Problem Analysis:
The task involves analyzing the number \( K = \frac{2}{3} \times 10^7 \) and determining its properties based on the given options. Let's break it down step by step.
---
Step 1: Understand the Number \( K \)
The number \( K \) is given as:
\[
K = \frac{2}{3} \times 10^7
\]
#### Simplify \( K \):
First, calculate the value of \( K \):
\[
K = \frac{2}{3} \times 10^7 = \frac{2 \times 10^7}{3}
\]
\[
K = \frac{20,000,000}{3}
\]
Perform the division:
\[
20,000,000 \div 3 = 6,666,666.\overline{6}
\]
Thus,
\[
K = 6,666,666.\overline{6}
\]
This means \( K \) is a repeating decimal with the digit \( 6 \) repeating indefinitely.
---
Step 2: Analyze the Properties of \( K \)
#### Option (a): Natural Numbers
Natural numbers are positive integers starting from 1 (i.e., \( 1, 2, 3, \ldots \)). Since \( K = 6,666,666.\overline{6} \) is not an integer (it has a repeating decimal part), \( K \) is
not a natural number.
#### Option (b): Whole Numbers
Whole numbers include all natural numbers and zero (i.e., \( 0, 1, 2, 3, \ldots \)). Since \( K \) is not an integer, it is also
not a whole number.
#### Option (c): Rational Numbers
Rational numbers are numbers that can be expressed as the ratio of two integers \( \frac{p}{q} \), where \( q \neq 0 \). The number \( K \) is explicitly given as \( \frac{2}{3} \times 10^7 \), which simplifies to \( \frac{20,000,000}{3} \). This is a ratio of two integers, so \( K \) is a
rational number.
#### Option (d): Irrational Numbers
Irrational numbers cannot be expressed as the ratio of two integers and have non-repeating, non-terminating decimal expansions. Since \( K \) is a repeating decimal (\( 6,666,666.\overline{6} \)), it is
not an irrational number.
#### Option (e): Real Numbers
Real numbers include all rational and irrational numbers. Since \( K \) is a rational number, it is also a
real number.
---
Step 3: Determine Which Options Apply
From the analysis above:
- \( K \) is
not a natural number.
- \( K \) is
not a whole number.
- \( K \) is a
rational number.
- \( K \) is
not an irrational number.
- \( K \) is a
real number.
Thus, the correct options are:
- (c) Rational Numbers
- (e) Real Numbers
---
Final Answer:
\[
\boxed{\text{c, e}}
\]
Parent Tip: Review the logic above to help your child master the concept of classifying numbers worksheet answers.
