Samacheer Kalvi Class 9 Maths Solutions Chapter 2 Real Numbers Exercise 2.1

Question 1. Which arrow best shows the position of \( \frac{11}{3} \) on the number line?
Answer: (C) Arrow C best shows the position of \( \frac{11}{3} \) on the number line. This is because \( \frac{11}{3} \) can be written as \( 3 \frac{2}{3} \), which means it is between the numbers 3 and 4, closer to 4.
In simple words: The fraction \( \frac{11}{3} \) is the same as \( 3 \) whole ones and \( \frac{2}{3} \) more. This number is found on the number line between 3 and 4, specifically at the point marked by arrow C.

🎯 Exam Tip: To locate an improper fraction on a number line, convert it to a mixed number first. This clearly shows the whole number part and the fractional part, making it easier to place.

Question 2. Find any three rational numbers between \( \frac{-7}{11} \) and \( \frac{2}{11} \)
Answer: We need to find three rational numbers that lie between \( \frac{-7}{11} \) and \( \frac{2}{11} \). Since both fractions already have the same denominator, we can simply look for integers between the numerators -7 and 2. The integers between -7 and 2 are -6, -5, -4, -3, -2, -1, 0, and 1. We can choose any three of these. For example, three rational numbers between \( \frac{-7}{11} \) and \( \frac{2}{11} \) are \( \frac{-6}{11}, \frac{-5}{11}, \frac{-4}{11} \).
In simple words: When the bottom numbers (denominators) are the same, just pick any three numbers between the top numbers (-7 and 2) and put them over the same bottom number (11).

🎯 Exam Tip: When finding rational numbers between two fractions with the same denominator, simply choose numerators that fall between the given numerators. If denominators are different, first find a common denominator.

Question 3. Find any five rational numbers between
(i) \( \frac{1}{4} \) and \( \frac{1}{5} \)
(ii) 0.1 and 0.11
(iii) -1 and -2
Answer:
(i) To find five rational numbers between \( \frac{1}{4} \) and \( \frac{1}{5} \), we first make their denominators the same. \( \frac{1}{4} = \frac{1 \times 30}{4 \times 30} = \frac{30}{120} \)
\( \frac{1}{5} = \frac{1 \times 24}{5 \times 24} = \frac{24}{120} \)
Now we need to find five rational numbers between \( \frac{24}{120} \) and \( \frac{30}{120} \). We can pick numerators between 24 and 30. Thus, five rational numbers are \( \frac{25}{120}, \frac{26}{120}, \frac{27}{120}, \frac{28}{120}, \frac{29}{120} \).
(ii) To find five rational numbers between 0.1 and 0.11, we can write them as fractions or extend their decimal places. \( 0.1 = \frac{1}{10} = \frac{1 \times 100}{10 \times 100} = \frac{100}{1000} \)
\( 0.11 = \frac{11}{100} = \frac{11 \times 10}{100 \times 10} = \frac{110}{1000} \)
Now, we need to find five rational numbers between \( \frac{100}{1000} \) and \( \frac{110}{1000} \). The five rational numbers are \( \frac{101}{1000}, \frac{102}{1000}, \frac{103}{1000}, \frac{104}{1000}, \frac{105}{1000} \).
In decimal form, these are 0.101, 0.102, 0.103, 0.104, and 0.105. These numbers are very close to the original values.
(iii) To find five rational numbers between -1 and -2, we can convert them to fractions with a common denominator. Let's use 10 as the denominator, so we can easily find numbers between them. \( -1 = \frac{-10}{10} \)
\( -2 = \frac{-20}{10} \)
Now, we need to find five rational numbers between \( \frac{-20}{10} \) and \( \frac{-10}{10} \). Remember that for negative numbers, a smaller absolute value means a larger number. Thus, five rational numbers between -2 and -1 are \( \frac{-11}{10}, \frac{-12}{10}, \frac{-13}{10}, \frac{-14}{10}, \frac{-15}{10} \).
In simple words: To find numbers between fractions or decimals, make their bottom numbers (denominators) the same or add more zeros after the decimal point. Then, pick numbers in between. For negative numbers, remember that -1 is larger than -2.

🎯 Exam Tip: When asked to find multiple rational numbers between two given numbers, multiply both the numerator and denominator by a number larger than the count needed (e.g., for 5 numbers, multiply by 6 or 10) to create enough 'space' between the fractions.