GSEB Class 8 Maths Solutions Chapter 1 સંમેય સંખ્યાઓ InText Questions

Try It (Textbook Page No. 4)

Fill in the blanks in the table below:

 

Question 1. Complete the following table about closure property.

Answer: We have already *studied* the *closure properties* for addition, subtraction, multiplication, and division of rational numbers, integers, whole numbers, and natural numbers. Based on that, the answer will be written as follows:
Note: Observe that rational numbers are not *closed* under division. For example, \( \frac {2}{3} \div 0 \) is not *defined*. Therefore, the answer in the table is “No”.
In simple words: We have studied closure properties for adding, subtracting, multiplying, and dividing rational numbers, integers, whole numbers, and natural numbers. The completed table shows if each set of numbers stays within its type after these operations.

Exam Tip: Remember that division by zero is undefined, which is a key reason why rational numbers are not closed under division.

 

Try It (Textbook Page No. 6)

Complete the table below:

 

Question 2. Complete the following table about the commutative property.

Answer: We have studied the *commutative properties* for addition, subtraction, multiplication, and division of rational numbers, integers, whole numbers, and natural numbers. Based on that, the above table will be completed as follows:

In simple words: We have learned about the commutative property for different number types and operations. The filled-in table shows which operations let you swap numbers around and still get the same answer.

Exam Tip: Remember that commutative property means \( a + b = b + a \) or \( a \times b = b \times a \). Subtraction and division are generally not commutative for any set of numbers.

 

Try It (Textbook Page No. 9)

Complete the table below:

 

Question 3. Complete the following table about the associative property.

Answer: We have studied the *associative properties* for addition, subtraction, multiplication, and division of rational numbers, integers, whole numbers, and natural numbers. Based on that, the above table will be completed as follows:

In simple words: We have studied the associative property for operations with different types of numbers. The completed table shows for which operations you can group numbers differently and still get the same result.

Exam Tip: Understand that associative property means \( (a + b) + c = a + (b + c) \) or \( (a \times b) \times c = a \times (b \times c) \). Similar to commutative property, subtraction and division are usually not associative.

 

Think, Discuss, and Write (Textbook Page No. 11)

Question 4. If a property holds true for rational numbers, what can be said about it for integers? When does it hold true and when does it not?
Answer:
(i) If a certain property holds true for rational numbers, then it *generally* holds true for integers as well. An *exception* is that for 'a' and 'b', the sum \( (a + b) \) is not *always* required to be an integer.
(ii) If a certain property holds true for rational numbers, then it *generally* holds true for whole numbers as well. An *exception* is:
(a) If 'a' and 'b' are whole numbers, then \( (a - b) \) is not *always* a whole number.
(b) If 'a' and 'b' are whole numbers, then \( (b \neq 0) \), \( a \div b \) is not *always* a whole number.
In simple words: If a math rule works for rational numbers, it usually works for integers too. But there are times it doesn't, like when adding two numbers doesn't always stay an integer. The same goes for whole numbers, especially with subtraction or division, where the answer might not be a whole number anymore.

Exam Tip: When comparing number properties, always consider specific examples for operations like subtraction and division, as these often show exceptions to the general rules.

 

Try It (Textbook Page No. 13)

Find using the distributive property:

 

Question 5. (i) \( \left\{\frac{7}{5} \times\left(\frac{-3}{12}\right)\right\}+\left\{\frac{7}{5} \times \frac{5}{12}\right\} \)
Answer:
\( = \frac{7}{5} \times \left(\frac{-3}{12} + \frac{5}{12}\right) \)
\( = \frac{7}{5} \times \left(\frac{-3+5}{12}\right) \)
\( = \frac{7}{5} \times \frac{2}{12} \)
\( = \frac{7}{5} \times \frac{1}{6} \)
\( = \frac{7}{30} \)
In simple words: We used the distributive property to simplify the expression. First, we took out the common fraction, then added the remaining fractions, and finally multiplied to get the answer.

Exam Tip: The distributive property \( a \times (b + c) = (a \times b) + (a \times c) \) can often simplify calculations by factoring out the common term first.

 

Question 6. (ii) \( \left\{\frac{9}{16} \times \frac{4}{12}\right\}+\left\{\frac{9}{16} \times \frac{-3}{9}\}\right\} \)
Answer:
\( = \frac{9}{16} \times \left(\frac{4}{12} + \frac{-3}{9}\right) \)
\( = \frac{9}{16} \times \left(\frac{1}{3} + \frac{-1}{3}\right) \)
\( = \frac{9}{16} \times \left(\frac{12+(-12)}{36}\right) \)
\( = \frac{9}{16} \times \frac{0}{36} \)
\( = \frac{9}{16} \times 0 \)
\( = 0 \)
In simple words: We applied the distributive property, first pulling out the common fraction. Then, we combined the terms inside the parentheses, which resulted in zero. Any number multiplied by zero is zero.

Exam Tip: Always look for opportunities to simplify fractions before combining them. Recognizing that \( \frac{4}{12} \) simplifies to \( \frac{1}{3} \) and \( \frac{-3}{9} \) simplifies to \( \frac{-1}{3} \) can make the sum much easier to calculate.

 

Try It (Textbook Page No. 17)

Represent the points marked with English letters on the number line below using rational numbers:

 

Question 7. (i)

Answer: The rational number corresponding to A is \( \frac {1}{5} \), to B is \( \frac {4}{5} \), to C is \( \frac {5}{5} \) which means 1, to D is \( \frac {8}{5} \), and to E is \( \frac {9}{5} \).
In simple words: On the number line, each letter stands for a specific fraction. A is at \( \frac{1}{5} \), B is at \( \frac{4}{5} \), C is at \( \frac{5}{5} \) (which is 1), D is at \( \frac{8}{5} \), and E is at \( \frac{9}{5} \).

Exam Tip: When reading number lines, first identify the unit fractions (e.g., 1/5) and then count the ticks to find the value of each marked point.

 

Question 8. (ii)

Answer: Here, the rational number corresponding to F is \( \frac {-2}{6} \) which means \( \frac {-1}{3} \). For G, it is \( \frac {-5}{6} \). For H, it is \( \frac {-7}{6} \). For I, it is \( \frac {-8}{6} \) which means \( \frac {-4}{3} \). And for J, it is \( \frac {-11}{6} \).
In simple words: On this number line, each letter shows a negative fraction. F is \( \frac{-2}{6} \) (or \( \frac{-1}{3} \)), G is \( \frac{-5}{6} \), H is \( \frac{-7}{6} \), I is \( \frac{-8}{6} \) (or \( \frac{-4}{3} \)), and J is \( \frac{-11}{6} \).

Exam Tip: Remember that for negative number lines, values decrease as you move to the left. Simplify fractions to their lowest terms when representing them.