Samacheer Kalvi Class 8 Maths Solutions Chapter 1 Numbers Exercise 1.1

 

Question 1. Fill in the blanks:
(i) \( \frac{-19}{5} \) lies between the integers __________ and __________.
(ii) The decimal form of the rational number \( \frac{15}{-4} \) is __________.
(iii) The rational numbers \( \frac{-8}{3} \) and \( \frac{8}{3} \) are equidistant from __________.
(iv) The next rational number in the sequence \( \frac{-15}{24}, \frac{20}{-32}, \frac{-25}{40} \) is __________.
(v) The standard form of \( \frac{58}{-78} \) is __________.
Answer:
(i) \( \frac{-19}{5} \) lies between the integers -4 and -3.
(ii) The decimal form of the rational number \( \frac{15}{-4} \) is -3.75.
(iii) The rational numbers \( \frac{-8}{3} \) and \( \frac{8}{3} \) are equidistant from 0. They are like mirror images across zero.
(iv) The next rational number in the sequence \( \frac{-15}{24}, \frac{20}{-32}, \frac{-25}{40} \) is \( \frac{30}{-48} \). Each numerator increases by 5, and each denominator increases by 8, while alternating signs.
(v) The standard form of \( \frac{58}{-78} \) is \( \frac{-29}{39} \). To find the standard form, you divide both numerator and denominator by their greatest common divisor.
In simple words: Rational numbers have many properties. They can be positive or negative, shown as decimals, and simplified to their simplest form. Finding patterns helps to predict the next number in a sequence.

🎯 Exam Tip: Always simplify fractions to their lowest terms for the standard form and remember that negative fractions can be placed on the number line by converting them to mixed numbers or decimals first.

 

Question 2. Say True or False
(i) 0 is the smallest rational number.
(ii) \( \frac{-4}{5} \) lies to the left of \( \frac{-3}{4} \).
(iii) \( \frac{-19}{5} \) is greater than \( \frac{15}{-4} \).
(iv) The average of two rational numbers lies between them.
(v) There are an unlimited number of rational numbers between 10 and 11.
Answer:
(i) False. Zero is not the smallest rational number, as negative rational numbers are smaller than zero. For example, -1 is a rational number and is smaller than 0.
(ii) True. To compare, convert them to a common denominator: \( \frac{-4}{5} = \frac{-16}{20} \) and \( \frac{-3}{4} = \frac{-15}{20} \). Since -16 is less than -15, \( \frac{-16}{20} \) (or \( \frac{-4}{5} \)) lies to the left of \( \frac{-15}{20} \) (or \( \frac{-3}{4} \)) on the number line.
(iii) False. Convert to decimals: \( \frac{-19}{5} = -3.8 \) and \( \frac{15}{-4} = -3.75 \). Since -3.8 is smaller than -3.75, \( \frac{-19}{5} \) is not greater than \( \frac{15}{-4} \). On a number line, a number to the left is smaller.
(iv) True. The average of any two distinct rational numbers will always fall exactly in the middle of those two numbers. This is a fundamental property of real numbers.
(v) True. Between any two distinct rational numbers, there are infinitely many other rational numbers. You can always find a new rational number by taking their average.
In simple words: We check if statements about numbers are true or false. Negative numbers are smaller than zero. When comparing fractions, it helps to make their bottom numbers (denominators) the same. The average of two numbers always sits right between them.

🎯 Exam Tip: For True/False questions involving comparisons, converting fractions to decimals or finding a common denominator can make the comparison clearer and prevent mistakes.

 

Question 3. Find the rational numbers represented by each of the question marks marked on the following number lines.
(i)

Answer: The number lies between -3 and -4. The part between -3 and -4 is divided into 3 equal sections. The question mark is on the first mark from -4, which is equivalent to the second part when counting from -3. This represents \( -3 \frac{2}{3} \).
\( -3 \frac{2}{3} = -\frac{11}{3} \)
In simple words: The line between -3 and -4 is split into 3 small parts. The '?' is on the second mark when you count from -3 towards -4. So, it is -3 and two-thirds.

🎯 Exam Tip: When reading number lines for negative mixed numbers like \( -3 \frac{2}{3} \), remember that as you move left from -3 towards -4, the value becomes more negative. The fraction part \( \frac{2}{3} \) means you've moved two-thirds of the way from -3 towards -4.

 

Question 3. Find the rational numbers represented by each of the question marks marked on the following number lines.
(ii)

Answer: The number lies between 0 and -1. This section is divided into 5 equal parts. The question mark is on the third mark from 0, moving towards -1, which is the second part if we consider the full fraction. So it represents \( -\frac{2}{5} \). This is because when moving from 0 to -1, the first mark is -1/5, the second is -2/5, and so on.
In simple words: The line between 0 and -1 is split into 5 small parts. The '?' is on the second mark from the number 0, when going to the left. So, it is negative two-fifths.

🎯 Exam Tip: Remember that on a number line, values decrease as you move to the left. When counting divisions from the right (like from 0 towards -1), the fraction will be negative.

 

Question 3. Find the rational numbers represented by each of the question marks marked on the following number lines.
(iii)

Answer: The required number lies between 1 and 2. This unit section is divided into 4 equal parts. The question mark is on the third mark when counting from 1 towards 2. So it represents \( 1 \frac{3}{4} \). To write it as an improper fraction, we calculate \( 1 \times 4 + 3 = 7 \), so it is \( \frac{7}{4} \).
In simple words: The line between 1 and 2 is split into 4 small parts. The '?' is on the third mark when you count from 1 towards 2. So, it is 1 and three-fourths.

🎯 Exam Tip: For positive mixed numbers like \( 1 \frac{3}{4} \), the integer part tells you which main segment it's in (between 1 and 2), and the fraction tells you how far along that segment it is from the smaller integer.

 

Question 4. The points S, Y, N, C, R, A, T, I and O on the number line are such that CN=NY=YS and RA=AT=TI=IO. Find the rational numbers represented by the letters Y, N, A, T and I.

Answer:
For segment SYNC: C is at -1 and S is at -2. The segment from -1 to -2 is divided into 3 equal parts (CN, NY, YS). Each part is \( \frac{1}{3} \). So, Y is at \( -2 + \frac{1}{3} = \frac{-6+1}{3} = \frac{-5}{3} \). And N is at \( -2 + \frac{2}{3} = \frac{-6+2}{3} = \frac{-4}{3} \).
For segment RATIO: R is at 2 and O is at 3. The segment from 2 to 3 is divided into 4 equal parts (RA, AT, TI, IO). Each part is \( \frac{1}{4} \). So, A is at \( 2 + \frac{1}{4} = \frac{8+1}{4} = \frac{9}{4} \). T is at \( 2 + \frac{2}{4} = \frac{8+2}{4} = \frac{10}{4} \). And I is at \( 2 + \frac{3}{4} = \frac{8+3}{4} = \frac{11}{4} \).
In simple words: We looked at the number line to find the values of Y, N, A, T, and I. The distance between whole numbers was split into smaller equal parts. We then counted these parts from a known number to find the position of each letter.

🎯 Exam Tip: When points are equally spaced on a number line, identify the full unit length and how many divisions are made. Then, each division represents a fraction of that unit, which can be added or subtracted from the nearest whole number.

 

Question 5. Draw a number line and represent the following rational numbers on it.
(i) \( \frac{9}{4} \)
Answer: First, convert \( \frac{9}{4} \) to a mixed number: \( 9 \div 4 = 2 \) with a remainder of \( 1 \), so \( \frac{9}{4} = 2 \frac{1}{4} \). This means the number is between 2 and 3, and it's one-fourth of the way from 2 towards 3.
In simple words: \( \frac{9}{4} \) is the same as 2 and a quarter. So, on the number line, find 2, then go a quarter of the way towards 3 and mark it.

🎯 Exam Tip: Always convert improper fractions to mixed numbers first to easily locate them on the number line. The whole number tells you the main interval, and the fraction tells you the specific position within that interval.

 

Question 5. Draw a number line and represent the following rational numbers on it.
(ii) \( \frac{-8}{3} \)
Answer: First, convert \( \frac{-8}{3} \) to a mixed number: \( -8 \div 3 = -2 \) with a remainder of \( 2 \), so \( \frac{-8}{3} = -2 \frac{2}{3} \). This means the number is between -2 and -3, and it's two-thirds of the way from -2 towards -3.
In simple words: \( \frac{-8}{3} \) is the same as negative 2 and two-thirds. So, on the number line, find -2, then go two-thirds of the way towards -3 and mark it.

🎯 Exam Tip: When dealing with negative fractions, the 'direction' on the number line is to the left. \( -2 \frac{2}{3} \) means it's past -2, moving towards -3.

 

Question 5. Draw a number line and represent the following rational numbers on it.
(iii) \( \frac{-17}{-5} \)
Answer: First, simplify the fraction: \( \frac{-17}{-5} = \frac{17}{5} \). Then, convert it to a mixed number: \( 17 \div 5 = 3 \) with a remainder of \( 2 \), so \( \frac{17}{5} = 3 \frac{2}{5} \). This means the number is between 3 and 4, and it's two-fifths of the way from 3 towards 4.
In simple words: \( \frac{-17}{-5} \) becomes positive \( \frac{17}{5} \), which is 3 and two-fifths. On the number line, find 3, then move two-fifths of the way towards 4 and mark it.

🎯 Exam Tip: Remember that a negative number divided by a negative number results in a positive number. Simplifying the fraction first makes it easier to locate on the number line.

 

Question 5. Draw a number line and represent the following rational numbers on it.
(iv) \( \frac{15}{-4} \)
Answer: First, write the negative sign correctly: \( \frac{15}{-4} = -\frac{15}{4} \). Then, convert it to a mixed number: \( -15 \div 4 = -3 \) with a remainder of \( 3 \), so \( -\frac{15}{4} = -3 \frac{3}{4} \). This means the number is between -3 and -4, and it's three-fourths of the way from -3 towards -4.
In simple words: \( \frac{15}{-4} \) is the same as negative 3 and three-fourths. On the number line, find -3, then move three-fourths of the way towards -4 and mark it.

🎯 Exam Tip: A negative sign can be in the numerator, denominator, or in front of the fraction. It always means the entire fraction is negative, so always represent it on the negative side of the number line.

 

Question 6. Write the decimal form of the following rational numbers.
(i) \( \frac{1}{11} \)
(ii) \( \frac{13}{4} \)
(iii) \( \frac{-18}{7} \)
(iv) \( 1 \frac{2}{5} \)
(v) \( -3 \frac{1}{2} \)
Answer:
(i) To find the decimal form of \( \frac{1}{11} \), divide 1 by 11. \[ \begin{array}{r} 0.0909\dots \\ 11 \overline{\text{) } 1.0000} \\ -0\downarrow \\ \hline 10\text{ } \\ -0\downarrow \\ \hline 100 \\ -99\downarrow \\ \hline 10\text{ } \\ -0\downarrow \\ \hline 100 \\ -99 \\ \hline 1 \end{array} \] The division shows that 09 repeats. So, \( \frac{1}{11} = 0.\overline{09} \). The bar indicates that both digits, 0 and 9, repeat indefinitely.
(ii) To find the decimal form of \( \frac{13}{4} \), divide 13 by 4. \[ \begin{array}{r} 3.25 \\ 4 \overline{\text{) } 13.00} \\ -12\downarrow \\ \hline 10 \\ -8\downarrow \\ \hline 20 \\ -20 \\ \hline 0 \end{array} \] The division stops with a remainder of 0. So, \( \frac{13}{4} = 3.25 \). This is a terminating decimal.
(iii) To find the decimal form of \( \frac{-18}{7} \), we first divide 18 by 7 and then add the negative sign. \[ \begin{array}{r} 2.571428\dots \\ 7 \overline{\text{) } 18.000000} \\ -14\downarrow \\ \hline 40 \\ -35\downarrow \\ \hline 50 \\ -49\downarrow \\ \hline 10 \\ -7\downarrow \\ \hline 30 \\ -28\downarrow \\ \hline 20 \\ -14\downarrow \\ \hline 60 \\ -56 \\ \hline 4 \end{array} \] The digits 571428 repeat. So, \( \frac{-18}{7} = -2.\overline{571428} \).
(iv) To find the decimal form of \( 1 \frac{2}{5} \), first convert the mixed number to an improper fraction: \( 1 \frac{2}{5} = \frac{(1 \times 5) + 2}{5} = \frac{7}{5} \). Now, divide 7 by 5. \[ \begin{array}{r} 1.4 \\ 5 \overline{\text{) } 7.0} \\ -5\downarrow \\ \hline 20 \\ -20 \\ \hline 0 \end{array} \] The division stops. So, \( 1 \frac{2}{5} = 1.4 \).
(v) To find the decimal form of \( -3 \frac{1}{2} \), first convert the mixed number to an improper fraction: \( -3 \frac{1}{2} = -\frac{(3 \times 2) + 1}{2} = -\frac{7}{2} \). Now, divide 7 by 2 and add the negative sign. \[ \begin{array}{r} 3.5 \\ 2 \overline{\text{) } 7.0} \\ -6\downarrow \\ \hline 10 \\ -10 \\ \hline 0 \end{array} \] The division stops. So, \( -3 \frac{1}{2} = -3.5 \).
In simple words: To change a fraction to a decimal, just divide the top number by the bottom number. If the numbers keep repeating, put a bar over them. If it's a mixed number, change it into a single fraction first.

🎯 Exam Tip: Pay close attention to negative signs throughout the calculation and ensure they are applied correctly to the final decimal form. For repeating decimals, identify the exact block of digits that repeat and place the bar only over those digits.

 

Question 7. Write 5 rational numbers between the given rational numbers.
(i) 2 and 0
(ii) \( \frac{-1}{2} \) and \( \frac{3}{5} \)
(iii) \( \frac{1}{4} \) and \( \frac{7}{20} \)
(iv) \( \frac{-6}{4} \) and \( \frac{-23}{10} \)
Answer:
(i) Between 2 and 0.
We can write 2 as \( \frac{20}{10} \) and 0 as \( \frac{0}{10} \). Now, we can list rational numbers between \( \frac{20}{10} \) and \( \frac{0}{10} \). Since we are looking for numbers *between* them, we consider numbers from \( \frac{19}{10} \) down to \( \frac{1}{10} \). Five rational numbers between 2 and 0 are \( \frac{19}{10}, \frac{18}{10}, \frac{17}{10}, \frac{16}{10}, \frac{15}{10} \). (The source example used -2 and 0, so I will stick with positive 2 and 0 as per the question, but the general approach is similar).
(ii) Between \( \frac{-1}{2} \) and \( \frac{3}{5} \).
First, find the Least Common Multiple (LCM) of the denominators 2 and 5, which is 10. Convert the fractions to equivalent fractions with denominator 10: \( \frac{-1}{2} = \frac{-1 \times 5}{2 \times 5} = \frac{-5}{10} \) \( \frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10} \) Now we need to find 5 rational numbers between \( \frac{-5}{10} \) and \( \frac{6}{10} \). Five rational numbers are \( \frac{-4}{10}, \frac{-3}{10}, \frac{0}{10}, \frac{1}{10}, \frac{2}{10} \). We can pick any 5 numbers in this range.
The numbers are between -0.5 and 0.6.
(iii) Between \( \frac{1}{4} \) and \( \frac{7}{20} \).
First, find the LCM of the denominators 4 and 20, which is 20. Convert \( \frac{1}{4} \) to an equivalent fraction with denominator 20: \( \frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20} \) Now we need to find rational numbers between \( \frac{5}{20} \) and \( \frac{7}{20} \). Since there's only one integer between 5 and 7 (which is 6), we can multiply both fractions by a common factor to create more space. Let's multiply by 3 (so LCM becomes 60): \( \frac{5}{20} = \frac{5 \times 3}{20 \times 3} = \frac{15}{60} \) \( \frac{7}{20} = \frac{7 \times 3}{20 \times 3} = \frac{21}{60} \) Now we need to find 5 rational numbers between \( \frac{15}{60} \) and \( \frac{21}{60} \). Five rational numbers are \( \frac{16}{60}, \frac{17}{60}, \frac{18}{60}, \frac{19}{60}, \frac{20}{60} \).
(iv) Between \( \frac{-6}{4} \) and \( \frac{-23}{10} \).
First, find the LCM of the denominators 4 and 10, which is 20. Convert the fractions to equivalent fractions with denominator 20: \( \frac{-6}{4} = \frac{-6 \times 5}{4 \times 5} = \frac{-30}{20} \) \( \frac{-23}{10} = \frac{-23 \times 2}{10 \times 2} = \frac{-46}{20} \) Now we need to find 5 rational numbers between \( \frac{-30}{20} \) and \( \frac{-46}{20} \). When dealing with negative numbers, remember that -40 is smaller than -35. Five rational numbers are \( \frac{-31}{20}, \frac{-32}{20}, \frac{-33}{20}, \frac{-34}{20}, \frac{-35}{20} \).
In simple words: To find numbers between two fractions, make sure they have the same bottom number (common denominator). If there aren't enough numbers between them, make the bottom number even bigger by multiplying. Then, just pick any numbers that fit in between.

🎯 Exam Tip: The key to finding rational numbers between two given fractions is to convert them to equivalent fractions with a common denominator. If the gap between the numerators is not large enough, multiply both fractions by a suitable integer (e.g., 10/10) to create more space.

 

Question 8. Use the method of averages to write 2 rational numbers between \( \frac{14}{5} \) and \( \frac{16}{3} \).
Answer: Let the two given rational numbers be \( a = \frac{14}{5} \) and \( b = \frac{16}{3} \). The method of averages states that the average of two numbers \( \frac{1}{2}(a+b) \) is a rational number between them.
First rational number (\( C_1 \)): \( C_1 = \frac{1}{2} \left( \frac{14}{5} + \frac{16}{3} \right) \) To add the fractions, find a common denominator, which is 15. \( C_1 = \frac{1}{2} \left( \frac{14 \times 3}{5 \times 3} + \frac{16 \times 5}{3 \times 5} \right) \) \( C_1 = \frac{1}{2} \left( \frac{42}{15} + \frac{80}{15} \right) \) \( C_1 = \frac{1}{2} \left( \frac{42 + 80}{15} \right) \) \( C_1 = \frac{1}{2} \times \frac{122}{15} \) \( C_1 = \frac{122}{30} = \frac{61}{15} \) So, \( \frac{14}{5} < \frac{61}{15} < \frac{16}{3} \).
Second rational number (\( C_2 \)): We can find another rational number between \( \frac{14}{5} \) and \( \frac{61}{15} \). \( C_2 = \frac{1}{2} \left( \frac{14}{5} + \frac{61}{15} \right) \) To add the fractions, find a common denominator, which is 15. \( C_2 = \frac{1}{2} \left( \frac{14 \times 3}{5 \times 3} + \frac{61}{15} \right) \) \( C_2 = \frac{1}{2} \left( \frac{42}{15} + \frac{61}{15} \right) \) \( C_2 = \frac{1}{2} \left( \frac{42 + 61}{15} \right) \) \( C_2 = \frac{1}{2} \times \frac{103}{15} \) \( C_2 = \frac{103}{30} \) So, \( \frac{14}{5} < \frac{103}{30} < \frac{61}{15} \). The two rational numbers between \( \frac{14}{5} \) and \( \frac{16}{3} \) are \( \frac{61}{15} \) and \( \frac{103}{30} \). This method can be used to find as many rational numbers as needed.
In simple words: To find a number between two other numbers, you can find their average. This means you add them up and divide by two. We did this once to get the first number, then did it again with one of the original numbers and our new average to get a second number.

🎯 Exam Tip: The average method is a reliable way to find rational numbers between any two given rational numbers. You can repeat the process to find as many numbers as you need, as there are infinite rational numbers between any two distinct ones.

 

Question 9. Compare the following pairs of rational numbers.
(i) \( \frac{-11}{5}, \frac{-21}{8} \)
(ii) \( \frac{3}{-4}, \frac{-1}{2} \)
(iii) \( \frac{2}{3}, \frac{4}{5} \)
Answer:
(i) Compare \( \frac{-11}{5} \) and \( \frac{-21}{8} \).
First, find the LCM of the denominators 5 and 8, which is 40. Convert both fractions to equivalent fractions with denominator 40: \( \frac{-11}{5} = \frac{-11 \times 8}{5 \times 8} = \frac{-88}{40} \) \( \frac{-21}{8} = \frac{-21 \times 5}{8 \times 5} = \frac{-105}{40} \) Now, compare the numerators: -88 and -105. Since -88 is greater than -105, we have \( \frac{-88}{40} > \frac{-105}{40} \). Therefore, \( \frac{-11}{5} > \frac{-21}{8} \). When comparing negative numbers, the one closer to zero is larger.
(ii) Compare \( \frac{3}{-4} \) and \( \frac{-1}{2} \).
First, write \( \frac{3}{-4} \) as \( \frac{-3}{4} \). Now, find the LCM of the denominators 4 and 2, which is 4. Convert \( \frac{-1}{2} \) to an equivalent fraction with denominator 4: \( \frac{-1}{2} = \frac{-1 \times 2}{2 \times 2} = \frac{-2}{4} \) Now, compare the numerators: -3 and -2. Since -3 is smaller than -2, we have \( \frac{-3}{4} < \frac{-2}{4} \). Therefore, \( \frac{3}{-4} < \frac{-1}{2} \).
(iii) Compare \( \frac{2}{3} \) and \( \frac{4}{5} \).
First, find the LCM of the denominators 3 and 5, which is 15. Convert both fractions to equivalent fractions with denominator 15: \( \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \) \( \frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15} \) Now, compare the numerators: 10 and 12. Since 10 is smaller than 12, we have \( \frac{10}{15} < \frac{12}{15} \). Therefore, \( \frac{2}{3} < \frac{4}{5} \).
In simple words: To see which fraction is bigger or smaller, we make their bottom numbers the same. Once the bottom numbers are equal, we just look at the top numbers to compare them. For negative numbers, the one closer to zero is bigger.

🎯 Exam Tip: When comparing rational numbers, always find a common denominator first. For negative numbers, remember that the number with the smaller absolute value is actually the larger number (e.g., -2 is greater than -3).

 

Question 10. Arrange the following rational numbers in ascending and descending order.
(i) \( \frac{-5}{12}, \frac{-11}{8}, \frac{-15}{24}, \frac{-7}{-9}, \frac{12}{36} \)
Answer:
First, simplify the given rational numbers: \( \frac{-5}{12} \) (already in simplest form) \( \frac{-11}{8} \) (already in simplest form) \( \frac{-15}{24} = \frac{-15 \div 3}{24 \div 3} = \frac{-5}{8} \) \( \frac{-7}{-9} = \frac{7}{9} \) (negative divided by negative is positive) \( \frac{12}{36} = \frac{12 \div 12}{36 \div 12} = \frac{1}{3} \)
The numbers are: \( \frac{-5}{12}, \frac{-11}{8}, \frac{-5}{8}, \frac{7}{9}, \frac{1}{3} \). Now, find the LCM of the denominators 12, 8, 9, 3. The LCM is 72. Convert each fraction to an equivalent fraction with denominator 72: \( \frac{-5}{12} = \frac{-5 \times 6}{12 \times 6} = \frac{-30}{72} \) \( \frac{-11}{8} = \frac{-11 \times 9}{8 \times 9} = \frac{-99}{72} \) \( \frac{-5}{8} = \frac{-5 \times 9}{8 \times 9} = \frac{-45}{72} \) \( \frac{7}{9} = \frac{7 \times 8}{9 \times 8} = \frac{56}{72} \) \( \frac{1}{3} = \frac{1 \times 24}{3 \times 24} = \frac{24}{72} \)
The equivalent fractions are: \( \frac{-30}{72}, \frac{-99}{72}, \frac{-45}{72}, \frac{56}{72}, \frac{24}{72} \).
Now, compare the numerators: -30, -99, -45, 56, 24.
**Ascending order (smallest to largest):** The smallest negative number is the one furthest from zero. So, -99 is the smallest, followed by -45, then -30. After that come the positive numbers, 24 then 56. \( -99 < -45 < -30 < 24 < 56 \) So, the ascending order of the original numbers is: \( \frac{-11}{8} < \frac{-15}{24} < \frac{-5}{12} < \frac{12}{36} < \frac{-7}{-9} \)
**Descending order (largest to smallest):** \( 56 > 24 > -30 > -45 > -99 \) So, the descending order of the original numbers is: \( \frac{-7}{-9} > \frac{12}{36} > \frac{-5}{12} > \frac{-15}{24} > \frac{-11}{8} \)
In simple words: First, make sure all fractions are as simple as possible. Then, change all fractions so they have the same bottom number. After that, line them up from smallest to biggest (ascending) or biggest to smallest (descending) by just looking at their top numbers. Remember that bigger negative numbers are actually smaller.

🎯 Exam Tip: Always simplify fractions and handle negative signs (like \( \frac{-a}{-b} = \frac{a}{b} \)) before finding the LCM. This reduces calculation errors and makes the comparison straightforward.

 

Question 10. Arrange the following rational numbers in ascending and descending order.
(ii) \( \frac{-17}{10}, \frac{-7}{5}, 0, \frac{-2}{4}, \frac{-19}{20} \)
Answer:
First, simplify the given rational numbers: \( \frac{-17}{10} \) (already in simplest form) \( \frac{-7}{5} \) (already in simplest form) \( 0 \) \( \frac{-2}{4} = \frac{-2 \div 2}{4 \div 2} = \frac{-1}{2} \) \( \frac{-19}{20} \) (already in simplest form)
The numbers are: \( \frac{-17}{10}, \frac{-7}{5}, 0, \frac{-1}{2}, \frac{-19}{20} \). Now, find the LCM of the denominators 10, 5, 2, 20. The LCM is 20. Convert each fraction to an equivalent fraction with denominator 20: \( \frac{-17}{10} = \frac{-17 \times 2}{10 \times 2} = \frac{-34}{20} \) \( \frac{-7}{5} = \frac{-7 \times 4}{5 \times 4} = \frac{-28}{20} \) \( 0 = \frac{0}{20} \) \( \frac{-1}{2} = \frac{-1 \times 10}{2 \times 10} = \frac{-10}{20} \) \( \frac{-19}{20} \) (already with denominator 20)
The equivalent fractions are: \( \frac{-34}{20}, \frac{-28}{20}, \frac{0}{20}, \frac{-10}{20}, \frac{-19}{20} \).
Now, compare the numerators: -34, -28, 0, -10, -19.
**Ascending order (smallest to largest):** The smallest negative number is -34. Then -28, -19, -10. Zero is larger than all negative numbers. \( -34 < -28 < -19 < -10 < 0 \) So, the ascending order of the original numbers is: \( \frac{-17}{10} < \frac{-7}{5} < \frac{-19}{20} < \frac{-2}{4} < 0 \)
**Descending order (largest to smallest):** \( 0 > -10 > -19 > -28 > -34 \) So, the descending order of the original numbers is: \( 0 > \frac{-2}{4} > \frac{-19}{20} > \frac{-7}{5} > \frac{-17}{10} \)
In simple words: We rewrite all fractions to have the same bottom number. Then, we look at the top numbers to put them in order. Remember that positive numbers are always bigger than zero and negative numbers. Also, a negative number like -10 is bigger than -30.

🎯 Exam Tip: Always include 0 in your ordering if it's part of the set. It acts as a separator between positive and negative numbers, making the ordering process clearer.

 

Question 11. The number which is subtracted from \( \frac{-6}{11} \) to get \( \frac{8}{9} \) is
(A) \( \frac{34}{99} \)
(B) \( \frac{-142}{99} \)
(C) \( \frac{142}{99} \)
(D) \( \frac{-34}{99} \)
Answer: (B) \( \frac{-142}{99} \)
In simple words: We need to find a number that, when taken away from \( \frac{-6}{11} \), leaves \( \frac{8}{9} \). We can find this number by subtracting \( \frac{8}{9} \) from \( \frac{-6}{11} \).

🎯 Exam Tip: Set up an algebraic equation to represent the problem. If 'x' is the unknown number, then "subtracted from A to get B" means \( A - x = B \), which rearranges to \( x = A - B \).

 

Question 12. Which of the following pairs is equivalent?
(A) \( \frac{-20}{12}, \frac{5}{3} \)
(B) \( \frac{16}{-30}, \frac{-8}{15} \)
(C) \( \frac{-18}{36}, \frac{-20}{44} \)
(D) \( \frac{7}{-5}, \frac{-5}{7} \)
Answer: (B) \( \frac{16}{-30}, \frac{-8}{15} \)
In simple words: We are looking for two fractions that are equal when simplified. We check each pair by simplifying them to their smallest form.

🎯 Exam Tip: To check if two fractions are equivalent, simplify both to their lowest terms. If their lowest terms are identical, the fractions are equivalent. Pay close attention to negative signs; \( \frac{a}{-b} \) is equivalent to \( \frac{-a}{b} \).

 

Question 13. \( \frac{-5}{4} \) is a rational number which lies between __________ .
(A) 0 and \( \frac{-5}{4} \)
(B) -1 and 0
(C) -1 and -2
(D) -4 and -5
Answer: (C) -1 and -2
In simple words: To find where \( \frac{-5}{4} \) is on the number line, we change it to a mixed number. \( \frac{-5}{4} \) is the same as -1 and one-quarter, so it's between -1 and -2.

🎯 Exam Tip: Convert improper fractions to mixed numbers or decimals to easily identify the two consecutive integers it lies between. For negative numbers, \( -1 \frac{1}{4} \) means it's further left than -1, but not as far as -2.

 

Question 14. Which of the following rational numbers is the greatest?
(A) \( \frac{-17}{24} \)
(B) \( \frac{-13}{16} \)
(C) \( \frac{7}{-8} \)
(D) \( \frac{-31}{32} \)
Answer: (A) \( \frac{-17}{24} \)
In simple words: We want to find the largest negative fraction. To do this, we make all the bottom numbers the same. Then, the negative fraction with the smallest top number (closest to zero) is the largest.

🎯 Exam Tip: To compare several rational numbers, find the Least Common Multiple (LCM) of all denominators. Convert each fraction to an equivalent fraction with this common denominator. For negative numbers, the one with the smallest absolute value (closest to zero) is the greatest.

 

Question 15. The sum of the digits of the denominator in the simplest form of is \( \frac{112}{528} \) is
(a) 4
(b) 5
(c) 6
(d) 7
Answer: (c) 6
To find the sum of the digits of the denominator in its simplest form, we first simplify the given fraction \( \frac{112}{528} \). We divide both the numerator and the denominator by their common factors.
\( \frac{112}{528} = \frac{112 \div 8}{528 \div 8} = \frac{14}{66} \)
Next, we simplify further by dividing both numbers by 2:
\( \frac{14}{66} = \frac{14 \div 2}{66 \div 2} = \frac{7}{33} \)
The simplest form of the fraction is \( \frac{7}{33} \). The denominator of this simplified fraction is 33. To find the sum of its digits, we add 3 and 3, which gives us 6. Simplifying fractions helps us understand their value more clearly.
In simple words: First, make the fraction as simple as possible by dividing the top and bottom numbers by any numbers that go into both. Then, take the new bottom number and add its individual digits together.

🎯 Exam Tip: Always ensure the fraction is fully simplified before finding the sum of the digits of its denominator. Missing a common factor will lead to an incorrect answer.