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1. List five rational numbers between
2. Write four more rational numbers in each of the following patterns.
(i)\(\frac{-3}{5},\ \frac{-6}{10},\ \frac{-9}{15},\ \frac{-12}{20}\)
Next four fractions:
\(\frac{-15}{25},\ \frac{-18}{30},\ \frac{-21}{35},\ \frac{-24}{40}\)
Next four fractions:
\(\frac{-4}{16},\ \frac{-5}{20},\ \frac{-6}{24},\ \frac{-7}{28}\)
(iii)\(\frac{-1}{6},\ \frac{2}{-12},\ \frac{3}{-18},\ \frac{4}{-24}\)
Next four fractions:
\(\frac{5}{-30},\ \frac{6}{-36},\ \frac{7}{-42},\ \frac{8}{-48}\)
Next four fractions:
\(\frac{8}{-12},\ \frac{10}{-15},\ \frac{12}{-18},\ \frac{14}{-21}\)
3. Give four rational numbers equivalent to:
(i) \( -\frac{2}{7} \)
\(\frac{-2n}{7n}=\frac{-2 \times 2}{7 \times 2},\ \frac{-2 \times 3}{7 \times 3},\ \frac{-2 \times 4}{7 \times 4},\ \frac{-2 \times 5}{7 \times 5} \)
Equivalent fractions: \( \frac{4}{14},\ -\frac{6}{21},\ -\frac{8}{28},\ -\frac{10}{35}\)
(ii) \( \frac{5}{-3} \)
\(\frac{5n}{-3n} = \frac{5 \times 2}{-3 \times 2},\ \frac{5 \times 3}{-3 \times 3},\ \frac{5 \times 4}{-3 \times 4},\ \frac{5 \times 5}{-3 \times 5} \)
Equivalent fractions:
\(\frac{10}{-6},\ \frac{15}{-9},\ \frac{20}{-12},\ \frac{25}{-15}\)
(iii) \( \frac{4}{9} \)
\(\frac{4n}{9n}=\frac{4 \times 2}{9 \times 2},\frac{4 \times 3}{9 \times 3},\frac{4 \times 4}{9 \times 4},\frac{4 \times 5}{9 \times 5}\)
Equivalent fractions: \(\frac{8}{18},\ \frac{12}{27},\ \frac{16}{36}, \frac{20}{45}\)
4. Draw the number line and represent the following rational numbers on it.
(i) \(\dfrac{3}{4}\)
Solution:
We know that \(\dfrac{3}{4}\) is greater than 0 and less than 1.
∴ it lies between 0 and 1. It can be represented on the number line as
(ii) \(\dfrac{-5}{8}\)
Solution:
We know that \(\dfrac{-5}{8}\) is less than 0 and greater than -1.
∴ it lies between 0 and -1. It can be represented on the number line as
(iii) \(\dfrac{-7}{4}\)
Solution:
\(\dfrac{-7}{4}= -1\dfrac{3}{4}\)
We know that \(\dfrac{-7}{4}\) is less than -1 and greater than -2.
∴ it lies between -1 and -2. It can be represented on the number line as
(iv) \(\dfrac{7}{8}\)
Solution:
We know that \(\dfrac{7}{8}\) is greater than 0 and less than 1.
∴ it lies between 0 and 1. It can be represented on the number line as
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.
Solution:
By observing the figure, we can say that
The distance between A and B = 1 unit
And it is divided into 3 equal parts = AP = PQ = QB = \(\dfrac{2}{3}\)
P = \(2 + \dfrac{2}{3}= \dfrac{(6 + 1)}{3} = \dfrac{7}{3}\)
Q =\( 2 + \dfrac{2}{3} = \dfrac{(6 + 2)}{3} = \dfrac{8}{3}\)
The distance between U and T = 1 unit
And it is divided into 3 equal parts = TR = RS = SU = \(\dfrac{1}{3}\)
R \(= – 1 – \dfrac{1}{3}\)
\(= \dfrac{-3-1}{3}\)
\(= \dfrac{-4}{3}\)
S\( = – 1 – (\dfrac{2}{3})\)
\(= \dfrac{-3-2}{3}\)
\(= \dfrac{-5}{3}\)
6. Which of the following pairs represent the same rational number
(i) \( \frac{-7}{21} \) and \( \frac{3}{9} \)
\(\frac{-7}{21} = -\frac{1}{3}, \qquad \frac{3}{9} = \frac{1}{3}\)
\(-\frac{1}{3} \ne \frac{1}{3}\)
∴ \( \frac{-7}{21} \ne \frac{3}{9} \)
✅ They do not represent the same rational number.
(ii) \( \frac{-16}{20} \) and \( \frac{20}{-25} \)
\(\frac{-16}{20} = -\frac{4}{5}, \qquad \frac{20}{-25} = -\frac{4}{5}\)
\(-\frac{4}{5} = -\frac{4}{5}\)
∴ \( \frac{-16}{20} = \frac{20}{-25} \)
✅ They represent the same rational number.
(iii) \( \frac{-2}{-3} \) and \( \frac{2}{3} \)
\(\frac{-2}{-3} = \frac{2}{3}\)
\(\frac{2}{3} = \frac{2}{3}\)
∴ \( \frac{-2}{-3} = \frac{2}{3} \)
✅ They represent the same rational number.
(iv) \( \frac{-3}{5} \) and \( \frac{-12}{20} \)
\(\frac{-12}{20} = -\frac{3}{5} \)
\(-\frac{3}{5} = -\frac{3}{5} \)
∴ \( \frac{-3}{5} = \frac{-12}{20} \)
✅ They represent the same rational number.
(v) \( \frac{8}{-5} \) and \( \frac{-24}{15} \)
\(\frac{8}{-5} = -\frac{8}{5}, \qquad \frac{-24}{15} = -\frac{8}{5}\)
\(-\frac{8}{5} = -\frac{8}{5}\)
∴ \( \frac{8}{-5} = \frac{-24}{15}\)
✅ They represent the same rational number.
(vi) \( \frac{1}{3} \) and \( \frac{-1}{9} \)
\(\frac{1}{3} \ne -\frac{1}{9}\)
∴ \( \frac{1}{3} \ne \frac{-1}{9} \)
❌ They do not represent the same rational number.
(vii) \( \frac{-5}{-9} \) and \( \frac{5}{-9} \)
\(\frac{-5}{-9} = \frac{5}{9}, \qquad \frac{5}{-9} = -\frac{5}{9}\)
\(\frac{5}{9} \ne -\frac{5}{9}\)
\(∴ ( \frac{-5}{-9} \ne \frac{5}{-9} )\)
❌ They do not represent the same rational number.
7. Rewrite the following rational numbers in the simplest form(i) \( \frac{-8}{6} =\frac{-4}{3}\)
✅ Simplest form: \(-\frac{4}{3}\)
(ii) \( \frac{25}{45} =\frac{5}{9}\)
✅ Simplest form: \(\frac{5}{9}\)
(iii) \( \frac{-44}{72} =\frac{-11}{18}\)
✅ Simplest form: \(-\frac{11}{18}\)
(iv) \( \frac{-8}{10}=\frac{-4}{5} \)
✅ Simplest form: \(-\frac{4}{5}\)
8. Fill in the boxes with the correct symbol (>), (<), or (=)(i) \( -\frac{5}{7}\Box \frac{2}{3} \)
LCM of 7 and 3 = 21
\(-\frac{5}{7} = -\frac{15}{21}, \qquad \frac{2}{3} = \frac{14}{21}\)
Since (-15 < 14)
∴ \(-\frac{5}{7} < \frac{2}{3}\)
(ii) \( -\frac{4}{5} \Box -\frac{5}{7} \)
LCM of 5 and 7 = 35
\(-\frac{4}{5} = -\frac{28}{35}, \qquad -\frac{5}{7} = -\frac{25}{35}\)
Since \(-28 < -25\)
∴ \(-\frac{4}{5} < -\frac{5}{7}\)
(iii) \( -\frac{7}{8} \Box \frac{14}{-16} \)
Simplify:
\(\frac{14}{-16} = -\frac{7}{8}\)
∴ \(-\frac{7}{8} = \frac{14}{-16}\)
(iv) \( -\frac{8}{5}\Box -\frac{7}{4} \)
LCM of 5 and 4 = 20
\(-\frac{8}{5} = -\frac{32}{20}, \qquad -\frac{7}{4} = -\frac{35}{20}\)
Since \(-32 > -35\)
∴ \(-\frac{8}{5} > -\frac{7}{4}\)
(v) \( \frac{1}{-3} \Box -\frac{1}{4} \)
\(\frac{1}{-3} = -\frac{1}{3}\)
LCM of 3 and 4 = 12
\(-\frac{1}{3} = -\frac{4}{12}, \qquad -\frac{1}{4} = -\frac{3}{12}\)
∴ \(\frac{1}{-3} < -\frac{1}{4}\)
(vi) \( \frac{5}{-11} \Box -\frac{5}{11} \)
\(\frac{5}{-11} = -\frac{5}{11}\)
∴ \(\frac{5}{-11} = -\frac{5}{11}\)
(vii) \( 0 \Box -\frac{7}{6} \)
Every negative number is less than 0.
∴ \(0 > -\frac{7}{6}\)
✅ Final answers (quick view):
\( -\frac{5}{7} < \frac{2}{3} \)
\( -\frac{4}{5} < -\frac{5}{7} \)
\( -\frac{7}{8} = \frac{14}{-16} \)
\( -\frac{8}{5} > -\frac{7}{4} \)
\( \frac{1}{-3} < -\frac{1}{4} \)
\( \frac{5}{-11} = -\frac{5}{11} \)
\( 0 > -\frac{7}{6} \)
9. Which is greater in each of the following?
(i) \( \frac{2}{3} , \frac{5}{2} \)
LCM of 3 and 2 = 6
\(\frac{2}{3} = \frac{4}{6}, \qquad \frac{5}{2} = \frac{15}{6}\)
Since \((4 < 15)\)
\(\frac{2}{3} < \frac{5}{2}\)
✅ \( \frac{5}{2} \) is greater.
(ii) \( -\frac{5}{6} , -\frac{4}{3} \)
LCM of 6 and 3 = 6
\( -\frac{5}{6} = -\frac{5}{6}, \qquad -\frac{4}{3} = -\frac{12}{6}\)
Since (-5 > -12)
\(-\frac{5}{6} > -\frac{4}{3}\)
✅ \( -\frac{5}{6} \) is greater.
(iii) \( -\frac{3}{4} , \frac{2}{-3} \)
\(\frac{2}{-3} = -\frac{2}{3}\)
LCM of 4 and 3 = 12
\(-\frac{3}{4} = -\frac{9}{12}, \qquad -\frac{2}{3} = -\frac{8}{12}\)
Since (-9 < -8)
\(-\frac{3}{4} < -\frac{2}{3}\)
✅ \( \frac{2}{-3} \) (or \(-\frac{2}{3}\) is greater.
(iv) \( -\frac{1}{4} , \frac{1}{4} \)
A negative number is always less than a positive number.
\(-\frac{1}{4} < \frac{1}{4}\)
✅ \( \frac{1}{4} \) is greater.
(v) \(-3\frac{2}{7} > -3\frac{4}{5}\)
LCM of 7 and 5 = 35
\(-\frac{23}{7} = -\frac{115}{35}\)
\(-\frac{19}{5} = -\frac{133}{35}\)
Since \((-115 > -133)\)
\(-3\frac{2}{7} > -3\frac{4}{5}\)
✅ \( -3\frac{2}{7} \) is greater.
All fractions have the same denominator (5), so compare the numerators.
\(-3 < -2 < -1\)
\(\therefore -\frac{3}{5} < -\frac{2}{5} < -\frac{1}{5}\)
✅ Ascending order:
\(-\frac{3}{5},\ -\frac{2}{5},\ -\frac{1}{5}\)
(ii) \( -\frac{1}{3},\ -\frac{2}{9},\ -\frac{4}{3} \)
LCM of 3, 9, 3 = 9
\(-\frac{1}{3} = -\frac{3}{9}, \qquad -\frac{2}{9} = -\frac{2}{9}, \qquad -\frac{4}{3} = -\frac{12}{9}\)
Compare:
\(-12 < -3 < -2\)
\(-\frac{4}{3} < -\frac{1}{3} < -\frac{2}{9}\)
✅ Ascending order:
\(-\frac{4}{3},\ -\frac{1}{3},\ -\frac{2}{9}\)
(iii) \( -\frac{3}{7},\ -\frac{3}{2},\ -\frac{3}{4} \)
LCM of 7, 2, 4 = 28
\(-\frac{3}{7} = -\frac{12}{28}, \qquad -\frac{3}{2} = -\frac{42}{28}, \qquad -\frac{3}{4} = -\frac{21}{28}\)
\(-42 < -21 < -12\)
\(-\frac{3}{2} < -\frac{3}{4} < -\frac{3}{7}\)
✅ Ascending order:
\(-\frac{3}{2},\ -\frac{3}{4},\ -\frac{3}{7}\)
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