Rational and Irrational Numbers

Which of the following numbers is a rational number?

Answer explanation

0.75 is a rational number because it can be expressed as the fraction 3/4. In contrast, \sqrt{3}, \pi, and e are irrational numbers, as they cannot be expressed as a fraction of two integers.

Answer explanation

To convert \( \frac{5}{8} \) to a decimal using long division, divide 5 by 8. 8 goes into 50 six times (48), leaving a remainder of 2. Bring down a 0 to make 20. 8 goes into 20 two times (16), leaving a remainder of 4. Bring down another 0 to make 40. 8 goes into 40 five times (40), leaving no remainder. Thus, \( \frac{5}{8} = 0.625 \). The correct answer is \( 0.625 \).

Tags

CCSS.8.NS.A.1

CCSS.7.NS.A.2D

Which of the following decimals is a repeating decimal?

Answer explanation

The repeating decimal is 0.333..., which represents the fraction 1/3. The other options, 0.5, 0.25, and 0.8, are terminating decimals and do not repeat.

Tags

CCSS.8.NS.A.1

CCSS.7.NS.A.2D

• Rational

• Irrational

• Both

• Neither

Answer explanation

The number -2.75 is a rational number because it can be expressed as a fraction (-11/4). Rational numbers include all integers, fractions, and terminating or repeating decimals, making 'Rational' the correct classification.

Which of the following numbers is irrational?

Answer explanation

The number \( \sqrt{5} \) is irrational because it cannot be expressed as a fraction of two integers. In contrast, \( \frac{7}{9} \), \( 0.121212\ldots \) (a repeating decimal), and \( -3 \) are all rational.

Tags

CCSS.8.NS.A.1

Answer explanation

The fraction \( \frac{1}{4} \) represents one part of four equal parts. When converted to decimal form, it equals 0.25. Therefore, the correct answer is \( 0.25 \).

Which of the following statements is true about rational numbers?

• Their decimal form always terminates.

• Their decimal form always repeats.

• Their decimal form either terminates or repeats.

• Their decimal form never repeats.

Answer explanation

Rational numbers can be expressed as fractions. Their decimal forms either terminate (like 0.5) or repeat (like 0.333...). Thus, the correct statement is that their decimal form either terminates or repeats.

Tags

CCSS.8.NS.A.1

CCSS.7.NS.A.2D

Answer explanation

To convert \( \frac{2}{3} \) to a decimal using long division, divide 2 by 3. The result is 0.666..., which is a repeating decimal. Thus, the correct answer is \( 0.66\ldots \).

Tags

CCSS.8.NS.A.1

CCSS.7.NS.A.2D

Which of the following is NOT a rational number?

Answer explanation

A rational number can be expressed as a fraction of two integers. -1.5, 0.272727..., and \sqrt{16} (which equals 4) are rational. However, \sqrt{2} is irrational as it cannot be expressed as a fraction.

Answer explanation

To find the decimal expansion of \( \frac{7}{20} \), divide 7 by 20. This results in 0.35, making \( 0.35 \) the correct answer. The other options do not equal \( \frac{7}{20} \).

Tags

CCSS.8.NS.A.1

CCSS.7.NS.A.2D