Grade 9 Rational and Irrational Numbers ... - Free Printable

I'll work through each problem step by step to ensure accuracy.

Problem (1): If -7/X = X/7, then X is _______

Let me solve this equation:
-7/X = X/7
Cross multiply: -7 × 7 = X × X
-49 = X²
X² = -49

Wait, let me reconsider. If X² = -49, there's no real number solution. Let me check if I made an error.

Actually, looking at the equation again: -7/X = X/7
Cross multiplying: -7 × 7 = X × X
-49 = X²

This gives X² = -49, which has no real solutions. But wait - maybe I need to think about this differently. Let me check if there's a typo in my understanding or if we're working with complex numbers.

For Grade 9 level, let me reconsider: Perhaps the question means something different. Let me try X = ±√(-49), but that's imaginary.

Actually, let me re-read: If -7/X = X/7
Multiply both sides by 7X: -49 = X²

Hmm, this seems problematic for real numbers. Let me check the answer choices:
a. not a rational number
b. a natural number
c. a prime number
d. a whole number

If X² = -49, then X would be imaginary, which is "not a rational number." So the answer would be (a).

Problem (2): If p/q lies to the left of r/s on the number line, then

When one number lies to the left of another on the number line, it means it's smaller.
So: p/q < r/s

Answer: (c)

Problem (3): Which statement is false?

a. Every prime number is a whole number - TRUE (primes are 2, 3, 5, 7, 11... all whole numbers)
b. Every prime number is a rational number - TRUE (all integers are rational)
c. Any rational number, when multiplied by 9 is a rational number - TRUE (rational × rational = rational)
d. Every rational number is a natural number - FALSE (e.g., 1/2 is rational but not natural)

Answer: (d)

Problem (4): The rational number lying between 1/7 and 1/5

Let me convert to decimals:
1/7 ≈ 0.142857
1/5 = 0.2

Now check the options:
a. 6/35 = 0.1714... ✓ (between 0.142857 and 0.2)
b. 1/9 ≈ 0.111... ✗ (less than 1/7)
c. 1/2 = 0.5 (greater than 1/5)
d. 1/11 ≈ 0.0909... ✗ (less than 1/7)

Answer: (a)

Problem (5): Which statement is true for a rational number a/b?

a. The numerator a cannot be a prime number - FALSE (a can be any integer, including primes)
b. The denominator b cannot be 0 - TRUE (division by zero is undefined)
c. The denominator b cannot be a prime - FALSE (b can be any non-zero integer)
d. The denominator b cannot be 1 - FALSE (b can be 1, giving us integers)

Answer: (b)

Problem (6): If A = 3^x, B = 9^y, C = 27^z, where x = 0.̅5, y = 0.̅8, z = 0.̅2

First, convert repeating decimals to fractions:
- 0.̅5 = 5/9
- 0.̅8 = 8/9
- 0.̅2 = 2/9

Now:
A = 3^(5/9)
B = 9^(8/9) = (3²)^(8/9) = 3^(16/9)
C = 27^(2/9) = (3³)^(2/9) = 3^(6/9) = 3^(2/3)

A × B × C = 3^(5/9) × 3^(16/9) × 3^(6/9)
= 3^(5/9 + 16/9 + 6/9)
= 3^(27/9)
= 3³
= 27

Answer: 27

Problem (7): Examine if 'x' represents a rational or irrational number

Looking at the image, I don't see a specific value given for x in problem 7. This appears to be incomplete or refers to information not visible. However, based on typical problems of this type, if x was defined earlier or in context, I would need to determine if it can be expressed as a fraction.

Since no specific value is shown for x in problem 7, I cannot complete this without more information.

Final Answer:
(1) a
(2) c
(3) d
(4) a
(5) b
(6) 27
(7) [Insufficient information provided]