The concept of a decimal number

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) $0.614$

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 a sports store, 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 pesos and fifteen cents.
2. b) A tennis ball costing two pesos and eight cents.
3. c) Running shorts costing twelve pesos and fifty cents.

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.

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 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.

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) $6.1$ ____ $6.099$

In the school Olympics, the times recorded in the $100$ meters 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.

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

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 $C$).

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 $5.679$.

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.

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?

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:

Lucía is packing her backpack for a trip. She has bought a bottle of water for $1.25$ pesos, a bag of nuts for $3.050$ pesos and a cap for $9.9$ pesos.

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 pesos cents does the bag of nuts cost?

A nanometer ($nm$) is equivalent to $0.000000001$ meters. If the thickness of a human hair is approximately $0.08$ millimeters, express this thickness in meters using powers of $10$ and determine how many nanometers the hair measures.

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) $\frac{1}{25}$

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:

$1.2\widehat{5}$ ; $\frac{5}{4}$ ; $1.2525...$ ; $1.25$ ; $\frac{11}{9}$