Exercises: Concept

In a sporting goods store, the prices of three items have been written in words. Write in your notebook the decimal number corresponding to each one using digits:

1. a) Swimming goggles that cost eight dollars and fifteen cents.
2. b) A tennis ball that costs two dollars and eight cents.
3. c) Running shorts that cost twelve dollars and fifty cents.

Indicate what place value (ones, tenths, hundredths, or thousandths) the digit $7$ has in each of the following numbers:

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

Decompose the following decimal numbers according to their place values, as in the example:

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

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

Observe 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 signs $<$ 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 ($0,01$). Observe the following image and write what decimal number the orange shaded area represents:

Complete these equivalences between place values:

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

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

Indicate what 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 ones and decimal fractions.

In the school Olympics, the times recorded in the final of the $100$ meters sprint 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 the fastest (lowest time) to the slowest (highest time) using the symbol $<$ and write how the decimal number corresponding to the gold medal winner is read.

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

Find the "mystery" number following these clues:

1. It has three decimal places.

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

3. The hundredths digit is double the tenths digit.

4. The thousandths digit is the sum of the tenths and hundredths.

5. The tenths digit is $3$.

Write the resulting number as a decimal fraction.

Classify the following decimal numbers indicating whether they are terminating decimals, pure repeating, or mixed repeating. For the repeating ones, clearly indicate which is the repeating part and, if there is one, the non-repeating 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 repeating decimal?)

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

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

Hint: Look at the prime factors of the denominators.

Ashley is organizing her backpack for a trip. She 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 a fraction in simplest form.
3. c) How many cents does the bag of nuts cost?

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

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

A decimal number is composed of $4$ ones, $15$ tenths, $27$ hundredths, and $105$ thousandths. Perform the necessary groupings to write the number correctly in its standard decimal form, and then indicate how many total thousandths this number contains.

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

Classify the following fractions according to the type of decimal expression they generate (Terminating, Pure repeating, or Mixed repeating) without performing the division, based solely on the prime factorization 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 generating fraction in simplest form of the following repeating decimals, 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 least to greatest. To compare them, convert all the numbers into their decimal form with at least four decimal places:

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

Find the number that meets the following conditions:

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

2. It has four decimal places that do not repeat.

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 the hundredths digit.

One 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.

Represent on a number line the interval between $5,67$ and $5,68$. Divide this interval into ten equal parts and mark the exact position of the numbers $5,674$ and $5,679$. Then, find a decimal number that has five decimal places and is located between those two marked points.