CBSE Class 8 Mathematics Rational Numbers Assignment Set 09

Question. Verify the following closure property if \( \frac{1}{2}, \frac{1}{4} \) are two rational numbers then
(i) \( \left(\frac{1}{2} + \frac{1}{4}\right) \) is also a rational number
(ii) \( \left(\frac{1}{2} \times \frac{1}{4}\right) \) is also a rational number
Answer:
(i) \( \frac{1}{2} + \frac{1}{4} = \frac{2 + 1}{4} = \frac{3}{4} \), which is a rational number.
(ii) \( \frac{1}{2} \times \frac{1}{4} = \frac{1}{8} \), which is a rational number.

Question. Verify the following commutative property if \( \frac{3}{5} \) and \( \frac{2}{3} \) are two rational numbers then
(i) \( \left(\frac{3}{5} + \frac{2}{3}\right) = \left(\frac{2}{3} + \frac{3}{5}\right) \)
(ii) \( \left(\frac{3}{5} \times \frac{2}{3}\right) = \left(\frac{2}{3} \times \frac{3}{5}\right) \)
Answer:
(i) Addition commutative property:
L.H.S. = \( \frac{3}{5} + \frac{2}{3} = \frac{9 + 10}{15} = \frac{19}{15} \)
R.H.S. = \( \frac{2}{3} + \frac{3}{5} = \frac{10 + 9}{15} = \frac{19}{15} \)
Since L.H.S. = R.H.S., the commutative property of addition is verified.

(ii) Multiplication commutative property:
L.H.S. = \( \frac{3}{5} \times \frac{2}{3} = \frac{6}{15} = \frac{2}{5} \)
R.H.S. = \( \frac{2}{3} \times \frac{3}{5} = \frac{6}{15} = \frac{2}{5} \)
Since L.H.S. = R.H.S., the commutative property of multiplication is verified.

Question. Verify the following associative property. If \( \frac{7}{5}, \frac{2}{7} \) and \( \frac{5}{3} \) are three rational numbers then
(i) \( \left(\frac{7}{5} + \frac{2}{7}\right) + \frac{5}{3} = \frac{7}{5} + \left(\frac{2}{7} + \frac{5}{3}\right) \)
(ii) \( \left(\frac{7}{5} \times \frac{2}{7}\right) \times \frac{5}{3} = \frac{7}{5} \times \left(\frac{2}{7} \times \frac{5}{3}\right) \)
Answer:
(i) Addition associative property:
L.H.S. = \( \left(\frac{7}{5} + \frac{2}{7}\right) + \frac{5}{3} = \left(\frac{49 + 10}{35}\right) + \frac{5}{3} = \frac{59}{35} + \frac{5}{3} = \frac{177 + 175}{105} = \frac{352}{105} \)
R.H.S. = \( \frac{7}{5} + \left(\frac{2}{7} + \frac{5}{3}\right) = \frac{7}{5} + \left(\frac{6 + 35}{21}\right) = \frac{7}{5} + \frac{41}{21} = \frac{147 + 205}{105} = \frac{352}{105} \)
Since L.H.S. = R.H.S., the associative property of addition is verified.

(ii) Multiplication associative property:
L.H.S. = \( \left(\frac{7}{5} \times \frac{2}{7}\right) \times \frac{5}{3} = \frac{2}{5} \times \frac{5}{3} = \frac{2}{3} \)
R.H.S. = \( \frac{7}{5} \times \left(\frac{2}{7} \times \frac{5}{3}\right) = \frac{7}{5} \times \frac{10}{21} = \frac{2}{3} \)
Since L.H.S. = R.H.S., the associative property of multiplication is verified.

Question. Verify the distributive property. If \( \frac{1}{2}, \frac{1}{4}, \frac{1}{8} \) are three rational numbers, then \( \frac{1}{2} \times \left(\frac{1}{4} + \frac{1}{8}\right) = \left(\frac{1}{2} \times \frac{1}{4}\right) + \left(\frac{1}{2} \times \frac{1}{8}\right) \)
Answer:
L.H.S. = \( \frac{1}{2} \times \left(\frac{1}{4} + \frac{1}{8}\right) = \frac{1}{2} \times \left(\frac{2 + 1}{8}\right) = \frac{1}{2} \times \frac{3}{8} = \frac{3}{16} \)
R.H.S. = \( \left(\frac{1}{2} \times \frac{1}{4}\right) + \left(\frac{1}{2} \times \frac{1}{8}\right) = \frac{1}{8} + \frac{1}{16} = \frac{2 + 1}{16} = \frac{3}{16} \)
Since L.H.S. = R.H.S., the distributive property is verified.

Question. Find the additive inverse of the following:
(i) \( \left(-\frac{5}{8}\right) \)
(ii) \( \left(\frac{0}{1}\right) \)
(iii) \( -\left(-\frac{3}{2}\right) \)
(iv) \( \left(\frac{5}{12}\right) \)
Answer:
(i) Additive inverse of \( -\frac{5}{8} \) is \( \frac{5}{8} \)
(ii) Additive inverse of \( \frac{0}{1} \) is \( 0 \) (or \( \frac{0}{1} \))
(iii) Additive inverse of \( -\left(-\frac{3}{2}\right) = \frac{3}{2} \) is \( -\frac{3}{2} \)
(iv) Additive inverse of \( \frac{5}{12} \) is \( -\frac{5}{12} \)

Question. Find the multiplicative inverse of the following:
(i) \( \left(\frac{3}{2}\right) \)
(ii) \( \left(-\frac{3}{2}\right) \)
(iii) \( \left(\frac{0}{1}\right) \)
(iv) \( \left(-\frac{5}{12}\right) \)
Answer:
(i) Multiplicative inverse of \( \frac{3}{2} \) is \( \frac{2}{3} \)
(ii) Multiplicative inverse of \( -\frac{3}{2} \) is \( -\frac{2}{3} \)
(iii) Multiplicative inverse of \( \frac{0}{1} \) does not exist.
(iv) Multiplicative inverse of \( -\frac{5}{12} \) is \( -\frac{12}{5} \)

Question. Simplify using properties:
(i) \( \frac{2}{5} \times \frac{-3}{7} - \frac{1}{14} - \frac{3}{7} \times \frac{3}{5} \)
(ii) \( -\frac{2}{3} \times \frac{3}{5} + \frac{5}{2} - \frac{3}{5} \times \frac{1}{6} \)
Answer:
(i)
Rearranging the terms:
\( = \frac{2}{5} \times \left(-\frac{3}{7}\right) - \frac{3}{7} \times \frac{3}{5} - \frac{1}{14} \)
Using the distributive property:
\( = -\frac{3}{7} \times \left(\frac{2}{5} + \frac{3}{5}\right) - \frac{1}{14} \)
\( = -\frac{3}{7} \times (1) - \frac{1}{14} \)
\( = -\frac{6}{14} - \frac{1}{14} = -\frac{7}{14} = -\frac{1}{2} \)

(ii)
Rearranging the terms:
\( = -\frac{2}{3} \times \frac{3}{5} - \frac{3}{5} \times \frac{1}{6} + \frac{5}{2} \)
Using the distributive property:
\( = \frac{3}{5} \times \left(-\frac{2}{3} - \frac{1}{6}\right) + \frac{5}{2} \)
\( = \frac{3}{5} \times \left(\frac{-4 - 1}{6}\right) + \frac{5}{2} \)
\( = \frac{3}{5} \times \left(-\frac{5}{6}\right) + \frac{5}{2} \)
\( = -\frac{1}{2} + \frac{5}{2} = \frac{4}{2} = 2 \)

Question. Simplify the following:
(i) \( \frac{2}{5} + \frac{8}{3} + \frac{-11}{15} + \frac{4}{5} + \frac{-2}{3} \)
(ii) \( \frac{4}{7} + \frac{0}{1} + \frac{-8}{9} + \frac{-13}{7} + \frac{17}{21} \)
Answer:
(i) Grouping terms with same denominators:
\( = \left(\frac{2}{5} + \frac{4}{5}\right) + \left(\frac{8}{3} - \frac{2}{3}\right) - \frac{11}{15} \)
\( = \frac{6}{5} + 2 - \frac{11}{15} \)
\( = \frac{18 + 30 - 11}{15} = \frac{37}{15} \)

(ii) The LCM of 7, 9, and 21 is 63:
\( = \frac{4 \times 9}{63} + 0 - \frac{8 \times 7}{63} - \frac{13 \times 9}{63} + \frac{17 \times 3}{63} \)
\( = \frac{36 - 56 - 117 + 51}{63} = \frac{-86}{63} \)

Question. Simplify the rational numbers and examine whether they are equal:
\( \frac{7}{12} \times \left(\frac{28}{13} - \frac{5}{11}\right) \) OR \( \frac{7}{12} \times \frac{28}{13} - \frac{7}{12} \times \frac{5}{11} \)
Answer:
First Expression:
\( \frac{7}{12} \times \left(\frac{28}{13} - \frac{5}{11}\right) = \frac{7}{12} \times \left(\frac{308 - 65}{143}\right) = \frac{7}{12} \times \frac{243}{143} = \frac{1701}{1716} \text{ (or } \frac{567}{572}\text{)} \)

Second Expression:
\( \frac{7}{12} \times \frac{28}{13} - \frac{7}{12} \times \frac{5}{11} = \frac{196}{156} - \frac{35}{132} \)
Using the LCM of 156 and 132, which is 1716:
\( = \frac{196 \times 11 - 35 \times 13}{1716} = \frac{2156 - 455}{1716} = \frac{1701}{1716} \text{ (or } \frac{567}{572}\text{)} \)

Since both expressions evaluate to the same value, they are equal.

Question. Simplify using suitable grouping:
\( \frac{3}{5} \times \frac{1}{2} \times \frac{7}{3} \times \frac{5}{4} \times \frac{2}{7} \times \frac{4}{1} \)
Answer:
Grouping terms and their reciprocals:
\( = \left(\frac{3}{5} \times \frac{5}{4} \times \frac{4}{1}\right) \times \left(\frac{1}{2} \times \frac{7}{3} \times \frac{2}{7}\right) \)
\( = (3) \times \left(\frac{1}{3}\right) = 1 \)