A teacher provides a list of eight numerical values on the board for a classification exer… — State PSC Exam

A teacher provides a list of eight numerical values on the board for a classification exercise:

(I) $\sqrt{144}$
(II) $\frac{22}{7}$
(III) $\pi$
(IV) $0.\overline{36}$
(V) $\sqrt{12}$
(VI) $\frac{\sqrt{3}}{\sqrt{27}}$
(VII) $3.14159$
(VIII) $0.121221222...$ (where the number of 2s increases by one each time)

What is the exact count of rational numbers in this list?

Answer: 5

Answer

There are exactly 5 rational numbers in the provided list.

By simplifying each expression and applying the definitions of real numbers, exactly 5 of the 8 values (values I, II, IV, VI, and VII) satisfy the conditions of being a rational number.

Step-by-Step Solution

Define the criteria for a rational number.

A rational number is any number that can be expressed as a fraction of two integers (p/q, where q is not zero). This includes all integers, terminating decimals, and repeating decimals.

Establishing the definition is necessary to accurately classify each item.

Evaluate the square roots and fractions in the list.

\sqrt{144} = 12

, which is an integer (Rational).

\frac{22}{7}

is a ratio of two integers (Rational).

\sqrt{12} = 2\sqrt{3}

, which contains the root of a non-perfect square (Irrational).

\frac{\sqrt{3}}{\sqrt{27}}

simplifies to

\sqrt{\frac{3}{27}} = \sqrt{\frac{1}{9}} = \frac{1}{3}

(Rational).

Radicals must be simplified to their lowest terms to reveal their true mathematical classification.

Evaluate the decimal representations and constants in the list.

0.\overline{36}

is a repeating decimal (Rational).

3.14159

is a terminating decimal (Rational).

\pi

is a transcendental constant with infinite, non-repeating digits (Irrational).

0.121221222...

has a changing pattern, making it non-terminating and non-recurring (Irrational).

Decimals must be analyzed by their termination or repetition properties.

Count the total number of items identified as rational.

The rational values are (I), (II), (IV), (VI), and (VII). The total count is 5.

The question asks for the exact numerical count of rational numbers.

Key Concept

Identifying rational and irrational numbers by their fractional, radical, and decimal properties.

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