Exercises: Concept

In a biology laboratory, a student is measuring the thickness of different plant leaves. Write the following decimal numbers obtained in his measurements in figures:

1. a) Twelve units and eight tenths.
2. b) Seven units and twenty-five hundredths.
3. c) Zero units and one hundred and four thousandths.
4. d) Five units and nine ten-thousandths.

Classify the following decimal numbers indicating whether they are exact, pure recurring or mixed recurring:

1. a) $4,25$
2. b) $7,\widehat{3}$
3. c) $1,02\widehat{5}$
4. d) $12,444...$
5. e) $0,123$
6. f) $5,6\widehat{12}$

In the Olympic Games, swimming times are measured with great precision. A swimmer has taken $48,052$ seconds to complete an event. Indicate what place each digit occupies in this number:

• The $4$ occupies the:

• The $8$ occupies the:

• The $0$ occupies the:

• The $5$ occupies the:

• The $2$ occupies the:

Observe the following grid representing the unit divided into one hundred equal parts. Indicate what decimal number the orange shaded part represents and how this number is read.

A group of friends is comparing the price of their favorite video games in different online stores. Copy into your notebook and use the signs $<$ , $>$ or $=$ to compare the prices found:

1. a) $\$14,50$ _____ $\$14,05$
2. b) $\$19,99$ _____ $\$19,990$
3. c) $\$7,025$ _____ $\$7,25$
4. d) $\$32,1$ _____ $\$32,098$

Identify which decimal numbers are represented by the letters $A$, $B$ and $C$ on the following number line:

Three siblings measure their height. Ashley is $1,54$ feet tall and Mark is $1,55$ feet tall. Find a decimal number that lies between Ashley's and Mark's heights.

Order the following decimal numbers from smallest to largest:

A nanotechnology laboratory measures the thickness of a new synthetic fiber obtaining a value of $0,034825\text{ mm}$. Indicate the place value of each of the digits in this number and fill in the blanks:

1. a) The thousandths digit is $...$
2. b) The number $8$ occupies the place of the $...$
3. c) The millionths digit is $...$
4. d) Express the number as a sum of decimal fractions (additive decomposition).

Classify the following decimal numbers in the attached table (Exact, Pure Recurring, Mixed Recurring or Irrational):

• $3,141592...$ (Number $\pi$)

• $2,15$

• $7,444...$

• $0,1232323...$

• $\sqrt{2}$

• $5,0616161...$

In a Formula 1 championship, the qualifying times of four drivers for one lap of the circuit were as follows:

• Driver A: $1:24,058\text{ s}$

• Driver B: $1:24,5\text{ s}$

• Driver C: $1:24,06\text{ s}$

• Driver D: $1:24,0578\text{ s}$

Order the drivers from fastest to slowest using the signs $>$ or $<$ as appropriate.

Draw a number line in your notebook and represent the following decimal numbers. For negative numbers, use a different color.

$A = 0,4$

$B = -0,25$

$C = 0,85$

$D = -0,7$

Three friends buy a gift that costs $\$45,75$ and decide to pay in equal parts. When dividing the price, they obtain a decimal number.

1. a) Calculate how much each one must pay.
2. b) Round the result to the nearest hundredth (cents).
3. c) Truncate the result to the hundredths.
4. d) Is there any difference between rounding and truncating in this case? Justify your answer.

A carpenter needs to cut a wooden plank with an exact length of $2,4568\text{ meters}$. However, his tape measure only reads to the nearest millimeter.

1. a) What measurement will he take if he decides to round to the nearest millimeter?
2. b) Calculate the absolute error made when making this approximation.
3. c) Would the absolute error be larger or smaller if he decided to truncate to the hundredths? Explain why.

Without performing the division of their terms, indicate what type of decimal number each of the following fractions will generate (exact or recurring). Justify your answer by analyzing the prime factors of the denominator:

1. a) $\dfrac{7}{40}$
2. b) $\dfrac{5}{12}$
3. c) $\dfrac{11}{21}$
4. d) $\dfrac{3}{250}$

Observe the following figure, where a square of area $1\text{ u}^2$ has been divided into sections.

1. a) Express the orange shaded part in decimal form.
2. b) Express the part left blank in decimal form.
3. c) What type of decimal numbers have you obtained (exact or recurring)?

Order the following decimal numbers from least to greatest, justifying your answer by expanding at least four of their decimal places:

• $A = 2,3\hat{5}$

• $B = 2,\widehat{35}$

• $C = 2,35$

• $D = 2,35\hat{3}$

• $E = 2,355$

Once ordered, find a terminating decimal number and a pure repeating decimal number that lie between the two central values of your list.

Find the generating fraction of the following decimal numbers. You must express the result as an irreducible fraction:

1. a) $1,02\widehat{45}$
2. b) $0,1\hat{9}$ (Analyze the result obtained and reflect on its value).
3. c) $3,4\hat{6} + 1,\hat{3}$ (Perform the addition using their generating fractions).

In a biology laboratory, the size of a microorganism has been measured, obtaining a value of $0,04586$ in. Due to a software error, the microscope shows this approximate value in two different ways:

• Approximation X: truncated to the thousandths.

• Approximation Y: rounded to the thousandths.

Calculate the relative error made in each of the two approximations. Which of the two estimates is more precise for the scientific study? Justify your answer by comparing the errors obtained.

Express the following measurements in scientific notation, which have been taken with different levels of precision. Afterwards, order them from largest to smallest mass:

1. a) The weight of a pollen grain: $0,00000000062$ kg.
2. b) The mass of a dust particle: $0,00000075$ g.
3. c) The mass of a virus: $0,0000000000000023$ hg.

Note: Express all quantities in the same unit (kg) before making the comparison.

Determine the value of the digit $x$ so that the generating fraction of the decimal number $0,x\hat{6}$ is equivalent to $\frac{7}{15}$. Explain step by step the logical process and the operations necessary to find this digit.

Insert three rational numbers (in fraction form) between the decimal numbers $0,\hat{7}$ and $0,7\hat{8}$.

To solve it, follow these steps:

1. Obtain the generating fractions of both numbers.

2. Reduce these fractions to a common denominator.

3. Find three fractions that lie between them and check that their decimal expressions meet the insertion condition.

A surveyor measures the length of a bridge and obtains $125,48$ ft. However, the true measurement is $125,4836$ ft.

1. a) Calculate the absolute error and the relative error of the surveyor's measurement.
2. b) If for a city hall plan it is decided to round the true measurement to the nearest tenth, what would be the new relative error?
3. c) Compare both relative errors and explain which of the two measurements (the surveyor's or the plan's) offers a more accurate representation of reality.

Represent the following numbers approximately on the number line, but determine exactly which of them is larger using their generating fractions:

• $X = 1,4\hat{6}$

• $Y = \sqrt{2}$ (Use the approximation $1,4142...$)

• $Z = \frac{11}{7}$

Justify the position of each one on the line by performing the necessary divisions to at least three decimal places.