GSEB Class 8 Maths Solutions Chapter 1 સંમેય સંખ્યાઓ Exercise 1.2

Exercise 1.2

Question 1. Represent these numbers on the number line.
1. \( \frac { 7 }{ 4 } \)
2. \( \frac { -5 }{ 6 } \)
Answer:
(i) To represent \( \frac { 7 }{ 4 } \), we make 7 markings each at a distance equal to \( \frac { 1 }{ 4 } \) on the right of 0. The 7th point represents the rational number \( \frac { 7 }{ 4 } \).
(ii) To represent \( \frac { -5 }{ 6 } \) on the number line, we make 5 markings each at a distance equal to \( \frac { 1 }{ 6 } \) on the left of 0. We consider the 5th point.
In simple words: To show a fraction on a number line, divide the space between whole numbers into equal parts based on the denominator, then count the steps from zero.

Exam Tip: Always ensure your markings are equally spaced to maintain accuracy on the number line.

Question 2. Represent \( \frac { -2 }{ 11 } \), \( \frac { -5 }{ 11 } \), \( \frac { -9 }{ 11 } \) on the number line.
Answer: To represent \( \frac { -2 }{ 11 } \), \( \frac { -5 }{ 11 } \), and \( \frac { -9 }{ 11 } \) on a number line, we make 11 markings each being equal to a distance of \( \frac { 1 }{ 11 } \) on the left of 0. The points are marked accordingly on the left side of zero.
In simple words: Since these are negative fractions, count the required number of steps to the left of zero on a line divided into 11 parts.

Exam Tip: Label the points clearly as A, B, and C to correspond with the given fractions for better presentation.

Question 3. Write five rational numbers which are smaller than 2.
Answer: There can be unlimited rational numbers smaller than 2. Five of them are: 0, -1, \( \frac { 1 }{ 2 } \), \( \frac { 1 }{ 4 } \), 1.
In simple words: Any number to the left of 2 on the number line is smaller than 2.

Exam Tip: You can choose any simple integers or fractions as long as they are less than 2.

Question 4. Find ten rational numbers between \( \frac { -2 }{ 5 } \) and \( \frac { 1 }{ 2 } \).
Answer: To convert \( \frac { -2 }{ 5 } \) and \( \frac { 1 }{ 2 } \) to have the same denominators:
We have \( \frac { -2 }{ 5 } = \frac{-2 \times 4}{5 \times 4} = \frac { -8 }{ 20 } \) and \( \frac { 1 }{ 2 } = \frac{1 \times 10}{2 \times 10} = \frac { 10 }{ 20 } \).
The rational numbers between \( \frac { -8 }{ 20 } \) and \( \frac { 10 }{ 20 } \) are \( \frac { 9 }{ 20 } \), \( \frac { 8 }{ 20 } \), \( \frac { 7 }{ 20 } \), \( \frac { 6 }{ 20 } \), \( \frac { 5 }{ 20 } \), \( \frac { 4 }{ 20 } \), \( \frac { 3 }{ 20 } \), \( \frac { 2 }{ 20 } \), \( \frac { 1 }{ 20 } \), 0.
In simple words: Make the denominators the same first, then pick any ten numbers that fall between the two new numerators.

Exam Tip: Converting to a common denominator is the most reliable method to find multiple rational numbers between two fractions.

Question 5. Find five rational numbers between:
1. \( \frac { 2 }{ 3 } \) and \( \frac { 4 }{ 5 } \)
2. \( \frac { -3 }{ 2 } \) and \( \frac { 5 }{ 3 } \)
3. \( \frac { 1 }{ 4 } \) and \( \frac { 1 }{ 2 } \)
Answer:
(i) Converting \( \frac { 2 }{ 3 } \) and \( \frac { 4 }{ 5 } \) to have the same denominator: \( \frac { 2 }{ 3 } = \frac { 40 }{ 60 } \) and \( \frac { 4 }{ 5 } = \frac { 48 }{ 60 } \). Five rational numbers are \( \frac { 41 }{ 60 } \), \( \frac { 42 }{ 60 } \), \( \frac { 43 }{ 60 } \), \( \frac { 44 }{ 60 } \), \( \frac { 45 }{ 60 } \).
(ii) Converting \( \frac { -3 }{ 2 } \) and \( \frac { 5 }{ 3 } \) to have the same denominator: \( \frac { -3 }{ 2 } = \frac { -9 }{ 6 } \) and \( \frac { 5 }{ 3 } = \frac { 10 }{ 6 } \). Five rational numbers are \( \frac { -8 }{ 6 } \), \( \frac { -7 }{ 6 } \), \( \frac { -6 }{ 6 } \), \( \frac { -5 }{ 6 } \), \( \frac { -4 }{ 6 } \).
(iii) Converting \( \frac { 1 }{ 4 } \) and \( \frac { 1 }{ 2 } \) to have the same denominator: \( \frac { 1 }{ 4 } = \frac { 8 }{ 32 } \) and \( \frac { 1 }{ 2 } = \frac { 16 }{ 32 } \). Five rational numbers are \( \frac { 9 }{ 32 } \), \( \frac { 10 }{ 32 } \), \( \frac { 11 }{ 32 } \), \( \frac { 12 }{ 32 } \), \( \frac { 13 }{ 32 } \).
In simple words: Find a common denominator for both fractions, then list the numbers that appear between the two resulting fractions.

Exam Tip: If you cannot find enough numbers, multiply the numerator and denominator by a larger factor to increase the gap.

Question 6. Write five rational numbers greater than – 2.
Answer: Five rational numbers greater than – 2 are: \( \frac { -3 }{ 2 } \), -1, \( \frac { -1 }{ 2 } \), 0, \( \frac { 1 }{ 2 } \).
In simple words: Any number to the right of -2 on the number line is greater than -2.

Exam Tip: Remember that for negative numbers, a smaller absolute value means the number is actually larger.

Question 7. Find the rational numbers between \( \frac { 3 }{ 5 } \) and \( \frac { 3 }{ 4 } \).
Answer: Converting \( \frac { 3 }{ 5 } \) and \( \frac { 3 }{ 4 } \) to have common denominators: \( \frac { 3 }{ 5 } = \frac { 3 \times 20 }{ 5 \times 20 } = \frac { 60 }{ 100 } \) and \( \frac { 3 }{ 4 } = \frac { 3 \times 25 }{ 4 \times 25 } = \frac { 75 }{ 100 } \). The rational numbers between them are \( \frac { 61 }{ 100 } \), \( \frac { 62 }{ 100 } \), \( \frac { 63 }{ 100 } \), \( \frac { 64 }{ 100 } \), \( \frac { 65 }{ 100 } \).
In simple words: By making the denominators 100, we can easily see the numbers that lie between 60/100 and 75/100.

Exam Tip: Always show the conversion steps clearly to demonstrate how you arrived at the common denominator.