Exercises on Rational Numbers

Study with the list of step-by-step exercises on rational numbers that Toda Matéria has prepared for you.

question 1

Then, from left to right, classify the following numbers as rational or non-rational.

a) Rational, rational, non-rational, non-rational, non-rational.
b) Rational, rational, non-rational, rational, rational.
c) Rational, rational, non-rational, non-rational, rational.
d) Rational, rational, rational, non-rational, rational.
e) Not rational, rational, not rational, rational, not rational.

See Answer

Correct answer: c) Rational, rational, non-rational, non-rational, rational.

-5 is rational because, being an integer, it is also contained in the set of rational numbers.

3/4 is rational because it is a number defined as a quotient of two integers, with a non-zero denominator.

it is irrational because there is no perfect square number, that is, a number that multiplied by itself results in three. Since there is no exact result, its decimal places are infinite rather than periodic.

it is irrational because it has infinitely many non-periodic decimal places.

it is rational because it represents the decimal decimal of a period equal to 4. Like this: 1.44444444... Although it has infinitely many decimal places, it can be written as the fraction 13/9.

question 2

Represent fractions in decimal form.

a) 12/5
b) 8/47
c) 9/4

See Answer

The)

B)

ç)

question 3

Represent decimal numbers as fractions.

a) 3.41
b) 154,461
c) 0.2

See Answer

The)

B)

ç)

Note: If possible, the answer can be simplified with an equivalent fraction. Ex: 2/10 = 1/5.

question 4

Considering the following rational numbers on a number line, write between which whole numbers they are located.

a) 6/4
b) -15/2
c) 4/21

See Answer

The) , so 1.5 is between 1 and 2.

1< 1,5 <2

B) , so -7.5 is between -8 and -7.

-8 < -7,5 < -7

ç) , so 5.25 is between 5 and 6.

question 5

Read the statements and check the option that correctly classifies them as true (T) or false (F).

1 - Every natural number is also a rational number.
2 - Rational numbers cannot be written as a fraction.
3 - There are numbers that are integers but are not natural, even though they are rational.
4 - A rational number can have infinite decimal places.

a) 1-F, 2-F, 3-V, 4-V.
b) 1-V, 2-F, 3-V, 4-F.
c) 1-V, 2-F, 3-V, 4-V.
d) 1-V, 2-V, 3-V, 4-V.
e) 1-V, 2-F, 3-F, 4-V.

See Answer

Correct answer: c) 1-V, 2-F, 3-V, 4-V.

1 - True. The set of natural numbers is contained in the set of whole numbers which, in turn, is contained in the set of rational numbers. Also, every natural number can be written as a fraction between two natural numbers, with a non-zero denominator.

2 - False. Every rational number can be written as a fraction.

3 - True. Negative numbers are integers and are not natural, although they can be expressed as a fraction.

4 - True. A rational number can have infinitely many decimal places, as long as it is a periodic decimal.

question 6

Compare the following rational numbers and rank them higher or lower.

See Answer

There are two ways to compare fractions, equating denominators or writing in the form of a decimal number.

Equating the denominators

The MMC (Least Common Multiple) between 3 and 2 is 6. This will be the new denominator of fractions. To determine the numerators, we divide 6 by the denominators of the original fractions and multiply by the numerators.

MMC(3,2)=6

the fraction we have: , so 2 multiplied by 5 is 10. The fraction looks like this: .

the fraction we have: , so 3 multiplied by 8 is 24. The fraction looks like this:

Since the two fractions have the same denominators, we compare the numerators.

Like is an equivalent fraction that originated from , we can conclude that it is less than .

Writing fractions as decimal numbers

Like , we concluded that .

question 7

Represent fractions in the form of decimal numbers, specifying, if any, their periodic decimals.

a) 1/3
b) 5/33
c) 7/9

See Answer

The)

B)

ç)

question 8

Add and subtract the rational numbers.

a) 4/6 + 2/6
b) 8/3 - 5/7
c) 13.45 + 0.3
d) 46.89 - 34.9

See Answer

The)

B)

The Equating the Denominators

c) 13.45 + 0.3 = 13.75

d) 46.89 - 34.9 =

question 9

Multiply the rational numbers.

a) 15/4 x 6/2
b) 8/7 x 9/5
c) 12.3 x 2.3
d) 3.02 x 6.2

See Answer

The)

B)

c) 12.3 x 2.3 = 28.29

d) 3.02 x 6.2 = 18.724

question 10

Perform rational number divisions.

The)

B)

ç)

d)

See Answer

The)

B)

ç)

d)

question 11

Power up the rational numbers.

The)
B)
ç)
d)

See Answer

The)

B)

ç)

d)

Enem questions about rational numbers

question 12

(Enem 2018) Article 33 of the Brazilian drug law provides for a prison sentence of 5 to 15 years for anyone who is convicted of illicit trafficking or unauthorized production of drugs. However, if the convict is a first-time offender, with a good criminal record, this penalty may be reduced from one-sixth to two-thirds.

Suppose a first offender, with a good criminal record, was convicted under article 33 of the Brazilian drug law.

After benefiting from the penalty reduction, your penalty may vary from

a) 1 year and 8 months to 12 years and 6 months.
b) 1 year and 8 months to 5 years.
c) 3 years and 4 months to 10 years.
d) 4 years and 2 months to 5 years.
e) 4 years and 2 months to 12 years and 6 months.

See Answer

Correct answer: a) 1 year and 8 months to 12 years and 6 months.

We must find the shortest and the longest time of confinement. As the options show counts in months, we used the time of the sentence described in the article for months, to facilitate the calculation.

5 years = 5. 12 months = 60 months
15 years = 15. 12 months = 180 months

Greatest possible reduction in the shortest seclusion time.

The biggest reduction is 2/3 of 60 months.

Applying a 40-month reduction to a 60-month sentence, 20 months are left over.

60 - 40 = 20 months

20 months is equal to 12 + 8, that is, 1 year and eight months.

Smallest possible reduction in the longest seclusion time.

The smallest reduction is 1/6 of 180 months.

Applying a 30-month reduction to a 180-month sentence, 150 months remain.

180 - 30 = 150 months

150 months is equal to 12 years and six months.

question 13

(Enem 2021) A survey was carried out on the educational level of a company's employees. It was found that 1/4 of the men who work there have completed high school, while 2/3 of the women who work in the company have completed high school. It was also found that among all those who have completed high school, half are men.

The fraction that represents the number of male employees in relation to the total employees of this company is

a) 1/8
b) 11/3
c) 11/24
d) 2/3
e) 11/8

See Answer

Correct answer: e) 8/11

If h is the total number of men and m is the total number of women, the total number of employees is h + m. The problem wants the number of men divided by the total number.

Half of those who have high school are men, so the other half are women, so one number equals another.

• 2/3 of women have high school
• 1/4 of men have high school

isolating m

Substituting m for this value in equation 1, we have

Therefore, the fraction that represents the number of male employees in relation to the total number of employees in this company is .

question 14

For one season of Formula 1 racing, each car's fuel tank capacity is now 100 kg of gasoline. One team chose to use a gasoline with a density of 750 grams per liter, starting the race with a full tank. At the first refueling stop, a car of this team presented a record in its on-board computer showing the consumption of four-tenths of the gasoline originally contained in the tank. To minimize the weight of this car and ensure the end of the race, the support team refueled the car with a third of what was left in the tank upon arrival for refueling.

Available at: www.superdanilof1page.com.br. Accessed on: July 6th 2015 (adapted).

The amount of gasoline used, in liters, in refueling was

The)

B)

ç)

d) 20 x 0.075

e) 20 x 0.75

See Answer

Correct answer: b)

The total amount of fuel in the tank is 100 kg or 100,000 g.

Each 750 g corresponds to 1 liter. In this way, the total amount of liters in the tank is:

4/10 of fuel was consumed until the stop, that is to say that there were 6/10 of 100,000 / 750 left over.

In replenishment, 1/3 of the remaining quantity was placed. This way we have:

Leftover fuel

quantity replenished

When reorganizing the fractions, we arrive more easily or result, like this:

You may be interested in:

• Rational Numbers
• Operations with decimal numbers
• Numerical sets
• fractions
• Multiplication and Division of Fractions