GSEB Class 8 Maths Solutions Chapter 1 Rational Numbers InText Questions

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Question 1. Fill in the table:

Answer: Using the closure property for addition, subtraction, multiplication, and division concerning rational numbers, integers, whole numbers, and natural numbers, we observe:
In simple words: The table above shows if rational numbers, integers, whole numbers, and natural numbers stay within their group (are "closed") when you do addition, subtraction, multiplication, or division.

Exam Tip: Remember the closure property by testing a few examples for each number set to quickly confirm if they stay within the set after an operation.

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Question 1. Complete the following table:

Answer: Here is the completed table showing the commutative property for different number sets:
In simple words: This table explains which kinds of numbers can be swapped around in an operation (like \( a + b = b + a \)) and still give the same result. This is called the commutative property.

Exam Tip: To remember commutativity, think of 'commuting' or moving numbers. If they can move and the answer stays the same, it's commutative.

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Question 1. Complete the following table:

Answer: The associative property describes whether grouping changes the result of an operation. Here is the completed table:
In simple words: This table indicates if you can group numbers differently (like \( (a + b) + c \) versus \( a + (b + c) \)) and still get the same result for different math operations.

Exam Tip: Associativity deals with grouping in operations. Addition and multiplication are generally associative for most number systems, while subtraction and division are not.

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Question 1. If a property holds for rational numbers, will it also hold for integers? For whole numbers? Which will? Which will not?

Answer:
1. Any property which is true for rational numbers is also generally true for integers, except when we consider any two integers 'a' and 'b', where their sum \( (a + b) \) is not necessarily an integer.
2. All properties which are true for rational numbers are also true for whole numbers, with these exceptions:
• If 'a' and 'b' are whole numbers, their difference \( (a - b) \) might not always be a whole number.
• If 'a' and 'b' are whole numbers (and 'b' is not equal to zero), their sum \( (a + b) \) might not be a whole number.In simple words: Most math rules that work for rational numbers also work for integers and whole numbers. However, there are a few exceptions, especially with subtraction and a specific case for addition with whole numbers as noted above.

Exam Tip: When checking if a property holds, always test with examples from each number set (positive, negative, zero, fractions) to find counter-examples.

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Question 1. Find using distributivity?
(i) \( \left\{\frac{7}{5} \times\left(\frac{-3}{12}\right)\right\}+\left\{\frac{7}{5} \times \frac{5}{12}\right\} \)
(ii) \( \left\{\frac{9}{16} \times \frac{4}{12}\right\}+\left\{\frac{9}{16} \times \frac{-3}{9}\right\} \)
Answer:
(i) \( \left\{\frac{7}{5} \times\left(\frac{-3}{12}\right)\right\}+\left\{\frac{7}{5} \times \frac{5}{12}\right\} \)
\( \implies \frac{7}{5} \times\left[\frac{-3}{12}+\frac{5}{12}\right] \)
\( \implies \frac{7}{5} \times \frac{-3+5}{12} \)
\( \implies \frac{7}{5} \times \frac{2}{12} \)
\( \implies \frac{7}{5} \times \frac{1}{6} \)
\( \implies \frac{7}{30} \)
(ii) \( \left\{\frac{9}{16} \times \frac{4}{12}\right\}+\left\{\frac{9}{16} \times \frac{-3}{9}\right\} \)
\( \implies \frac{9}{16} \times\left[\frac{4}{12}+\left(\frac{-3}{9}\right)\right] \)
\( \implies \frac{9}{16} \times\left[\frac{12+(-12)}{36}\right] \) (LCM of 12 and 9 is 36)
\( \implies \frac{9}{16} \times \frac{0}{36} \)
\( \implies \frac{9}{16} \times 0 \)
\( \implies 0 \)In simple words: The distributive property allows us to take a common number that is multiplying parts of an addition or subtraction problem and factor it out. This often simplifies the calculations significantly before you multiply.

Exam Tip: Look for a common factor outside the brackets when applying distributivity. It's a key property for simplifying complex rational number expressions.

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Question 1. Write the rational number for each point labelled with a letter?
(i)
(ii)
Answer:
(i) For point A, the rational number is \( \frac{1}{5} \).
For point B, the rational number is \( \frac{4}{5} \).
The rational number for point C is \( \frac{5}{5} \) or \( 1 \).
For point D, the rational number is \( \frac{8}{5} \).
For point E, the rational number is \( \frac{9}{5} \).
(ii) The rational numbers are:
Point F is \( \frac{-2}{6} \) or \( -\frac{1}{3} \).
Point G is \( \frac{-5}{6} \).
Point H is \( \frac{-7}{6} \).
Point I is \( \frac{-8}{6} \) or \( \frac{-4}{3} \).
Point J is \( \frac{-11}{6} \).In simple words: To find the rational number for each point, simply count the marks from zero and write the fraction, making sure to include negative signs where needed for points on the left side of the number line. Simplify the fractions if possible.

Exam Tip: Pay close attention to the denominator (the total parts between whole numbers) and whether the points are to the left (negative) or right (positive) of zero on the number line.