Exercises: Generating fraction of a decimal number

To start working with generating fractions, it is essential to know what type of number we are dealing with. Classify the following decimal numbers into terminating decimal, pure recurring or mixed recurring:

1. a) $3.15$
2. b) $0.666...$
3. c) $12.4777...$
4. d) $5.232323...$
5. e) $8.02333...$

In the Natural Sciences laboratory, you have weighed a sample of sand and the scale reads $0.75$ grams. Write this number as a decimal fraction (with a power of 10 in the denominator) and then simplify it to obtain its irreducible generating fraction.

Identify the elements of the following repeating decimal numbers. For each one, state its integer part, its non-repeating part (if it has one) and its repeating part (period):

1. a) $5.888...$

1. b) $0.1232323...$

1. c) $14.5262626...$

Calculate the generating fraction of the following terminating decimal numbers. Remember to simplify the result wherever possible:

1. a) $0.5$
2. b) $1.2$
3. c) $0.25$
4. d) $0.08$

A video game indicates that you have completed $0.\overline{6}$ of a mission. To find out the exact part of the map you have covered, convert that pure recurring decimal number into its generating fraction.

Match each decimal number with its corresponding fraction. Long calculations are not required, use the logic of terminating decimals:

Find the generating fraction of the following pure recurring decimal numbers with a single digit in the recurring part. Simplify if you can:

1. a) $0,\overline{3}$
2. b) $0,\overline{7}$
3. c) $1,\overline{1}$
4. d) $0,\overline{5}$

On offer in a music shop, a digital album costs $9.90$ €. Express the price of the album as a fraction of natural numbers and simplify it until it cannot be reduced any further.

To express the pure recurring decimal $0,\widehat{4}$ as a fraction, Cian wrote on the board:

1. a) Without doing any division yet, explain what confusion Cian had between terminating and recurring decimals when choosing the denominator.
2. b) Check Cian's mistake by doing the decimal division of $2$ by $5$ by hand and explain why his result does not match the starting number.
3. c) Apply the rule for generating fractions to obtain the irreducible fraction that represents $0,\widehat{4}$ exactly.

We want to find the generating fraction of the pure recurring decimal $0,\widehat{7}$.

1. a) Estimate: Remember that $0.5 = \frac{1}{2}$ and that $0.75 = \frac{3}{4}$. Reason whether the value of $0,\widehat{7}$ must be greater or less than $\frac{3}{4}$.
2. b) Calculate: Obtain the generating fraction of $0,\widehat{7}$ by applying the rule for pure recurring decimals. Is it an irreducible fraction, or can it be simplified?
3. c) Check: Find a common denominator for the fraction $\frac{3}{4}$ and the fraction obtained in the previous section to compare their numerators. Do the calculations confirm your initial estimate?

Classify the following decimal numbers into terminating, pure recurring or mixed recurring. Once classified, calculate the generating fraction of each of them and simplify the result to obtain the irreducible fraction:

1. a) $3.15$
2. b) $0,\widehat{12}$
3. c) $2.1\widehat{6}$

In a strategy video game, a loading bar indicates that $0.85$ of a special mission has been completed. What irreducible fraction of the mission has already been completed? Calculate the generating fraction of that decimal number and simplify it.

In an individual time trial stage of the Rás Tailteann, the official stopwatch records that the winner took exactly $1.1\widehat{6}$ hours to complete the course.

1. a) Classify this decimal number (terminating, pure recurring, or mixed recurring) and state its integer part, its non-recurring part, and its repetend.
2. b) Find the irreducible generating fraction of this time in hours, calculating the fraction corresponding to the decimal part and then adding the integer part to it.
3. c) A television commentator rounds the figure to $1.16$ hours and states: “The cyclist took exactly $1$ hour and $16$ minutes”. Multiply the irreducible generating fraction from the previous section by the $60$ minutes in an hour to find the exact actual time, and explain the two mistakes made by the commentator.

To calculate the generating fraction of the pure recurring decimal $2,\widehat{54}$, two students follow different strategies:

• Aoife's strategy (mixed number): She separates the integer part and writes $2 + \frac{54}{99}$. Then she simplifies the recurring fraction and finally adds the integer part.

• Seán's strategy (direct fraction): In the numerator, she subtracts the integer part from the entire number without the decimal point: $\frac{254 - 2}{99}$, and then simplifies the resulting fraction.

1. a) Work through both methods step by step and verify that both lead to the same irreducible fraction.
2. b) Compare both approaches: which one do you consider more advantageous if you have to work with a number with a very large integer part, such as $340,\widehat{6}$? Justify your answer by assessing the difficulty of the calculations in each case.

We are looking for a decimal number greater than $1$ whose irreducible generating fraction strictly has the number $33$ as its denominator. We know that its integer part is $1$ and that its repetend has two digits starting with $4$; that is, the number is of the form $1,\widehat{4a}$, where $a$ is an unknown digit.

1. a) Reason why this decimal must necessarily be pure recurring and cannot be terminating or mixed recurring, by analyzing what unsimplified denominator $33$ comes from.
2. b) Write the fraction with denominator $99$ equivalent to the irreducible fraction with denominator $33$. By what factor had it been simplified?
3. c) Find out what possible values the digit $a$ can take so that, after simplifying by that factor, the resulting fraction is irreducible and retains the denominator $33$ (remember that the simplified numerator cannot share any factors with $33$). Write down the decimal numbers found and their respective irreducible fractions.

At a seismic monitoring station of the Geological Survey Ireland (GSI) in County Galway, two sensors record the movement of a fault:

• Sensor A detects a displacement of $0.2\widehat{6}$ millimeters.

• Sensor B detects a displacement of $0.\widehat{4}$ millimeters.

A novice technician states: “We add the digits directly: $0.2\widehat{6} + 0.\widehat{4} = 0.6\widehat{10}$, which equals a total displacement of exactly $0.7$ millimeters”.

1. a) Justify in detail why the technician's procedure is mathematically incorrect and explain what problem arises when trying to directly add digits that repeat to infinity.
2. b) Model the situation by calculating the irreducible generating fraction of each sensor's reading.
3. c) Add both fractions to obtain the irreducible generating fraction of the total displacement. Convert it back to decimal form, classify the resulting decimal, and calculate the exact difference between the actual value and the $0.7$ millimeters that the technician had estimated.