it is irreducible.

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: $12,\hat{5}$ seconds.

• Lucy: $12,0\hat{5}$ seconds.

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

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

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. Calculate the generating fraction of the initially occupied space.

2. Calculate the generating fraction of the freed space.

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

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:

1. Express the radius of the inner circle as a recurring decimal number.

2. 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.

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.

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:

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 $\frac{1}{40}$.

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.

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.

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.

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:

1. a) What volume of liquid is left in the container? (Use generating fractions for the calculation).
2. b) Is the result a terminating or recurring decimal?

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.

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?