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Question

Difficulty: MediumClassification of Numbers

Consider the following set of 8 numbers:

S={−7,0,π,227,169,50,2.5,0.45‾}S = \left\{ -7, 0, \pi, \frac{22}{7}, \sqrt{169}, \sqrt{50}, 2.5, 0.\overline{45} \right\}

Let:
P=P = the number of integers in SS
Q=Q = the number of rational numbers in SS
R=R = the number of irrational numbers in SS
T=T = the number of whole numbers in SS

Calculate the exact value of (Q×R)+(P×T)(Q \times R) + (P \times T).

Answer: 18

Answer

18
Based on mathematical definitions: P (integers) = 3 because -7, 0, and 13 are integers. Q (rationals) = 6 because it includes the 3 integers plus 22/7, 2.5, and 0.4545... R (irrationals) = 2 because it includes pi and sqrt(50). T (whole numbers) = 2 because it includes the non-negative integers 0 and 13. Plugging these into (Q * R) + (P * T) yields (6 * 2) + (3 * 2) = 18.

Step-by-Step Solution

1
Evaluate and simplify each number in the set to determine its properties.
\sqrt{169} simplifies to exactly 13. The repeating decimal 0.\overline{45} can be written as a fraction (45/99).
Numbers must be evaluated in their simplest form to avoid misclassification.
2
Determine the value of P by counting the integers.
The integers are -7, 0, and 13. Therefore, P = 3.
Integers include positive numbers, negative numbers, and zero, with no decimal or fractional parts.
3
Determine the value of Q by counting the rational numbers.
The rational numbers are -7, 0, 22/7, 13, 2.5, and 0.\overline{45}. Therefore, Q = 6.
Rational numbers are any numbers that can be expressed as a ratio of two integers.
4
Determine the value of R by counting the irrational numbers.
The irrational numbers are \pi and \sqrt{50}. Therefore, R = 2.
Irrational numbers have non-terminating, non-repeating decimal expansions.
5
Determine the value of T by counting the whole numbers.
The whole numbers are 0 and 13. Therefore, T = 2.
Whole numbers consist only of the non-negative integers.
6
Substitute the counts into the requested mathematical expression and calculate the final result.
(6 \times 2) + (3 \times 2) = 12 + 6 = 18.
To answer the specific question given in the stem.

Key Concept

Classification of real numbers into integers, rational numbers, irrational numbers, and whole numbers.
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Topics

Classification of Numbers

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