Exercises: Concept

In a sports shop, the prices of three items have been written in words. Write in your notebook the decimal number that corresponds to each one using digits:

1. a) Swimming goggles costing eight euros and fifteen cents.
2. b) A tennis ball costing two euros and eight cents.
3. c) Running shorts costing twelve euros and fifty cents.

State the place value (units, tenths, hundredths or thousandths) of the digit $7$ in each of the following numbers:

1. a) $15,72$
2. b) $0,007$
3. c) $7,431$
4. d) $2,075$

Expand the following decimal numbers according to their place value units, as in the example:

Example: $4,25 = 4$ Units + $2$ tenths + $5$ hundredths.

1. a) $8,39$
2. b) $12,045$
3. c) $0,614$

Look at the following number line and write which decimal number corresponds to each of the letters marked with an orange dot:

Copy into your notebook and use the symbols $<$ or $>$ to compare these pairs of decimal numbers:

1. a) $5,4$ ____ $5,39$
2. b) $0,25$ ____ $0,3$
3. c) $12,07$ ____ $12,071$
4. d) $6,1$ ____ $6,099$

Match each decimal number with its corresponding decimal fraction:

• Numbers: $0,7$ | $0,07$ | $7,7$ | $0,77$

• Fractions: $\frac{77}{100}$ | $\frac{7}{10}$ | $\frac{7}{100}$ | $\frac{77}{10}$

If we divide a square into $100$ equal parts, each part is one hundredth ($0,01$). Look at the following image and write which decimal number represents the orange shaded area:

Complete these equivalences between place values:

1. a) $4$ units = ____ tenths.
2. b) $70$ hundredths = ____ tenths.
3. c) $5$ tenths = ____ thousandths.
4. d) $300$ thousandths = ____ hundredths.

Lucía is ordering measurements for a school laboratory practical and confidently states: "The number $3,125$ is greater than $3,8$ because $125$ is much bigger than $8$".

1. a) Explain the conceptual error Lucía makes regarding the place value of the decimal digits after the comma.
2. b) Show how to correctly compare both numbers by matching their number of decimal places using zeros.
3. c) Write the correct order relation between them using the appropriate symbol ($<$ or $>$).

In an event of the Spanish U-14 Athletics Championship, the digital timing system recorded the times of three athletes in the final of the $80\text{ m}$ sprint: Darío took $10,18\text{ s}$, Lucas clocked $10,2\text{ s}$, and Mateo stopped the clock at $10,105\text{ s}$.

1. a) Write the three times with the same precision (in thousandths of a second), adding the necessary zeros so that they have the same number of decimal places.
2. b) Order the athletes from shortest to longest time (from fastest to slowest) to determine the podium positions.
3. c) How many hundredths of a second separated the winner of the race from the runner-up? Justify your answer based on the place value of each digit.

In a science laboratory, a group of students measured the mass of a mineral sample using a precision balance. The result obtained is $42,308$ grams.

State what place value each digit represents in this number:

• The digit $4$ represents the ............................

• The digit $2$ represents the ............................

• The digit $3$ represents the ............................

• The digit $0$ represents the ............................

• The digit $8$ represents the ............................

Also write its expanded form using units and decimal fractions.

In the school Olympics, the times recorded in the $100$ metres flat final were very close. The judges recorded the following results in seconds:

$12,45$ — $12,408$ — $12,5$ — $12,099$ — $12,41$

Order the runners from fastest (shortest time) to slowest (longest time) using the symbol $<$ and write out in words the decimal number that corresponds to the gold medal winner.

Observe the following number line where several points are represented. Identify which terminating decimal number corresponds to each of the marked letters ($A$, $B$ and $C$).

Classify the following decimal numbers, indicating whether they are terminating decimals, pure recurring decimals or mixed recurring decimals. For recurring decimals, clearly state what the recurring part is and, if present, the non-recurring part:

1. a) $5,08$
2. b) $2,333...$
3. c) $14,252525...$
4. d) $0,1666...$
5. e) $3,141592...$ (is this last one a recurring decimal number?)

Sometimes fractions help us better understand the type of decimal number we have. Without performing long division, classify the decimal expansion of the following fractions as terminating or recurring:

1. a) $\frac{7}{20}$
2. b) $\frac{4}{9}$
3. c) $\frac{13}{6}$
4. d) $\frac{1}{25}$

Lucía is packing her rucksack for a trip. She has bought a bottle of water for $1,25$ €, a bag of nuts for $3,050$ € and a cap for $9,9$ €.

1. a) Write all the prices so that they have the same number of decimal places (adding zeros if necessary).
2. b) Convert the price of the cap into an irreducible fraction.
3. c) How many euro cents does the bag of nuts cost?

One of the most interesting properties of decimal numbers is that we can always find another number between two given ones, no matter how close together they are.

Represent graphically on a number line the interval between $0,56$ and $0,57$ and mark on it three decimal numbers that lie between them.

You can do it on paper and send a photo with your answer.

To find out whether the fraction $\frac{18}{150}$ results in a terminating decimal, a pure recurring decimal, or an eventually recurring decimal, two students propose different approaches:

• Method A (Marcos): Directly carry out the division $18 : 150$ using the traditional paper algorithm until checking if the remainder is zero or if the digits start repeating.

• Method B (Sara): First simplify the fraction to its irreducible form and then factor the new denominator into prime factors to apply the rule of twos and fives.

1. a) Apply both methods step by step and classify the resulting decimal.
2. b) If instead of that fraction you were asked to classify $\frac{21}{8750}$, which of the two methods would you choose and why? Justify your decision by explaining the relationship between the prime factors of the denominator and the powers of $10$.

We propose studying the decimal value of the fraction $\frac{13}{32}$ by combining estimation and theoretical analysis:

1. a) Estimate: Without performing the division, reason between which two consecutive tenths ($0,1$; $0,2$; $0,3$; $0,4$; $0,5$...) this number must lie. To guide yourself, compare it with simpler benchmark fractions with that same denominator, such as $\frac{8}{32}$ and $\frac{16}{32}$.
2. b) Check: Carry out the division to find the first three decimal places and check whether your estimation was correct.
3. c) Reason: Looking at the prime factorization of the denominator of $\frac{13}{32}$, justify whether the complete decimal expansion will end up having a finite number of digits (terminating) or infinite repeating digits. If it is terminating, deduce how many decimal places it will have in total without needing to continue dividing.

A decimal number is made up of $4$ units, $15$ tenths, $27$ hundredths and $105$ thousandths. Make the necessary regroupings (for example, $10$ thousandths make $1$ hundredth, $10$ hundredths make $1$ tenth and $10$ tenths make $1$ unit) to write the number correctly in standard decimal form and then state how many thousandths in total this number contains.

Find a decimal number $x$ that is at exactly the same distance from $3,14$ and $3,15$. Express it both in decimal form and as an irreducible fraction. What type of decimal number is the result?

Classify the following fractions according to the type of decimal expansion they generate (Terminating, Pure Recurring or Mixed Recurring) without performing the division, based solely on the prime factorisation of their denominators:

1. a) $\frac{7}{40}$
2. b) $\frac{5}{18}$
3. c) $\frac{11}{21}$
4. d) $\frac{9}{250}$

Write the irreducible generating fraction of the following recurring decimal numbers, showing the step-by-step process:

1. a) $2,1\widehat{36}$
2. b) $0,00\widehat{5}$

Order the following decimal numbers mixed with fractions from smallest to largest. To compare them, convert all the elements to their decimal form with at least four decimal places:

$1,2\widehat{5}$ ; $\frac{5}{4}$ ; $1,2525...$ ; $1,25$ ; $\frac{11}{9}$

Find out which number it is, knowing that it satisfies the following conditions:

1. Its whole part is the smallest even prime number.

2. It has four decimal places.

3. The thousandths digit is three times the tenths digit.

4. The sum of all its digits is $15$.

5. The ten-thousandths digit is half of the hundredths digit.

A nanometre ($nm$) is equivalent to $0,000000001$ metres. If the thickness of a human hair is approximately $0,08$ millimetres, express this thickness in metres using powers of $10$ and determine how many nanometres the hair measures.

Represent on a number line the interval between $5,67$ and $5,68$. Divide this interval into ten equal parts (each part measures $0,001$) and mark the exact position of the numbers $5,674$ and $5,679$.

1. a) Draw the line with the interval divided into ten equal parts and write the corresponding numbers under the marks.

1. b) Mark the numbers $5,674$ and $5,679$ on your line.

1. c) Find a decimal number that has exactly five decimal places (the last digit different from $0$) and that lies between $5,674$ and $5,679$.

You are presented with the reverse challenge of constructing two different irreducible fractions, $\frac{A}{B}$ and $\frac{C}{D}$, that simultaneously satisfy these three conditions:

1. The decimal value of both fractions must strictly lie between $0,65$ and $0,75$.

2. The denominators $B$ and $D$ must be two-digit integers less than $30$.

3. The fraction $\frac{A}{B}$ must be a terminating decimal, while the fraction $\frac{C}{D}$ must be an eventually recurring decimal.

1. a) Deduce which prime factors can (and which cannot) make up the denominators $B$ and $D$ to guarantee the type of decimal requested in each case.
2. b) Find a pair of fractions $\frac{A}{B}$ and $\frac{C}{D}$ that satisfies all the conditions.
3. c) Check your answer by finding the decimal expression of both and identifying, in the case of the eventually recurring decimal, its non-recurring part and its recurring period.

In a debate about decimal numbers, Hugo defends the following rule: "To know if an irreducible fraction gives a terminating decimal, you only have to look at the last digit of the denominator: if it ends in $5$ or $0$, it is always terminating because numbers ending in $5$ or $0$ are divisible by $5$ or $10$. On the other hand, if it ends in any other digit, it can never be terminating".

1. a) Prove that Hugo's statement is false by finding:

• A counterexample of an irreducible fraction whose denominator ends in $5$ and is NOT terminating.

• A counterexample of an irreducible fraction whose denominator ends in $0$ and is NOT terminating.

• A counterexample of an irreducible fraction whose denominator does NOT end in either $5$ or $0$ and IS terminating.

1. b) Classify the type of decimal generated by each of your three counterexamples.
2. c) Explain Hugo's conceptual error: why is it not enough to look at the last digit, and what relationship do the prime factors of the simplified denominator have with the possibility of expanding the fraction to a power of $10$ ($10$, $100$, $1.000$...)?.