Exercises: Generating fraction of a decimal number

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).

1. a) $2.75$

1. b) $1,\widehat{6}$

1. c) $0.1\widehat{25}$

1. d) $34,\widehat{09}$

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) $2.1\widehat{45}$

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.

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?

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.

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

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.

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.

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 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.

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

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.

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 is $4.5$ cm:

1. Express the radius of the inner circle as a terminating decimal.

2. If the area of the outer circle is calculated as $A = \pi \times r^2$, and we use the approximation $\pi \approx 3.14$, calculate the area of the small circle using always the generating fraction to obtain the highest possible precision.

A number is said to be "curious" if adding the digits of its numerator and denominator in its irreducible generating fraction yields the same value. Check whether the numbers $0.4\hat{6}$ and $1,\hat{3}$ are "curious" numbers by calculating their generating fractions.

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.

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:

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?

1. a) $\frac{202}{99}$ meters.
2. b) $\frac{37}{18}$ meters.
3. c) $\frac{92}{45}$ meters.

Look at the following number line where three points have been marked: $A$, $B$ and $C$. Point $B$ represents the decimal number $1.2\widehat{7}$. Find its irreducible generating fraction and state which two integers this point lies between.

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:

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 $\frac{1}{40}$.

In an archaeological excavation in Ancient Egypt, a square tile was found whose surface area measures exactly $5.4\widehat{4}$ square decimeters. The archaeologists need to know the length of the tile's side to reconstruct the original mosaic.

Determine the side length as a fraction and then calculate the perimeter of the tile, expressing it as a decimal number.

Find a digit $x$ such that the pure recurring decimal $0,\widehat{x7}$ has a generating fraction whose simplified denominator is $33$. Is there more than one solution? Justify your answer by analysing the sum of the digits in the numerator.

A surveillance drone travels a distance of $14.2\widehat{7}$ kilometers 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 work with fractions.

Calculate the average speed of the drone in km/h using generating fractions. Express the final result as an integer or an irreducible fraction.

Two chemical tanks contain different mixtures. The first holds $2.1\widehat{6}$ liters of a reagent and the second holds $1.8\widehat{3}$ liters. 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 remains in the container? (Use generating fractions for the calculation).
2. b) Is the result a terminating or a repeating 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 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.

A satellite sends data in burst transmissions lasting microseconds. A burst lasts $0,\widehat{142857}$ seconds. The control center receives $2.1$ bursts per clock cycle.

If a clock cycle lasts exactly as long as the sum of those bursts, calculate the duration of a cycle expressing it as an irreducible fraction.

Hint: Look at the period $142857$, does it remind you of dividing a natural number by 7?

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:

An organic farmer has designed an irrigation tank with a rectangular base. The dimensions of the base are $2,\widehat{6}$ meters long and $1.125$ meters wide. If they want to install a protective cover that covers exactly the entire surface of the base, what should the area of the cover be? Express the result as an irreducible fraction of a square meter.

In a laboratory test, a scientist mixes three substances. The first weighs $\frac{5}{9}$ of a gram, the second weighs $0,\widehat{4}$ grams, and the third weighs $0,\widehat{09}$ grams. Prove, using generating fractions, that the total sum of the three substances is exactly $\frac{12}{11}$ of a gram.

Find the value of the unknown $x$ in the following equality, expressing the decimal numbers as fractions first:

A television game show sets the following challenge: "Find the smallest non-zero natural number that, when multiplied by the mixed recurring decimal $0.1\widehat{36}$, gives an integer as a result". Which number are they looking for? Explain the steps followed.

Three siblings have inherited an estate and divide it as follows: the eldest gets $0.4\widehat{6}$ of the estate, the middle one $0.3\widehat{18}$ and the youngest gets the rest.

1. a) Calculate the generating fraction of the share corresponding to each of the two older siblings.
2. b) What fraction of the total is left for the youngest sibling? Express the result as an irreducible fraction.

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:

A tanker truck is carrying $12,000$ liters of milk. At the first stop, it unloads $0.2\widehat{3}$ of the total. At the second stop, it unloads $\frac{3}{11}$ of what was left.

1. a) How many liters remain in the truck after the second stop?
2. b) Express the total amount unloaded across both stops, in liters, as a repeating decimal, stating whether it is pure or mixed.