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 $3,\widehat{3}$ litres per square metre. How would you express that exact amount of water using a fraction?

Aoife and Cian are comparing how much progress they have made reading a book. Aoife says she has read $\frac{3}{4}$ of the book, and Cian says he has read $0.75$ parts.

Calculate the generating fraction of Cian'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:

To find the generating fraction of the mixed recurring decimal $0.2\widehat{5}$, Cian writes the following procedure on the board:

When asked by the teacher, Cian justifies himself saying: "The numerator is the subtraction of the digits without the comma minus the non-recurring part, and in the denominator I put two nines because behind the comma there are two decimal places in total".

1. a) Explain in a reasoned way what conceptual error Cian has made when determining the denominator of the fraction.
2. b) Write the correct rule for this type of numbers and calculate the irreducible generating fraction of $0.2\widehat{5}$.
3. c) Check that your result is correct by carrying out by hand the division of your numerator by your denominator until you obtain at least three decimal places.

It is proposed to calculate the exact value of the following sum of pure recurring decimals:

1. a) Without using pencil and paper to calculate, make a quick mental estimate of the result by rounding each term to the tenths place (a single decimal place). What approximate value do you get?
2. b) Determine the generating fraction of each of the two terms and add them to obtain the exact result as an irreducible fraction.
3. c) Write the exact result as an integer or decimal number. Compare it with your estimate from section a) and explain why in mathematics it is essential to use generating fractions when operating with infinite recurring decimal places.

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 metres. The times obtained are:

• Aoife: $12.5$ seconds.

• Cian: $12,\hat{5}$ seconds.

• Niamh: $12.0\hat{5}$ 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 $2.0\widehat{4}$ metres of blue fabric. In the shop, 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) $\frac{202}{99}$ metres.
2. b) $\frac{37}{18}$ metres.
3. c) $\frac{92}{45}$ metres.

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.

On the water bill of a local authority area in Ireland, the smart meter of a household has recorded a two-month consumption of $12.8\widehat{3}\text{ m}^3$. The local council tariff sets a price of €1.80 for each cubic metre consumed, to which a fixed service fee of €5.50 must be added.

1. a) To prevent the bill from containing errors due to rounding infinite digits, the computer system must convert the meter reading into its generating fraction. Find said irreducible fraction of cubic metres.
2. b) Express the price per cubic metre (€1.80) as an irreducible fraction and calculate the exact cost of water consumption (without the fixed fee) by multiplying both fractions.
3. c) Add the fixed fee to obtain the exact total amount of the bill.
4. d) If the company had approximated the consumption to two decimal places ($12.83\text{ m}^3$), what would have been the final amount charged? Would the family or the utility company have benefited? Justify your answer by calculating the exact difference.

We are looking for a pure recurring decimal $D = 0,\widehat{a}$ (where $a$ is a digit from $1$ to $9$) that satisfies the following condition: when multiplying it by the terminating decimal $1.5$ and then adding the fraction $\frac{1}{6}$, exactly the integer $1$ is obtained.

1. a) Write the generating fraction of the number $D$ in terms of the unknown $a$.
2. b) Convert the terminating decimal $1.5$ to an irreducible fraction and set up an equation with all the generating fractions that translates the problem statement.
3. c) Solve the equation to find the value of the digit $a$ and write the pure recurring decimal $D$ that was sought.
4. d) Check the result by substituting the found value into the original combined operation and solving it step by step.

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 decimetres. 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 $14.2\widehat{7}$ kilometres 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 $2.1\widehat{6}$ litres of a reagent and the second holds $1.8\widehat{3}$ litres. 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 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 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 centre 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.

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 $2,\widehat{6}$ metres long and $1.125$ metres 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 metre.

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 $12,000$ litres 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 litres remain in the truck after the second stop?
2. b) Express the total amount unloaded across both stops, in litres, as a recurring decimal, stating whether it is pure or mixed.

To calculate the exact value of the combined expression:

Three 2nd Year students discuss which strategy is the best:

• Cian (Decimal shortcut): Observes that $1,\widehat{6}$ and $0,\widehat{6}$ have exactly the same repetend, so he proposes subtracting them directly in decimal format ($1,\widehat{6} - 0,\widehat{6} = 1$) before converting to fractions.

• Aoife (Standard method): Proposes converting each and every one of the four recurring decimals to its irreducible generating fraction from the very beginning and operating solely with fractions respecting the order of operations.

• Seán (Decimal approximation): Proposes rounding each number to two decimal places ($1.67$; $0.67$; $0.33$; $2.17$), calculating the result with decimals and then looking for which fraction it resembles.

1. a) Work through the complete procedure according to Aoife's method and find the exact result as an irreducible fraction.
2. b) Apply Cian's shortcut to the numerator and complete the rest of the operation. Do you arrive at the same result as with Aoife's method? Justify mathematically why Cian's shortcut is legitimate in this specific case and warn about what danger it would pose if the repetends were not identical.
3. c) Calculate the numerical result that Seán would obtain with his approximations. Compare its value with the exact result obtained in the previous sections and argue why Seán's method is not acceptable in an exact calculation.

Consider two pure recurring decimals less than $1$, formed by two digits in the repetend that are reversed relative to each other: $N_1 = 0,\widehat{ab}$ and $N_2 = 0,\widehat{ba}$, where $a$ and $b$ are distinct non-zero digits such that $a > b$.

1. a) Write the generating fraction of $N_1$ in terms of the digits $a$ and $b$. Remember that the number formed by two digits $ab$ is equivalent to $10a + b$.
2. b) Express the sum $N_1 + N_2$ as a single algebraic fraction and prove that this sum can always be simplified so that its denominator is $9$, regardless of what the digits $a$ and $b$ are.
3. c) It is known that the exact sum of the two numbers is $N_1 + N_2 = 1,\widehat{2}$ and that their exact difference is $N_1 - N_2 = 0,\widehat{27}$.

1. Set up a system of two equations with two unknowns ($a$ and $b$) from the generating fractions of these results.

2. Solve the system to uniquely determine the values of the digits $a$ and $b$.

3. Write the original recurring decimals $N_1$ and $N_2$ and check that all initial conditions are satisfied.