GSEB Class 7 Maths Solutions Chapter 9 Rational Numbers Exercise 9.1

Question 1. List five rational numbers between:
(i) -1 and 0
(ii) -2 and -1
(iii) \( \frac {-4}{5} \) and \( \frac { -2 }{ 3 } \)
(iv) \( \frac {1}{2 } \) and \( \frac { 2 }{ 3 } \)
Answer:
(i) -1 and 0
Since -1 equals \( \frac { -1 }{ 1 } \), which is also \( \frac {(-1) \times 10 }{ 1 \times 10 } = \frac { -10 }{ 10 } \), and 0 equals \( \frac { 0 }{ 1 } \), which is \( \frac { 0 \times 10 }{ 1 \times 10 } = \frac { 0 }{ 10 } \).
Also, we observe that \( \frac {-10 }{ 10 } \) is less than \( \frac { -9 }{ 10 } \), which is less than \( \frac { -8 }{ 10 } \), which is less than \( \frac { -7 }{ 10 } \), which is less than \( \frac { -6 }{ 10} \), which is less than \( \frac { -5}{10} \), which is less than \( \frac { 0 }{ 10 } \). This means \( \frac { -9 }{ 10 }, \frac { -8 }{ 10 }, \frac { -7 }{ 10 }, \frac {-6}{10} \), and \( \frac { -5 }{ 10 } \) are five rational numbers that fall between \( \frac { -10 }{10} \) and \( \frac { 0 }{ 10 } \) (or between -1 and 0).
Therefore, the five rational numbers between -1 and 0 include \( \frac { -9 }{ 10 }, \frac { -8 }{ 10 }, \frac {-6}{ 10 } \), and \( \frac { -5 }{ 10 } \). Alternatively, these numbers can be expressed as \( \frac { -9 }{ 10 }, \frac { -4 }{ 5 }, \frac { -7 }{ 10 }, \frac { -3 }{ 5 } \), and \( \frac { -1 }{ 2 } \).
(ii) -2 and -1
Since -2 equals \( \frac { -2}{ 1 } \), which is \( \frac { (-2) \times 10 }{ 1 \times 10 } = \frac { -20 }{ 10} \), and -1 equals \( \frac {-1}{ 1 } \), which is \( \frac { (-1) \times 10 }{ 1 \times 10 } = \frac { -10 }{ 10 } \).
So, we observe that \( \frac { -20 }{ 10 } \) is less than \( \frac { -19 }{ 10 } \), then \( \frac { -18 }{10} \), \( \frac { -17 }{ 10 } \), \( \frac { -16 }{ 10 } \), \( \frac { -15 }{10} \), and finally \( \frac {-10 }{ 10 } \). Alternatively, this shows that -2 is less than \( \frac { -19 }{ 10 } \), then \( \frac { -9 }{5} \), \( \frac { -17 }{ 10 } \), \( \frac { -8}{5} \), and \( \frac { -3}{2} \), which is less than -1.
Therefore, the five rational numbers lying between -2 and -1 are \( \frac { -19 }{ 10 }, \frac { -9 }{ 10 }, \frac { -17 }{ 10 }, \frac { -8 }{5} \), and \( \frac { -3 }{ 2 } \).
(iii) \( \frac {-4}{5} \) and \( \frac { -2 }{ 3 } \)
We start by making denominators the same: \( \frac {-4}{5} \) becomes \( \frac {(-4) \times 12}{5 \times 12} = \frac {-48}{60} \), and \( \frac {-2}{3} \) becomes \( \frac {(-2) \times 20}{3 \times 20} = \frac {-40}{60} \).
Now, we can clearly see that \( \frac {-48}{60} \) is less than \( \frac {-47}{60} \), \( \frac {-46}{60} \), \( \frac {-45}{60} \), \( \frac {-44}{60} \), \( \frac {-43}{60} \), and \( \frac {-40}{60} \). This can also be written as \( \frac {-4}{5} < \frac {-47}{60} < \frac {-46}{60} < \frac {-45}{60} < \frac {-44}{60} < \frac {-43}{60} < \frac {-2}{3} \). An alternative list of numbers in between is \( \frac {-4}{5} < \frac {-47}{60} < \frac {-23}{30} < \frac {-3}{4} < \frac {-11}{15} < \frac {-43}{60} < \frac {-2}{3} \).
Therefore, the five rational numbers found between \( \frac {-4}{5} \) and \( \frac {-2}{3} \) are \( \frac { -47 }{ 60 }, \frac { -23 }{ 30 }, \frac { -3 }{ 4 }, \frac { -11 }{15} \), and \( \frac { -43 }{ 60 } \).
(iv) \( \frac {1}{2 } \) and \( \frac { 2 }{ 3 } \)
We begin by converting the fractions to have a common denominator: \( \frac {1}{2} \) becomes \( \frac {1 \times 18}{2 \times 18} = \frac {18}{36} \), and \( \frac {2}{3} \) becomes \( \frac {2 \times 12}{3 \times 12} = \frac {24}{36} \).
We can then list numbers in between: \( \frac {18}{36} < \frac {19}{36} < \frac {20}{36} < \frac {21}{36} < \frac {22}{36} < \frac {23}{36} < \frac {24}{36} \). This means \( \frac {1}{2} < \frac {19}{36} < \frac {5}{9} < \frac {7}{12} < \frac {11}{18} < \frac {23}{36} < \frac {2}{3} \).
So, five rational numbers located between \( \frac {1}{2} \) and \( \frac {2}{3} \) are \( \frac {10}{18}, \frac {5}{9}, \frac {7}{12}, \frac {11}{18} \), and \( \frac {23}{36} \).
In simple words: To find rational numbers between two fractions, first make sure they have the same bottom number (denominator). Then, you can easily pick numbers by increasing the top number (numerator) by one each time.

Exam Tip: When finding rational numbers between two given numbers, you can always multiply both the numerator and denominator by a larger number (e.g., 10) to create more "space" and find more numbers easily.

Question 2. Write four more rational numbers in each of the following patterns:
(i) \( \frac {-3}{ 5 }, \frac { -6 }{ 10 }, \frac { -9 }{ 15 }, \frac { -12 }{ 20 }, .... \)
(ii) \( \frac {-1}{ 4 }, \frac { -2 }{ 8 }, \frac { -3 }{ 12 }, ..... \)
(iii) \( \frac {-1}{ 6 }, \frac { 2 }{ -12 }, \frac { 3 }{ -18 }, \frac { 4 }{ -24 }, .... \)
(iv) \( \frac {-2}{ 3 }, \frac { 2 }{ -3 }, \frac { 4 }{ -6 }, \frac { 6 }{ -9 }, .... \)
Answer:
(i) \( \frac {-3}{ 5 }, \frac { -6 }{ 10 }, \frac { -9 }{ 15 }, \frac { -12 }{ 20 }, .... \)
The given pattern starts with \( \frac {-3}{5}, \frac {-6}{10}, \frac {-9}{15} \), and \( \frac {-12}{20} \).
Here, \( \frac {-3}{5} = \frac {(-3)\times 1}{5\times 1} \), \( \frac {-6}{10} = \frac {(-3)\times 2}{5\times 2} \), \( \frac {-9}{15} = \frac {(-3)\times 3}{5\times 3} \), \( \frac {-12}{20} = \frac {(-3)\times 4}{5\times 4} \).
We can easily observe a clear pattern in these fractions. Clearly, the next four rational numbers in this sequence will be:
\( \frac {(-3)\times 5}{5\times 5} = \frac {-15}{25} \)
\( \frac {(-3)\times 6}{5\times 6} = \frac {-18}{30} \)
\( \frac {(-3)\times 7}{5\times 7} = \frac {-21}{35} \)
\( \frac {(-3)\times 8}{5\times 8} = \frac {-24}{40} \)
So, the next four required rational numbers are \( \frac { -15 }{ 25 }, \frac { -18 }{ 30 }, \frac { -21 }{ 35 } \), and \( \frac { -24 }{ 40 } \).
(ii) \( \frac {-1}{ 4 }, \frac { -2 }{ 8 }, \frac { -3 }{ 12 }, ..... \)
The given sequence is \( \frac {-1}{4}, \frac {-2}{8} \), and \( \frac {-3}{12} \).
Here, \( \frac {-1}{4} = \frac {(-1)\times 1}{4\times 1} \), \( \frac {-2}{8} = \frac {(-1)\times 2}{4\times 2} \), \( \frac {-3}{12} = \frac {(-1)\times 3}{4\times 3} \).
We can see a pattern emerging here. Thus, the next four rational numbers will be:
\( \frac {(-1)\times 4}{4\times 4} = \frac {-4}{16} \)
\( \frac {(-1)\times 5}{4\times 5} = \frac {-5}{20} \)
\( \frac {(-1)\times 6}{4\times 6} = \frac {-6}{24} \)
\( \frac {(-1)\times 7}{4\times 7} = \frac {-7}{28} \)
Therefore, the next four needed rational numbers are \( \frac {-4}{16}, \frac {-5}{20}, \frac {-6}{24} \), and \( \frac {-7}{28} \).
(iii) \( \frac {-1}{ 6 }, \frac { 2 }{ -12 }, \frac { 3 }{ -18 }, \frac { 4 }{ -24 }, .... \)
In this sequence, we can see that \( \frac {-1}{6} = \frac {1\times 1}{(-6)\times 1} \), \( \frac {2}{-12} = \frac {1\times 2}{(-6)\times 2} \), \( \frac {3}{-18} = \frac {1\times 3}{(-6)\times 3} \), and \( \frac {4}{-24} = \frac {1\times 4}{(-6)\times 4} \).
We clearly notice a pattern in these numbers. So, the next four rational numbers in this series will be:
\( \frac {1\times 5}{(-6)\times 5} = \frac {-5}{30} \)
\( \frac {1\times 6}{(-6)\times 6} = \frac {-6}{36} \)
\( \frac {1\times 7}{(-6)\times 7} = \frac {-7}{42} \)
\( \frac {1\times 8}{(-6)\times 8} = \frac {-8}{48} \)
Therefore, the next four rational numbers we need are \( \frac { -5 }{ 30 }, \frac { -6 }{ 36 }, \frac { -7 }{ 42 } \), and \( \frac { -8 }{ 48 } \).
(iv) \( \frac {-2}{ 3 }, \frac { 2 }{ -3 }, \frac { 4 }{ -6 }, \frac { 6 }{ -9 }, .... \)
We are given the numbers \( \frac {-2}{3}, \frac {2}{-3}, \frac {4}{-6} \), and \( \frac {6}{-9} \).
Here, \( \frac {-2}{3} \)
\( \frac {2}{-3} = \frac {(-2)\times 1}{3\times 1} = \frac {-2}{3} \)
\( \frac {4}{-6} = \frac {(-2)\times 2}{3\times 2} = \frac {-4}{6} \)
\( \frac {6}{-9} = \frac {(-2)\times 3}{3\times 3} = \frac {-6}{9} \)
Therefore, by observing the numerical pattern, we can determine the next four numbers as:
\( \frac {-2}{3} \times \frac {-4}{-4} = \frac {8}{-12} \)
\( \frac {-2}{3} \times \frac {-5}{-5} = \frac {10}{-15} \)
\( \frac {-2}{3} \times \frac {-6}{-6} = \frac {12}{-18} \)
\( \frac {-2}{3} \times \frac {-7}{-7} = \frac {14}{-21} \)
So, the next four rational numbers required are \( \frac {8}{ -12 }, \frac { 10 }{ -15 }, \frac { 12 }{ -18 } \), and \( \frac { 14 }{ -21 } \).
In simple words: Look closely at the pattern in the given fractions. See how the top and bottom numbers are changing. Then, follow that same rule to find the next fractions in the series.

Exam Tip: Always identify the common multiplier or divisor used to generate subsequent terms in the pattern to correctly extend the sequence.

Question 3. Give four rational numbers equivalent to:
(i) \( \frac { -2 }{ 7 } \)
(ii) \( \frac { 5 }{ -3 } \)
(iii) \( \frac { 4 }{ 9 } \)
Answer:
(i) \( \frac { -2 }{ 7 } \)
\( \frac {(-2)\times 2}{7\times 2} = \frac {-4}{14} \)
\( \frac {(-2)\times 3}{7\times 3} = \frac {-6}{21} \)
\( \frac {(-2)\times 4}{7\times 4} = \frac {-8}{28} \)
\( \frac {(-2)\times 5}{7\times 5} = \frac {-10}{35} \)
Thus, four rational numbers that are equivalent to \( \frac {-2}{7} \) are \( \frac {-4}{14}, \frac {-6}{21}, \frac {-8}{28} \), and \( \frac {-10}{35} \).
(ii) \( \frac { 5 }{ -3 } \)
\( \frac {5\times 2}{(-3)\times 2} = \frac {10}{-6} \)
\( \frac {5\times 3}{(-3)\times 3} = \frac {15}{-9} \)
\( \frac {5\times 4}{(-3)\times 4} = \frac {20}{-12} \)
\( \frac {5\times 5}{(-3)\times 5} = \frac {25}{-15} \)
Therefore, the four rational numbers needed that are equivalent to \( \frac {5}{-3} \) are \( \frac {10}{-6}, \frac {15}{-9}, \frac {20}{-12} \), and \( \frac {25}{-15} \).
(iii) \( \frac { 4 }{ 9 } \)
\( \frac {4\times 2}{9\times 2} = \frac {8}{18} \)
\( \frac {4\times 3}{9\times 3} = \frac {12}{27} \)
\( \frac {4\times 4}{9\times 4} = \frac {16}{36} \)
\( \frac {4\times 5}{9\times 5} = \frac {20}{45} \)
So, the four rational numbers equivalent to \( \frac {4}{9} \) are \( \frac {8}{18}, \frac {12}{27}, \frac {16}{36} \), and \( \frac {20}{45} \).
In simple words: To find equivalent fractions, multiply both the top number (numerator) and the bottom number (denominator) by the same non-zero whole number.

Exam Tip: Remember, you can multiply by any integer (other than zero) to get an equivalent rational number. Common choices are 2, 3, 4, 5, etc.

Question 4. Draw the number line and represent the following rational numbers on it:
(i) \( \frac { 3 }{ 4 } \)
(ii) \( \frac { -5 }{ 8 } \)
(iii) \( \frac { -7 }{ 4 } \)
(iv) \( \frac {7}{8} \)
Answer:
(i) \( \frac { 3 }{ 4 } \)
Draw a number line. Mark 0 and 1. Divide the segment between 0 and 1 into four equal parts. Mark the third point from 0 as \( \frac {3}{4} \).
(ii) \( \frac { -5 }{ 8 } \)
Draw a number line. Mark -1 and 0. Divide the segment between -1 and 0 into eight equal parts. Mark the fifth point to the left of 0 as \( \frac {-5}{8} \).
(iii) \( \frac { -7 }{ 4 } \)
Draw a number line. Mark -1 and -2. Divide the segment between -1 and -2 into four equal parts. Mark the third point to the left of -1 as \( \frac {-7}{4} \).
(iv) \( \frac {7}{8} \)
Draw a number line. Mark 0 and 1. Divide the segment between 0 and 1 into eight equal parts. Mark the seventh point from 0 as \( \frac {7}{8} \).
In simple words: To show a fraction on a number line, first decide if it's positive or negative. Then, divide the space between whole numbers into as many parts as the bottom number (denominator) says, and count over to the correct mark based on the top number (numerator).

Exam Tip: For negative fractions, remember to move left from zero. For fractions greater than 1, like \( \frac{7}{4} \), convert them to mixed numbers first (e.g., \( 1\frac{3}{4} \) or \( -1\frac{3}{4} \)) to make placement easier.

Question 5. The points P, Q, R, S, T, U, A and B on the number line are such that, TR = RS = SU and AP = PQ = QB. Name the rational numbers represented by P, Q, R and S.
Answer:
Since AP = PQ = QB, the total distance between points A and B is divided into three equal sections.
Similarly, the distance between -2 and -1 is also split into three equal portions.
So, P stands for the rational number \( 2 + \frac {1}{3} \), which is \( \frac {7}{3} \).
Q shows the rational number \( 2 + \frac {2}{3} \), meaning \( \frac {8}{3} \).
R denotes the rational number \( -1 - \frac {1}{3} \), which is \( \frac {-4}{3} \).
S indicates the rational number \( -1 - \frac {2}{3} \), meaning \( \frac {-5}{3} \).
In simple words: When a number line is divided into equal sections, each section represents a fraction. Count how many sections from a known point (like a whole number) to find the fraction for a given letter.

Exam Tip: Pay close attention to the number of equal parts each unit interval on the number line is divided into. This denominator will be crucial for identifying the rational numbers.

Question 6. Following pairs represent the same rational number?
(i) \( \frac { -7 }{21} \) and \( \frac { 3 }{ 9 } \)
(ii) \( \frac {-16 }{ 20 } \) and \( \frac { 20 }{ -25 } \)
(iii) \( \frac {-2}{-3} \) and \( \frac { 2 }{ 3 } \)
(iv) \( \frac {-3}{5} \) and \( \frac { -12 }{ 20 } \)
(v) \( \frac {8}{-5} \) and \( \frac { -24 }{ 15 } \)
(vi) \( \frac { 1 }{ 3 } \) and \( \frac { -1 }{ 9 } \)
(vii) \( \frac { -5 }{ -9 } \) and \( \frac { 5 }{ -9 } \)
Answer:
(i) \( \frac { -7 }{21} \) and \( \frac { 3 }{ 9 } \)
In this instance, \( \frac { -7 }{ 21 } \) is a rational number that is negative, while \( \frac { 3 }{9} \) is a rational number that is positive.
So, \( \frac { -7 }{ 21 } \ne \frac { 3 }{9} \).
(ii) \( \frac {-16 }{ 20 } \) and \( \frac { 20 }{ -25 } \)
We calculate \( \frac { -16 }{ 20 } \) by dividing both parts by 4, which gives \( \frac { -4 }{ 5 } \), or \( -\frac { 4 }{ 5 } \). Similarly, \( \frac {20}{ -25 } \) becomes \( \frac { 4 }{ -5 } \) when both parts are divided by 5, which is also \( -\frac { 4 }{ 5 } \).
Thus, both \( \frac { -16 }{ 20 } \) and \( \frac {20 }{ -25 } \) show the exact same rational number.
(iii) \( \frac {-2}{-3} \) and \( \frac { 2 }{ 3 } \)
The fraction \( \frac { -2 }{ -3 } \) simplifies to \( \frac { (-2)\div(-1) }{ (-3)\div(-1) } = \frac { 2 }{ 3 } \).
Therefore, \( \frac { -2 }{ -3 } = \frac { 2 }{ 3 } \).
Consequently, \( \frac {-2}{-3} \) and \( \frac { 2 }{ 3 } \) both depict the identical rational number.
(iv) \( \frac {-3}{5} \) and \( \frac { -12 }{ 20 } \)
We see that \( \frac {-3}{5} \) can be changed to \( \frac { (-3)\times4 }{ 5\times4} = \frac { -12 }{ 20 } \).
So, \( \frac { -3 }{5} \) equals \( \frac { -12 }{ 20 } \).
Hence, \( \frac {-3}{5} \) and \( \frac { -12 }{ 20 } \) both depict the same rational number.
(v) \( \frac {8}{-5} \) and \( \frac { -24 }{ 15 } \)
We observe that \( \frac {8}{- 5 } \) can be written as \( \frac { 8\times3 }{ (-5)\times3 } = \frac { 24 }{ -15 } \), which is also \( \frac { -24 }{ 15 } \).
Therefore, \( \frac {8}{ -5 } = \frac { -24 }{ 15 } \).
Consequently, \( \frac { 8 }{ -5 } \) and \( \frac { -24 }{ 15 } \) both show the same rational number.
(vi) \( \frac { 1 }{ 3 } \) and \( \frac { -1 }{ 9 } \)
In this case, \( \frac { 1 }{ 3 } \) is a positive rational number, while \( \frac {-1}{9} \) is a negative rational number.
So, \( \frac { 1 }{ 3 } \ne \frac { -1 }{9} \).
(vii) \( \frac { -5 }{ -9 } \) and \( \frac { 5 }{ -9 } \)
As \( \frac { -5 }{ -9 } \) is a positive rational number, and \( \frac { 5 }{ -9 } \) is a negative rational number.
Therefore, \( \frac { -5 }{ -9 } \ne \frac {5}{ -9 } \).
In simple words: To check if two rational numbers are the same, simplify both of them to their simplest form. If the simplified forms are identical, then the original numbers are equivalent. Also, a positive number can never equal a negative number.

Exam Tip: Always reduce fractions to their simplest form before comparing. Also, pay attention to the signs; a positive and a negative number can never be equal.

Question 7. Rewrite the following rational numbers in the simplest form:
(i) \( \frac { -8 }{ 6 } \)
(ii) \( \frac { 25 }{ 45 } \)
(iii) \( \frac { -44 }{ 72 } \)
(iv) \( \frac { -8 }{ 10 } \)
Answer:
(i) \( \frac { -8 }{ 6 } \)
The highest common factor (HCF) for 8 and 6 is 2.
So, \( \frac { -8 }{ 6 } \) becomes \( \frac { (-8)\div2 }{ 6\div2 } = \frac { -4 }{ 3 } \).
The most simplified version of \( \frac { -8 }{ 6 } \) is \( \frac { -4 }{ 3 } \).
(ii) \( \frac { 25 }{ 45 } \)
The greatest common divisor (HCF) for 25 and 45 is 5.
Thus, \( \frac { 25 }{ 45 } \) simplifies to \( \frac { 25\div5}{45\div5} = \frac {5}{9} \).
Therefore, the simplest form for \( \frac { 25 }{ 45 } \) is \( \frac {5}{9} \).
(iii) \( \frac { -44 }{ 72 } \)
The highest common factor for 44 and 72 is 4.
So, \( \frac { -44 }{ 72 } \) simplifies to \( \frac { (-44)\div4 }{ 72\div4} = \frac { -11 }{ 18 } \).
Consequently, the simplest form of \( \frac { -44 }{ 72 } \) is \( \frac { -11 }{ 18 } \).
(iv) \( \frac { -8 }{ 10 } \)
The highest common factor of 8 and 10 is 2.
So, \( \frac { -8 }{ 10 } \) simplifies to \( \frac { (-8)\div2 }{ 10\div2} = \frac { -4 }{ 5 } \).
Therefore, the most simplified form of \( \frac { -8 }{ 10 } \) is \( \frac {-4}{5} \).
In simple words: To simplify a rational number, find the largest number that divides evenly into both the top and bottom numbers, and then divide both by that number. Keep doing this until no common factor remains.

Exam Tip: Always divide by the Greatest Common Divisor (GCD) or HCF to reach the simplest form in a single step. If you can't find the GCD, divide by common prime factors repeatedly.

Question 8. Fill in the boxes with the correct symbol out of >, < and =.
(i) \( \frac {-5}{7} \Box \frac {2}{3} \)
(ii) \( \frac {-4}{5} \Box \frac {-5}{7} \)
(iii) \( \frac {-7}{8} \Box \frac {14}{-16} \)
(iv) \( \frac {-8}{5} \Box \frac {-7}{4} \)
(v) \( \frac {1}{-3} \Box \frac {-1}{4} \)
(vi) \( \frac {5}{-11} \Box \frac {-5}{11} \)
(vii) \( 0 \Box \frac {-7}{6} \)
Answer:
(i) \( \frac {-5}{7} < \frac {2}{3} \)
As \( \frac {-5}{7} \) represents a negative rational number, it is naturally less than \( \frac {2}{3} \), which is a positive number.
So, \( \frac {-5}{7} < \frac {2}{3} \).
(ii) \( \frac {-4}{5} < \frac {-5}{7} \)
The least common multiple (LCM) of 5 and 7 is 35.
We convert the fractions: \( \frac {-4}{5} \) becomes \( \frac {(-4)\times 7}{5\times 7} = \frac {-28}{35} \), and \( \frac {-5}{7} \) becomes \( \frac {(-5)\times 5}{7\times 5} = \frac {-25}{35} \).
Because \( \frac {-28}{35} \) is smaller than \( \frac {-25}{35} \).
Therefore, \( \frac {-4}{5} < \frac {-5}{7} \).
(iii) \( \frac {-7}{8} = \frac {14}{-16} \)
The least common multiple for 8 and 16 is 16.
We adjust the fractions: \( \frac {-7}{8} \) becomes \( \frac {(-7)\times 2}{8\times 2} = \frac {-14}{16} \), and \( \frac {14}{-16} \) becomes \( \frac {14\times (-1)}{(-16)\times (-1)} = \frac {-14}{16} \).
Hence, \( \frac {-7}{8} = \frac {14}{-16} \).
(iv) \( \frac {-8}{5} > \frac {-7}{4} \)
The least common multiple of 5 and 4 is 20.
We convert the fractions: \( \frac {-8}{5} \) becomes \( \frac {(-8)\times 4}{5\times 4} = \frac {-32}{20} \), and \( \frac {-7}{4} \) becomes \( \frac {(-7)\times 5}{4\times 5} = \frac {-35}{20} \).
Since \( \frac {-32}{20} \) is greater than \( \frac {-35}{20} \).
Therefore, \( \frac {-8}{5} > \frac {-7}{4} \).
(v) \( \frac {1}{-3} < \frac {-1}{4} \)
The least common multiple for 3 and 4 is 12.
We change the fractions: \( \frac {1}{-3} \) becomes \( \frac {1\times 4}{(-3)\times 4} = \frac {-4}{12} \), and \( \frac {-1}{4} \) becomes \( \frac {(-1)\times 3}{4\times 3} = \frac {-3}{12} \).
Given that \( \frac {-4}{12} \) is less than \( \frac {-3}{12} \).
So, \( \frac {1}{-3} < \frac {-1}{4} \).
(vi) \( \frac {5}{-11} = \frac {-5}{11} \)
Given that \( \frac {5}{-11} \) equals \( \frac {5\times (-1)}{(-11)\times (-1)} = \frac {-5}{11} \).
Therefore, \( \frac {5}{-11} = \frac {-5}{11} \).
(vii) \( 0 > \frac {-7}{6} \)
Because 0 is always larger than any negative number.
Thus, \( 0 > \frac {-7}{6} \).
In simple words: To compare fractions, make their bottom numbers (denominators) the same using the LCM. Then, compare their top numbers (numerators). Remember that negative numbers get smaller as they move further away from zero.

Exam Tip: For comparing rational numbers, finding a common denominator is the most reliable method. Also, always consider the sign; positive numbers are always greater than negative numbers, and zero is greater than all negative numbers.

Question 9. Which is greater in each of the following:
(i) \( \frac { 2 }{ 3 } \) and \( \frac {5}{ 2 } \)
(ii) \( \frac { -5 }{6} \) and \( \frac { -4 }{ 3 } \)
(iii) \( \frac {-3}{4} \) and \( \frac { 2 }{ -3 } \)
(iv) \( \frac {-1}{4} \) and \( \frac { 1 }{ 4 } \)
(v) \( -3\frac { 2 }{7}, -3\frac { 4 }{ 5 } \)
Answer:
(i) \( \frac { 2 }{ 3 } \) and \( \frac {5}{ 2 } \)
We transform the fractions to have a common denominator: \( \frac {2}{3} \) becomes \( \frac {2\times 2}{3\times 2} = \frac {4}{6} \), and \( \frac {5}{2} \) becomes \( \frac {5\times 3}{2\times 3} = \frac {15}{6} \).
Because \( \frac {15}{6} \) is larger than \( \frac {4}{6} \), it means \( \frac {5}{2} \) is greater than \( \frac {2}{3} \).
Therefore, \( \frac {5}{ 2 } \) is the greater rational number.
(ii) \( \frac { -5 }{6} \) and \( \frac { -4 }{ 3 } \)
The least common multiple for 6 and 3 is 6.
We convert the fractions: \( \frac {-5}{6} \) becomes \( \frac {(-5)\times 1}{6\times 1} = \frac {-5}{6} \), and \( \frac {-4}{3} \) becomes \( \frac {(-4)\times 2}{3\times 2} = \frac {-8}{6} \).
Given that \( \frac {-5}{6} \) is greater than \( \frac {-8}{6} \), this means \( \frac {-5}{6} \) is greater than \( \frac {-4}{3} \).
Hence, \( \frac { -5 }{ 6 } \) is the greater rational number.
(iii) \( \frac {-3}{4} \) and \( \frac { 2 }{ -3 } \)
We change the fractions to share a common denominator: \( \frac {-3}{4} \) becomes \( \frac {(-3)\times 3}{4\times 3} = \frac {-9}{12} \), and \( \frac {2}{-3} \) becomes \( \frac {2\times 4}{(-3)\times 4} = \frac {8}{-12} \), which is also \( \frac {-8}{12} \).
Since \( \frac {-8}{12} \) is larger than \( \frac {-9}{12} \), it means \( \frac {2}{-3} \) is greater than \( \frac {-3}{4} \).
Therefore, the rational number \( \frac {2}{-3} \) is the greater one.
(iv) \( \frac {-1}{4} \) and \( \frac { 1 }{ 4 } \)
Because a positive rational number will always be larger than a negative rational number.
So, between \( \frac { 1 }{ 4 } \) and \( \frac {-1}{4} \), the greater rational number is \( \frac {1}{ 4 } \).
(v) \( -3\frac { 2 }{7}, -3\frac { 4 }{ 5 } \)
We convert the mixed numbers to improper fractions with a common denominator: \( -3\frac {2}{7} \) becomes \( \frac {(-23)\times 5}{7\times 5} = \frac {-115}{35} \), and \( -3\frac {4}{5} \) becomes \( \frac {(-19)\times 7}{5\times 7} = \frac {-133}{35} \).
Given that \( \frac {-115}{35} \) is greater than \( \frac {-133}{35} \), it implies \( \frac {-23}{7} \) is larger than \( \frac {-19}{5} \).
Therefore, the rational number \( -3\frac {2}{7} \) is greater.
In simple words: To find the greater fraction, make sure both numbers have the same bottom part (denominator). Then, compare the top parts (numerators). For negative fractions, the one closer to zero is greater.

Exam Tip: Convert mixed numbers to improper fractions and find a common denominator for accurate comparison. Remember that for negative numbers, the one with the smaller absolute value is greater (e.g., -2 > -5).

Question 10. Write the following rational numbers in ascending order:
(i) \( \frac {-3}{ 5 }, \frac { -2 }{ 5 }, \frac { -1}{5} \)
(ii) \( \frac { -1 }{ 3 }, \frac { -2 }{ 9 }, \frac { -4 }{ 3 } \)
(iii) \( \frac { -3 }{ 7 }, \frac { -3 }{ 2 }, \frac { -3 }{ 4 } \)
Answer:
(i) \( \frac {-3}{ 5 }, \frac { -2 }{ 5 }, \frac { -1}{5} \)
Because -3 is less than -2, and -2 is less than -1.
Therefore, \( \frac { -3 }{ 5 } \) is less than \( \frac { -2 }{ 5 } \), which is less than \( \frac { -1 }{ 5 } \).
Thus, the numbers in increasing order are \( \frac { -3 }{ 5 } < \frac { -2 }{ 5 } < \frac {-1}{5} \).
(ii) \( \frac { -1 }{ 3 }, \frac { -2 }{ 9 }, \frac { -4 }{ 3 } \)
Because the least common multiple of 3 and 9 is 9.
We convert \( \frac {-1}{3} \) to \( \frac {(-1)\times 3}{3\times 3} = \frac {-3}{9} \).
Then \( \frac {-2}{9} \) stays as \( \frac {(-2)\times 1}{9\times 1} = \frac {-2}{9} \).
And \( \frac {-4}{3} \) becomes \( \frac {(-4)\times 3}{3\times 3} = \frac {-12}{9} \).
Because -12 is less than -3, which is less than -2.
So, \( \frac {-12}{9} < \frac {-3}{9} < \frac {-2}{9} \) or \( \frac {-4}{3} < \frac {-1}{3} < \frac {-2}{9} \).
Therefore, the increasing order of the provided rational numbers is \( \frac { -4 }{ 3 }, \frac { -1 }{ 3 } \), and \( \frac {-2}{9} \).
(iii) \( \frac { -3 }{ 7 }, \frac { -3 }{ 2 }, \frac { -3 }{ 4 } \)
The least common multiple for 7, 2, and 4 is 28.
We convert \( \frac {-3}{7} \) to \( \frac {(-3)\times 4}{7\times 4} = \frac {-12}{28} \).
Then \( \frac {-3}{2} \) becomes \( \frac {(-3)\times 14}{2\times 14} = \frac {-42}{28} \).
And \( \frac {-3}{4} \) changes to \( \frac {(-3)\times 7}{4\times 7} = \frac {-21}{28} \).
Because -42 is less than -21, which is less than -12.
So, \( \frac {-42}{28} < \frac {-21}{28} < \frac {-12}{28} \) or \( \frac {-3}{2} < \frac {-3}{4} < \frac {-3}{7} \).
Therefore, the increasing order of the given rational numbers is \( \frac { -3 }{ 2 }, \frac { -3 }{ 4 } \), and \( \frac {-3}{7} \).
In simple words: To put fractions in order, first make them all have the same bottom number (denominator). Then, arrange them by their top numbers (numerators). Remember, for negative numbers, the one with the biggest negative top number is actually the smallest.

Exam Tip: When dealing with negative rational numbers, remember that the number with the larger absolute value (ignoring the negative sign) is actually smaller. Convert all fractions to a common denominator to simplify ordering.