Rational Numbers Worksheet - Fill in the blanks on properties and operations of rational numbers.

Let's solve each question in the "Rational Numbers" Worksheet-1 step by step and explain the reasoning.

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(1) The product of two positive rational numbers is always ________.

✔ Answer: Positive

Explanation: When you multiply two positive numbers, the result is always positive. This holds true for rational numbers as well.

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(2) The product of a positive rational number and a negative rational number is always ________.

✔ Answer: Negative

Explanation: A positive × negative = negative. This rule applies to rational numbers too.

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(3) The product of two negative rational numbers is always ________.

✔ Answer: Positive

Explanation: Negative × negative = positive. So, two negative rationals give a positive product.

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(4) The reciprocal of a positive rational number is ________.

✔ Answer: Positive

Explanation: If a number is positive, its reciprocal (1/number) is also positive.

Example: Reciprocal of $ \frac{2}{3} $ is $ \frac{3}{2} $ — both positive.

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(5) The reciprocal of a negative rational number is ________.

✔ Answer: Negative

Explanation: The reciprocal of a negative number remains negative.

Example: Reciprocal of $ -\frac{3}{4} $ is $ -\frac{4}{3} $

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(6) Zero has ________ reciprocal.

✔ Answer: No

Explanation: Division by zero is undefined. So, $ \frac{1}{0} $ is not defined. Hence, zero has no reciprocal.

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(7) The product of a rational number and its reciprocal is ________.

✔ Answer: 1

Explanation: By definition, $ a \times \frac{1}{a} = 1 $, provided $ a \neq 0 $.

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(8) The numbers ________ and ________ are their own reciprocals.

✔ Answer: 1 and -1

Explanation:
- $ \frac{1}{1} = 1 $ → reciprocal of 1 is 1
- $ \frac{1}{-1} = -1 $ → reciprocal of -1 is -1

So, only 1 and -1 are their own reciprocals.

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(9) If $ a $ is reciprocal of $ b $, then the reciprocal of $ b $ is ________.

✔ Answer: $ a $

Explanation: If $ a = \frac{1}{b} $, then $ \frac{1}{b} = a $. So reciprocal of $ b $ is $ a $.

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(10) The number 0 is ________ the reciprocal of any number.

✔ Answer: Not

Explanation: Since $ \frac{1}{0} $ is undefined, 0 cannot be the reciprocal of any number.

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(11) Reciprocal of $ \frac{1}{r}, r \ne 0 $ is ________.

✔ Answer: $ r $

Explanation:
The reciprocal of $ \frac{1}{r} $ is $ \frac{1}{\frac{1}{r}} = r $

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(12) $ \frac{-4}{13} - \frac{-3}{26} = $ ________

Let’s simplify:

$$
\frac{-4}{13} - \left( \frac{-3}{26} \right) = \frac{-4}{13} + \frac{3}{26}
$$

Find common denominator: LCM of 13 and 26 is 26.

Convert:
$$
\frac{-4}{13} = \frac{-8}{26}
$$

Now:
$$
\frac{-8}{26} + \frac{3}{26} = \frac{-5}{26}
$$

✔ Answer: $ \boxed{\frac{-5}{26}} $

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(13) $ \frac{-9}{14} + \_\_\_\_\_\_ = -1 $

Let the missing number be $ x $:

$$
\frac{-9}{14} + x = -1
\Rightarrow x = -1 + \frac{9}{14} = \frac{-14}{14} + \frac{9}{14} = \frac{-5}{14}
$$

✔ Answer: $ \boxed{\frac{-5}{14}} $

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(14) $ \frac{-7}{9} + \_\_\_\_\_\_ = 3 $

Let $ x $ be the missing number:

$$
x = 3 - \left( \frac{-7}{9} \right) = 3 + \frac{7}{9} = \frac{27}{9} + \frac{7}{9} = \frac{34}{9}
$$

✔ Answer: $ \boxed{\frac{34}{9}} $

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(15) $ \_\_\_\_\_\_ + \frac{15}{23} = 4 $

Let $ x $ be the missing number:

$$
x = 4 - \frac{15}{23} = \frac{92}{23} - \frac{15}{23} = \frac{77}{23}
$$

✔ Answer: $ \boxed{\frac{77}{23}} $

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(16) $ (17 \times 12)^{-1} = 17^{-1} \times \_\_\_\_\_\_ $

We use the property:
$$
(ab)^{-1} = a^{-1} \times b^{-1}
$$

So,
$$
(17 \times 12)^{-1} = 17^{-1} \times 12^{-1}
$$

✔ Answer: $ \boxed{12^{-1}} $

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(17) $ -4 \times \frac{7}{9} = \frac{7}{9} \times \_\_\_\_\_\_ $

We can rewrite:
$$
-4 \times \frac{7}{9} = \frac{7}{9} \times (-4)
$$

Because multiplication is commutative.

✔ Answer: $ \boxed{-4} $

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✔ Final Answers Summary:

| Q.No | Answer |
|------|--------|
| (1) | Positive |
| (2) | Negative |
| (3) | Positive |
| (4) | Positive |
| (5) | Negative |
| (6) | No |
| (7) | 1 |
| (8) | 1 and -1 |
| (9) | $ a $ |
| (10) | Not |
| (11) | $ r $ |
| (12) | $ \frac{-5}{26} $ |
| (13) | $ \frac{-5}{14} $ |
| (14) | $ \frac{34}{9} $ |
| (15) | $ \frac{77}{23} $ |
| (16) | $ 12^{-1} $ |
| (17) | $ -4 $ |

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