GSEB Class 7 Maths Solutions Chapter 9 સંમેય સંખ્યાઓ Exercise 9.1

 

Question 1. નિગ્નલિખિત સંમેય સંખ્યાઓની વચ્ચે આવતી પાંચ સંમેય સંખ્યાઓ લખો:
(i) -1 અને 0
(ii) -2 અને -1
(iii) \( \frac{-4}{5} \) અને \( \frac{-2}{3} \)
(iv) \( \frac{-1}{2} \) અને \( \frac{2}{3} \)
Answer:
(i) -1 can be written as \( \frac{-1 \times 7}{1 \times 7} = \frac{-7}{7} \). Similarly, 0 can be written as \( \frac{0 \times 7}{1 \times 7} = \frac{0}{7} \). Therefore, we can find several rational numbers between \( \frac{-7}{7} \) and \( \frac{0}{7} \). For example, five rational numbers between -1 and 0 are \( \frac{-6}{7}, \frac{-5}{7}, \frac{-4}{7}, \frac{-3}{7}, \) and \( \frac{-2}{7} \).
(ii) -2 can be expressed as \( \frac{-2 \times 7}{1 \times 7} = \frac{-14}{7} \). Likewise, -1 can be expressed as \( \frac{-1 \times 7}{1 \times 7} = \frac{-7}{7} \). We can then list the integers in between. Thus, five rational numbers found between -2 and -1 are \( \frac{-13}{7}, \frac{-12}{7}, \frac{-11}{7}, \frac{-10}{7}, \) and \( \frac{-9}{7} \).
(iii) To compare \( \frac{-4}{5} \) and \( \frac{-2}{3} \), we first find a common denominator, which is 60. So, \( \frac{-4}{5} = \frac{-4 \times 12}{5 \times 12} = \frac{-48}{60} \). Similarly, \( \frac{-2}{3} = \frac{-2 \times 20}{3 \times 20} = \frac{-40}{60} \). Therefore, many rational numbers exist between \( \frac{-48}{60} \) and \( \frac{-40}{60} \). Five rational numbers between \( \frac{-4}{5} \) and \( \frac{-2}{3} \) are \( \frac{-47}{60}, \frac{-46}{60}, \frac{-45}{60}, \frac{-44}{60}, \) and \( \frac{-43}{60} \). These can also be shown in simplified forms as \( \frac{-47}{60}, \frac{-23}{30}, \frac{-3}{4}, \frac{-11}{15}, \) and \( \frac{-43}{60} \).
(iv) To find numbers between \( \frac{-1}{2} \) and \( \frac{2}{3} \), we get a common denominator, which is 36. So, \( \frac{-1}{2} = \frac{-1 \times 18}{2 \times 18} = \frac{-18}{36} \). And \( \frac{2}{3} = \frac{2 \times 12}{3 \times 12} = \frac{24}{36} \). Many numbers lie between \( \frac{-18}{36} \) and \( \frac{24}{36} \). Five rational numbers found between \( \frac{-1}{2} \) and \( \frac{2}{3} \) are \( \frac{-17}{36}, \frac{-16}{36}, \frac{-15}{36}, \frac{-14}{36}, \) and \( \frac{-13}{36} \). These can also be shown in simplified form as \( \frac{-17}{36}, \frac{-4}{9}, \frac{-5}{12}, \frac{-7}{18}, \) and \( \frac{-13}{36} \).
In simple words: To find rational numbers between two given numbers, first make their denominators the same. Then, you can list any fractions with that common denominator that fall between the original numerators.

Exam Tip: Always make sure to use a sufficiently large common denominator to find the required number of rational numbers. If you need more, just multiply the denominator by an even larger number.

 

Question 2. Write the next four equivalent rational numbers in the given pattern for each of the following:
(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) The pattern for the given rational numbers is to multiply the numerator and denominator by consecutive integers. So, starting from \( \frac{-3}{5} \), the next four equivalent rational numbers are determined by multiplying by 5, 6, 7, and 8.
\( \frac{-3 \times 5}{5 \times 5} = \frac{-15}{25} \)
\( \frac{-3 \times 6}{5 \times 6} = \frac{-18}{30} \)
\( \frac{-3 \times 7}{5 \times 7} = \frac{-21}{35} \)
\( \frac{-3 \times 8}{5 \times 8} = \frac{-24}{40} \)
Thus, the four required rational numbers are \( \frac{-15}{25}, \frac{-18}{30}, \frac{-21}{35}, \) and \( \frac{-24}{40} \).
(ii) The given sequence of rational numbers follows a pattern where the numerator and denominator are multiplied by increasing integers. For instance, starting with \( \frac{-1}{4} \), we obtain the next four equivalent rational numbers by multiplying by 4, 5, 6, and 7.
\( \frac{-1 \times 4}{4 \times 4} = \frac{-4}{16} \)
\( \frac{-1 \times 5}{4 \times 5} = \frac{-5}{20} \)
\( \frac{-1 \times 6}{4 \times 6} = \frac{-6}{24} \)
\( \frac{-1 \times 7}{4 \times 7} = \frac{-7}{28} \)
Therefore, the four equivalent rational numbers are \( \frac{-4}{16}, \frac{-5}{20}, \frac{-6}{24}, \) and \( \frac{-7}{28} \).
(iii) This series of rational numbers multiplies the numerator and denominator by consecutive whole numbers. Starting from \( \frac{-1}{6} \), the next four equivalent rational numbers are found by multiplying by 5, 6, 7, and 8.
\( \frac{1 \times 5}{-6 \times 5} = \frac{5}{-30} \)
\( \frac{1 \times 6}{-6 \times 6} = \frac{6}{-36} \)
\( \frac{1 \times 7}{-6 \times 7} = \frac{7}{-42} \)
\( \frac{1 \times 8}{-6 \times 8} = \frac{8}{-48} \)
Hence, the four desired rational numbers are \( \frac{5}{-30}, \frac{6}{-36}, \frac{7}{-42}, \) and \( \frac{8}{-48} \).
(iv) For this set of rational numbers, we find equivalent fractions by multiplying the numerator and denominator by integers. The initial fractions are based on \( \frac{-2}{3} \). The next four equivalent rational numbers are generated by multiplying by 4, 5, 6, and 7.
\( \frac{-2 \times 4}{3 \times 4} = \frac{-8}{12} \)
\( \frac{-2 \times 5}{3 \times 5} = \frac{-10}{15} \)
\( \frac{-2 \times 6}{3 \times 6} = \frac{-12}{18} \)
\( \frac{-2 \times 7}{3 \times 7} = \frac{-14}{21} \)
So, the four equivalent rational numbers are \( \frac{-8}{12}, \frac{-10}{15}, \frac{-12}{18}, \) and \( \frac{-14}{21} \).
In simple words: To continue a pattern of equivalent rational numbers, multiply both the top and bottom of the fraction by the next whole numbers in the sequence.

Exam Tip: Ensure that the sign of the fraction is handled correctly when writing equivalent rational numbers, especially if the negative sign moves between the numerator and denominator.

 

Question 3. નીચેના માટે ચાર સમાન સંમેય સંખ્યા લખોઃ
(i) \( \frac{-2}{7} \)
(ii) \( \frac{5}{-3} \)
(iii) \( \frac{4}{9} \)
Answer:
(i) To find four equivalent rational numbers for \( \frac{-2}{7} \), we can multiply both the numerator and denominator by the integers 2, 3, 4, and 5.
\( \frac{-2 \times 2}{7 \times 2} = \frac{-4}{14} \)
\( \frac{-2 \times 3}{7 \times 3} = \frac{-6}{21} \)
\( \frac{-2 \times 4}{7 \times 4} = \frac{-8}{28} \)
\( \frac{-2 \times 5}{7 \times 5} = \frac{-10}{35} \)
Hence, the four equivalent rational numbers are \( \frac{-4}{14}, \frac{-6}{21}, \frac{-8}{28}, \) and \( \frac{-10}{35} \).
(ii) To obtain four equivalent rational numbers for \( \frac{5}{-3} \), we multiply both the numerator and denominator by the integers 2, 3, 4, and 5.
\( \frac{5 \times 2}{-3 \times 2} = \frac{10}{-6} \)
\( \frac{5 \times 3}{-3 \times 3} = \frac{15}{-9} \)
\( \frac{5 \times 4}{-3 \times 4} = \frac{20}{-12} \)
\( \frac{5 \times 5}{-3 \times 5} = \frac{25}{-15} \)
Thus, the four equivalent rational numbers are \( \frac{10}{-6}, \frac{15}{-9}, \frac{20}{-12}, \) and \( \frac{25}{-15} \).
(iii) To find four equivalent rational numbers for \( \frac{4}{9} \), we multiply both the numerator and denominator by the integers 2, 3, 4, and 5.
\( \frac{4 \times 2}{9 \times 2} = \frac{8}{18} \)
\( \frac{4 \times 3}{9 \times 3} = \frac{12}{27} \)
\( \frac{4 \times 4}{9 \times 4} = \frac{16}{36} \)
\( \frac{4 \times 5}{9 \times 5} = \frac{20}{45} \)
Therefore, the four equivalent rational numbers 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 and bottom of the fraction by the same non-zero whole number.

Exam Tip: Remember that multiplying the numerator and denominator by the same non-zero number does not change the value of the rational number.

 

Question 4. સંખ્યારેખા દોરો અને નીચે આપેલી સંમેય સંખ્યાઓનું તેની પર નિરૂપણ કરો:
(i) \( \frac{3}{4} \)
(ii) \( \frac{-5}{8} \)
(iii) \( \frac{-7}{4} \)
(iv) \( \frac{7}{8} \)
Answer:
(i) To represent \( \frac{3}{4} \) on a number line, we first identify that it lies between 0 and 1. We divide the segment between 0 and 1 into four equal parts. The third mark from 0 will then show the position of \( \frac{3}{4} \).

(ii) To show \( \frac{-5}{8} \) on a number line, we first realize it is a negative fraction between -1 and 0. We then divide the segment between -1 and 0 into eight equal parts. Counting five marks to the left from 0 will locate the point for \( \frac{-5}{8} \).

(iii) To plot \( \frac{-7}{4} \) on a number line, which is equal to \( -1\frac{3}{4} \), we know it lies between -2 and -1. We divide the segment between -2 and -1 into four equal parts. The first mark to the right of -2 (or three marks to the left of -1) represents \( \frac{-7}{4} \).

(iv) To mark \( \frac{7}{8} \) on a number line, we recognize it is a positive fraction situated between 0 and 1. We divide the interval between 0 and 1 into eight equal segments. The seventh mark from 0 will accurately show the position of \( \frac{7}{8} \).

In simple words: To represent a fraction on a number line, locate which two whole numbers it falls between. Then divide that segment into equal parts based on the fraction's denominator and mark the correct position.

Exam Tip: Always extend the number line sufficiently to include the rational number, and clearly label the integers and the fractional subdivisions.

 

Question 5. બિંદુઓ P, Q, R, S, T, U, A અને B સંખ્યારેખા પર એવી રીતે આવેલાં છે કે જ્યાં TR = RS = SU અને AP = PQ = QB થાય. P, Q, R, S અને વડે દર્શાવાતી સંમેય સંખ્યા લખો.
Answer: Based on the given conditions, the segment between 2 and 3 on the number line is divided into three equal parts by points P and Q. Also, the segment between -2 and -1 is similarly divided into three equal parts by points R and S.
For points P and Q, starting from 2:
\( P = 2 + \frac{1}{3} = \frac{6+1}{3} = \frac{7}{3} \)
\( Q = 2 + \frac{2}{3} = \frac{6+2}{3} = \frac{8}{3} \)
For points R and S, starting from -1 and moving left:
\( R = -1 - \frac{1}{3} = \frac{-3-1}{3} = \frac{-4}{3} \)
\( S = -1 - \frac{2}{3} = \frac{-3-2}{3} = \frac{-5}{3} \)
Therefore, the rational numbers represented are P(\( \frac{7}{3} \)), Q(\( \frac{8}{3} \)), R(\( \frac{-4}{3} \)), and S(\( \frac{-5}{3} \)).

In simple words: When points divide a segment of a number line into equal parts, calculate the value of each part and add it to the starting point to find the coordinate of each dividing point.

Exam Tip: Be careful with signs when calculating points on the negative side of the number line; moving to the left means subtracting from the reference point.

 

Question 6. નીચે આપેલી જોડીઓમાંની કઈ જોડી સમાન સંમેય સંખ્યાઓનું નિરૂપણ કરે છે?
(i) \( \frac{-7}{21} \) અને \( \frac{3}{9} \)
(ii) \( \frac{-16}{20} \) અને \( \frac{20}{-25} \)
(iii) \( \frac{-2}{-3} \) અને \( \frac{2}{3} \)
(iv) \( \frac{-3}{5} \) અને \( \frac{-12}{20} \)
(v) \( \frac{8}{-5} \) અને \( \frac{-24}{15} \)
(vi) \( \frac{1}{3} \) અને \( \frac{-1}{9} \)
(vii) \( \frac{-5}{-9} \) અને \( \frac{5}{-9} \)
Answer:
(i) We need to check if \( \frac{-7}{21} \) and \( \frac{3}{9} \) are equivalent rational numbers. The first number, \( \frac{-7}{21} \), is a negative rational number. The second number, \( \frac{3}{9} \), is a positive rational number. Since one is negative and the other is positive, they cannot be equal. Thus, they do not represent the same rational number.
(ii) Let's simplify both rational numbers to their lowest terms.
For \( \frac{-16}{20} \), we divide the numerator and denominator by their greatest common factor, 4: \( \frac{-16 \div 4}{20 \div 4} = \frac{-4}{5} \).
For \( \frac{20}{-25} \), we divide the numerator and denominator by their greatest common factor, 5: \( \frac{20 \div 5}{-25 \div 5} = \frac{4}{-5} \). We can write \( \frac{4}{-5} \) as \( \frac{-4}{5} \).
Since both fractions simplify to \( \frac{-4}{5} \), they are equal and represent the same rational number.
(iii) To check if \( \frac{-2}{-3} \) and \( \frac{2}{3} \) are equivalent, we simplify the first fraction. Dividing both the numerator and denominator of \( \frac{-2}{-3} \) by -1 gives \( \frac{2}{3} \).
Since \( \frac{-2}{-3} \) is equal to \( \frac{2}{3} \), both rational numbers represent the same value.
(iv) To compare \( \frac{-3}{5} \) and \( \frac{-12}{20} \), we can express \( \frac{-3}{5} \) with a denominator of 20. By multiplying both the numerator and denominator by 4: \( \frac{-3 \times 4}{5 \times 4} = \frac{-12}{20} \).
Since \( \frac{-3}{5} \) is equivalent to \( \frac{-12}{20} \), both rational numbers represent the same value.
(v) To determine if \( \frac{8}{-5} \) and \( \frac{-24}{15} \) are equivalent, we multiply the numerator and denominator of \( \frac{8}{-5} \) by 3: \( \frac{8 \times 3}{-5 \times 3} = \frac{24}{-15} \).
We know that \( \frac{24}{-15} \) is the same as \( \frac{-24}{15} \) by relocating the negative sign to the numerator.
Thus, both given rational numbers are equal and represent the same value.
(vi) We are comparing \( \frac{1}{3} \) and \( \frac{-1}{9} \). The rational number \( \frac{1}{3} \) is positive, whereas \( \frac{-1}{9} \) is negative. A positive number can never be equal to a negative number. Therefore, these two rational numbers do not represent the same value.
(vii) Let's examine \( \frac{-5}{-9} \) and \( \frac{5}{-9} \). When we simplify \( \frac{-5}{-9} \), dividing both numerator and denominator by -1 gives \( \frac{5}{9} \), which is a positive rational number.
However, \( \frac{5}{-9} \) can be written as \( \frac{-5}{9} \), which is a negative rational number.
Since one is positive and the other is negative, these two rational numbers are not equal and do not represent the same value.
In simple words: To see if two rational numbers are the same, either simplify both to their simplest form and compare, or make their denominators equal and compare the numerators. Also, a positive number can never equal a negative number.

Exam Tip: Always check the sign of the rational numbers first; a positive and a negative number can never be equivalent, simplifying comparisons.

 

Question 7. નીચે આપેલી સંમેય સંખ્યાઓને અતિસંક્ષિપ્ત સ્વરૂપે ફરીથી લખો:
(i) \( \frac{-8}{6} \)
(ii) \( \frac{25}{45} \)
(iii) \( \frac{-44}{72} \)
(iv) \( \frac{-8}{10} \)
Answer:
(i) To simplify \( \frac{-8}{6} \), we find the greatest common factor (HCF) of the numerator 8 and the denominator 6, which is 2.
Then, we divide both the numerator and denominator by 2: \( \frac{-8 \div 2}{6 \div 2} = \frac{-4}{3} \).
Therefore, the simplest form of \( \frac{-8}{6} \) is \( \frac{-4}{3} \).
(ii) To express \( \frac{25}{45} \) in its simplest form, we first find the greatest common factor (HCF) of 25 and 45, which is 5.
Next, we divide both the numerator and the denominator by 5: \( \frac{25 \div 5}{45 \div 5} = \frac{5}{9} \).
Thus, the simplest form of \( \frac{25}{45} \) is \( \frac{5}{9} \).
(iii) To simplify \( \frac{-44}{72} \), we identify the greatest common factor (HCF) of 44 and 72, which is 4.
Then, we divide both the numerator and denominator by 4: \( \frac{-44 \div 4}{72 \div 4} = \frac{-11}{18} \).
The simplest form of \( \frac{-44}{72} \) is therefore \( \frac{-11}{18} \).
(iv) To find the simplest form of \( \frac{-8}{10} \), we first determine the greatest common factor (HCF) of 8 and 10, which is 2.
We then divide both the numerator and the denominator by 2: \( \frac{-8 \div 2}{10 \div 2} = \frac{-4}{5} \).
Hence, the simplest form of \( \frac{-8}{10} \) is \( \frac{-4}{5} \).
In simple words: To write a rational number in its simplest form, divide both the top and bottom numbers by their largest common factor until they cannot be divided evenly anymore.

Exam Tip: Always find the Greatest Common Factor (GCF) of the numerator and denominator to ensure the fraction is reduced to its absolute simplest form.

 

Question 8. >, < અને = માંથી યોગ્ય સંકેત પસંદ કરી ખાલી જગ્યામાં ભરોઃ
(i) \( \frac{-5}{7} \) _ \( \frac{2}{3} \)
(ii) \( \frac{-4}{5} \) _ \( \frac{-2}{3} \)
(iii) \( \frac{-7}{8} \) _ \( \frac{14}{-16} \)
(iv) \( \frac{-8}{5} \) _ \( \frac{-7}{4} \)
(v) \( \frac{1}{-3} \) _ \( \frac{1}{-4} \)
(vi) \( \frac{5}{-11} \) _ \( \frac{-5}{11} \)
(vii) \( 0 \) _ \( \frac{-7}{6} \)
Answer:
(i) We are comparing \( \frac{-5}{7} \) and \( \frac{2}{3} \). Since \( \frac{-5}{7} \) is a negative rational number and \( \frac{2}{3} \) is a positive rational number, a negative number is always smaller than a positive number.
Therefore, \( \frac{-5}{7} < \frac{2}{3} \).
(ii) To compare \( \frac{-4}{5} \) and \( \frac{-2}{3} \), we find a common denominator, which is the LCM of 5 and 3, which is 15.
Convert the fractions: \( \frac{-4}{5} = \frac{-4 \times 3}{5 \times 3} = \frac{-12}{15} \).
And \( \frac{-2}{3} = \frac{-2 \times 5}{3 \times 5} = \frac{-10}{15} \).
Now we compare \( \frac{-12}{15} \) and \( \frac{-10}{15} \). Since -12 is less than -10, it means \( \frac{-12}{15} < \frac{-10}{15} \).
Therefore, \( \frac{-4}{5} < \frac{-2}{3} \).
(iii) We need to compare \( \frac{-7}{8} \) and \( \frac{14}{-16} \). Let's simplify the second rational number, \( \frac{14}{-16} \).
Dividing both the numerator and denominator by 2 gives \( \frac{7}{-8} \), which can be written as \( \frac{-7}{8} \).
Since both fractions are equal to \( \frac{-7}{8} \), they are equivalent.
Therefore, \( \frac{-7}{8} = \frac{14}{-16} \).
(iv) To compare \( \frac{-8}{5} \) and \( \frac{-7}{4} \), we find a common denominator, which is the LCM of 5 and 4, which is 20.
Convert the fractions: \( \frac{-8}{5} = \frac{-8 \times 4}{5 \times 4} = \frac{-32}{20} \).
And \( \frac{-7}{4} = \frac{-7 \times 5}{4 \times 5} = \frac{-35}{20} \).
Now compare \( \frac{-32}{20} \) and \( \frac{-35}{20} \). Since -32 is greater than -35, it implies \( \frac{-32}{20} > \frac{-35}{20} \).
Therefore, \( \frac{-8}{5} > \frac{-7}{4} \).
(v) To compare \( \frac{1}{-3} \) and \( \frac{1}{-4} \), we first rewrite them as \( \frac{-1}{3} \) and \( \frac{-1}{4} \).
The least common multiple (LCM) of 3 and 4 is 12.
Convert to equivalent fractions: \( \frac{-1}{3} = \frac{-1 \times 4}{3 \times 4} = \frac{-4}{12} \).
And \( \frac{-1}{4} = \frac{-1 \times 3}{4 \times 3} = \frac{-3}{12} \).
Comparing \( \frac{-4}{12} \) and \( \frac{-3}{12} \), since -4 is less than -3, it means \( \frac{-4}{12} < \frac{-3}{12} \).
Therefore, \( \frac{1}{-3} < \frac{1}{-4} \).
(vi) We compare \( \frac{5}{-11} \) and \( \frac{-5}{11} \). We can simply move the negative sign from the denominator to the numerator in \( \frac{5}{-11} \) to get \( \frac{-5}{11} \).
Since both rational numbers are now in the same form \( \frac{-5}{11} \), they are equivalent.
Therefore, \( \frac{5}{-11} = \frac{-5}{11} \).
(vii) We are comparing 0 with \( \frac{-7}{6} \). Any positive number is greater than 0, and 0 is always greater than any negative number.
Since \( \frac{-7}{6} \) is a negative rational number, 0 will always be greater than it.
Therefore, \( 0 > \frac{-7}{6} \).
In simple words: To compare rational numbers, convert them to a common denominator first. For negative numbers, the one closer to zero is greater. A positive number is always greater than zero, and zero is always greater than a negative number.

Exam Tip: When comparing negative rational numbers, remember that the number with the smaller absolute value is actually greater (e.g., -2 > -5).

 

Question 9. નીચેના દરેકમાં કઈ સંખ્યા મોટી છે?
(i) \( \frac{2}{3}, \frac{5}{2} \)
(ii) \( \frac{-5}{6}, \frac{-4}{3} \)
(iii) \( \frac{-3}{4}, \frac{2}{-3} \)
(iv) \( \frac{-1}{4}, \frac{1}{4} \)
(v) \( -3\frac{2}{7}, -3\frac{4}{5} \)
Answer:
(i) To determine which is greater between \( \frac{2}{3} \) and \( \frac{5}{2} \), we find their common denominator. The least common multiple (LCM) of 3 and 2 is 6.
We convert the fractions: \( \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6} \).
And \( \frac{5}{2} = \frac{5 \times 3}{2 \times 3} = \frac{15}{6} \).
Comparing \( \frac{4}{6} \) and \( \frac{15}{6} \), since 15 is greater than 4, it means \( \frac{15}{6} > \frac{4}{6} \).
Therefore, \( \frac{5}{2} \) is the larger rational number.
(ii) To compare \( \frac{-5}{6} \) and \( \frac{-4}{3} \), we find their common denominator. The least common multiple (LCM) of 6 and 3 is 6.
We convert the fractions: \( \frac{-5}{6} = \frac{-5 \times 1}{6 \times 1} = \frac{-5}{6} \).
And \( \frac{-4}{3} = \frac{-4 \times 2}{3 \times 2} = \frac{-8}{6} \).
Comparing \( \frac{-5}{6} \) and \( \frac{-8}{6} \), since -5 is greater than -8, it means \( \frac{-5}{6} > \frac{-8}{6} \).
Therefore, \( \frac{-5}{6} \) is the larger rational number.
(iii) To compare \( \frac{-3}{4} \) and \( \frac{2}{-3} \), first rewrite \( \frac{2}{-3} \) as \( \frac{-2}{3} \).
Next, find a common denominator; the least common multiple (LCM) of 4 and 3 is 12.
Convert the fractions: \( \frac{-3}{4} = \frac{-3 \times 3}{4 \times 3} = \frac{-9}{12} \).
And \( \frac{-2}{3} = \frac{-2 \times 4}{3 \times 4} = \frac{-8}{12} \).
Comparing \( \frac{-9}{12} \) and \( \frac{-8}{12} \), since -8 is greater than -9, it implies \( \frac{-8}{12} > \frac{-9}{12} \).
Therefore, \( \frac{2}{-3} \) is the larger rational number.
(iv) To compare \( \frac{-1}{4} \) and \( \frac{1}{4} \), we recall that any positive rational number is always larger than any negative rational number.
Since \( \frac{1}{4} \) is a positive number and \( \frac{-1}{4} \) is a negative number, \( \frac{1}{4} \) is clearly greater.
Therefore, \( \frac{1}{4} \) is the larger rational number.
(v) To compare \( -3\frac{2}{7} \) and \( -3\frac{4}{5} \), we first convert these mixed numbers into improper fractions.
\( -3\frac{2}{7} = -\frac{(3 \times 7) + 2}{7} = -\frac{23}{7} \).
\( -3\frac{4}{5} = -\frac{(3 \times 5) + 4}{5} = -\frac{15+4}{5} = -\frac{19}{5} \).
Next, we find a common denominator, which is the LCM of 7 and 5, which is 35.
Convert to equivalent fractions: \( -\frac{23}{7} = -\frac{23 \times 5}{7 \times 5} = -\frac{115}{35} \).
And \( -\frac{19}{5} = -\frac{19 \times 7}{5 \times 7} = -\frac{133}{35} \).
When comparing negative numbers, the one with the smaller absolute value is greater. Since -115 is greater than -133, it follows that \( -\frac{115}{35} > -\frac{133}{35} \).
Therefore, \( -3\frac{2}{7} \) is the larger rational number.
In simple words: To find the greater number, change fractions to have the same bottom number. For positive numbers, the one with the bigger top number is greater. For negative numbers, the one with the smaller absolute value (closer to zero) is greater.

Exam Tip: Be cautious when comparing negative mixed numbers. Convert them to improper fractions and find a common denominator before comparing numerators.

 

Question 10. નીચે આપેલી સંમેય સંખ્યાઓને ચડતા ક્રમમાં લખો:
(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) To arrange \( \frac{-3}{5}, \frac{-2}{5}, \) and \( \frac{-1}{5} \) in ascending order, we observe that all denominators are the same. When comparing negative rational numbers with the same denominator, the number with the smallest (most negative) numerator is the smallest value.
Since -3 is less than -2, and -2 is less than -1, the order is:
\( -3 < -2 < -1 \)
Therefore, in ascending order, the rational numbers are \( \frac{-3}{5} < \frac{-2}{5} < \frac{-1}{5} \).
(ii) To arrange \( \frac{-1}{3}, \frac{-2}{9}, \) and \( \frac{-4}{3} \) in ascending order, we first find a common denominator. The least common multiple (LCM) of 3 and 9 is 9.
Convert the fractions:
\( \frac{-1}{3} = \frac{-1 \times 3}{3 \times 3} = \frac{-3}{9} \)
\( \frac{-2}{9} \) remains as \( \frac{-2}{9} \)
\( \frac{-4}{3} = \frac{-4 \times 3}{3 \times 3} = \frac{-12}{9} \)
Now, compare the numerators: -12, -3, -2. In ascending order, these are -12, -3, -2.
Therefore, the rational numbers in ascending order are \( \frac{-12}{9} < \frac{-3}{9} < \frac{-2}{9} \), which corresponds to \( \frac{-4}{3} < \frac{-1}{3} < \frac{-2}{9} \).
(iii) To arrange \( \frac{-3}{7}, \frac{-3}{2}, \) and \( \frac{-3}{4} \) in ascending order, we must find a common denominator. The least common multiple (LCM) of 7, 2, and 4 is 28.
Convert the fractions to equivalent forms:
\( \frac{-3}{7} = \frac{-3 \times 4}{7 \times 4} = \frac{-12}{28} \)
\( \frac{-3}{2} = \frac{-3 \times 14}{2 \times 14} = \frac{-42}{28} \)
\( \frac{-3}{4} = \frac{-3 \times 7}{4 \times 7} = \frac{-21}{28} \)
Now, compare the numerators: -42, -21, -12. In ascending order, these are -42, -21, -12.
Therefore, the rational numbers in ascending order are \( \frac{-42}{28} < \frac{-21}{28} < \frac{-12}{28} \), which translates to \( \frac{-3}{2} < \frac{-3}{4} < \frac{-3}{7} \).
In simple words: To put rational numbers in order from smallest to largest, first make sure they all have the same bottom number. Then, arrange them by their top numbers. For negative numbers, the biggest negative top number is the smallest value.

Exam Tip: When denominators are positive and common, comparing rational numbers is equivalent to comparing their numerators. For negative numbers, the smaller the numerator, the greater the value (closer to zero).