CBSE Class 8 Mathematics Rational Numbers Assignment Set 10

Question. Verify that \( |x+y| \le |x| + |y| \) if
(i) \( x = \frac{3}{5}, y = -\frac{4}{15} \)
(ii) \( x = -\frac{5}{12}, y = -\frac{7}{18} \)
Answer:
(i) LHS: \( |x + y| = \left| \frac{3}{5} + \left(-\frac{4}{15}\right) \right| = \left| \frac{9 - 4}{15} \right| = \left| \frac{5}{15} \right| = \frac{1}{3} \)
RHS: \( |x| + |y| = \left| \frac{3}{5} \right| + \left| -\frac{4}{15} \right| = \frac{3}{5} + \frac{4}{15} = \frac{9 + 4}{15} = \frac{13}{15} \)
Since \( \frac{1}{3} = \frac{5}{15} \) and \( \frac{5}{15} < \frac{13}{15} \), LHS \( \le \) RHS. (Verified)

(ii) LHS: \( |x + y| = \left| -\frac{5}{12} + \left(-\frac{7}{18}\right) \right| = \left| -\frac{15}{36} - \frac{14}{36} \right| = \left| -\frac{29}{36} \right| = \frac{29}{36} \)
RHS: \( |x| + |y| = \left| -\frac{5}{12} \right| + \left| -\frac{7}{18} \right| = \frac{5}{12} + \frac{7}{18} = \frac{15 + 14}{36} = \frac{29}{36} \)
Since \( \frac{29}{36} = \frac{29}{36} \), LHS \( \le \) RHS. (Verified)

Question. Verify that \( |x \times y| = |x| \times |y| \) if
(i) \( x = -\frac{5}{3}, y = \frac{2}{7} \)
(ii) \( x = -\frac{2}{3}, y = -\frac{9}{8} \)
Answer:
(i) LHS: \( |x \times y| = \left| -\frac{5}{3} \times \frac{2}{7} \right| = \left| -\frac{10}{21} \right| = \frac{10}{21} \)
RHS: \( |x| \times |y| = \left| -\frac{5}{3} \right| \times \left| \frac{2}{7} \right| = \frac{5}{3} \times \frac{2}{7} = \frac{10}{21} \)
LHS = RHS. (Verified)

(ii) LHS: \( |x \times y| = \left| -\frac{2}{3} \times \left(-\frac{9}{8}\right) \right| = \left| \frac{18}{24} \right| = \frac{3}{4} \)
RHS: \( |x| \times |y| = \left| -\frac{2}{3} \right| \times \left| -\frac{9}{8} \right| = \frac{2}{3} \times \frac{9}{8} = \frac{18}{24} = \frac{3}{4} \)
LHS = RHS. (Verified)

Question. Add:
(i) \( \frac{5}{4} \) and \( \frac{-11}{4} \)
(ii) \( \frac{-4}{5} \) and \( \frac{-7}{3} \)
Answer:
(i) \( \frac{5}{4} + \left(\frac{-11}{4}\right) = \frac{5 - 11}{4} = \frac{-6}{4} = -\frac{3}{2} \)
(ii) \( \frac{-4}{5} + \left(\frac{-7}{3}\right) = \frac{-12 - 35}{15} = -\frac{47}{15} \)

Question. If \( x = \frac{8}{5} \) then verify \( -(-x) = x \)
Answer:
LHS: \( -(-x) = -\left(-\frac{8}{5}\right) = \frac{8}{5} \)
RHS: \( x = \frac{8}{5} \)
LHS = RHS. (Verified)

Question. If \( x = \frac{2}{3}, y = \frac{5}{7} \) verify \( -(x + y) = (-x) + (-y) \)
Answer:
LHS: \( -(x + y) = -\left(\frac{2}{3} + \frac{5}{7}\right) = -\left(\frac{14 + 15}{21}\right) = -\frac{29}{21} \)
RHS: \( (-x) + (-y) = \left(-\frac{2}{3}\right) + \left(-\frac{5}{7}\right) = \frac{-14 - 15}{21} = -\frac{29}{21} \)
LHS = RHS. (Verified)

Question. Simplify by taking suitable order:
(i) \( \frac{2}{5} + \frac{8}{3} + \frac{-11}{3} + \frac{4}{5} - \frac{2}{15} \)
(ii) \( \frac{4}{7} + \frac{-8}{9} + \frac{-13}{7} + \frac{17}{21} \)
Answer:
(i) Grouping terms with similar denominators:
\( \left(\frac{2}{5} + \frac{4}{5}\right) + \left(\frac{8}{3} - \frac{11}{3}\right) - \frac{2}{15} \)
\( = \frac{6}{5} - 1 - \frac{2}{15} \)
\( = \frac{18 - 15 - 2}{15} = \frac{1}{15} \)

(ii) Grouping terms with similar denominators:
\( \left(\frac{4}{7} - \frac{13}{7}\right) + \frac{-8}{9} + \frac{17}{21} \)
\( = -\frac{9}{7} - \frac{8}{9} + \frac{17}{21} \)
LCM of denominators 7, 9, 21 is 63.
\( = \frac{-81 - 56 + 51}{63} = -\frac{86}{63} \)

Question. What should be added to \( \frac{19}{27} \) to make it \( \frac{-13}{15} \)?
Answer:
Let the required number to be added be \( a \).
\( \frac{19}{27} + a = \frac{-13}{15} \)
\( a = \frac{-13}{15} - \frac{19}{27} \)
LCM of 15 and 27 is 135.
\( a = \frac{-117 - 95}{135} = -\frac{212}{135} \)

Question. Name the property used in rational numbers.
If \( \frac{2}{5} \) and \( \frac{3}{2} \) are two rational numbers then
(i) \( \left(\frac{2}{5} + \frac{3}{2}\right) \) is also a rational number.
(ii) \( \left(\frac{2}{5} \times \frac{3}{2}\right) \) is also a rational number.
Answer:
(i) Closure Property for addition
(ii) Closure Property for multiplication

Question. Name the property used in rational number.
If \( \frac{5}{7} \) and \( \frac{3}{2} \) are two rational numbers then
(i) \( \left(\frac{5}{7} + \frac{3}{2}\right) = \left(\frac{3}{2} + \frac{5}{7}\right) = \frac{31}{14} \) is true
(ii) \( \left(\frac{5}{7} \times \frac{3}{2}\right) = \left(\frac{3}{2} \times \frac{5}{7}\right) = \frac{15}{14} \) is true
Answer:
(i) Commutative Property for addition
(ii) Commutative Property for multiplication

Question. State the property used in rational number.
If \( a, b, c \) are three rational numbers then
(i) \( (a+b)+c = a+(b+c) \) holds true
(ii) \( (a\times b)\times c = a\times (b\times c) \) holds true
Answer:
(i) Associative Property for addition
(ii) Associative Property for multiplication

Question. Name the property used in the following rational nos.
If \( \frac{1}{2}, \frac{1}{4}, \frac{1}{8} \) are three rational numbers then
\( \frac{1}{2} \times \left(\frac{1}{4} + \frac{1}{8}\right) = \frac{1}{2} \times \frac{1}{4} + \frac{1}{2} \times \frac{1}{8} \) holds true
Answer: Distributive Property

Question. Name the property holds in rational numbers
"If for the rational number \( \frac{5}{8} \), there exists another rational number \( \left(-\frac{5}{8}\right) \) such that \( \frac{5}{8} + \left(-\frac{5}{8}\right) = 0 \)"
Answer: Additive Inverse

Question. Give the name of property in rational number.
"If \( \frac{a}{b}, b \ne 0 \) is a rational number then there exist another rational number \( \frac{b}{a}, a \ne 0 \) such that \( \left(\frac{a}{b}\right) \times \left(\frac{b}{a}\right) = 1 \)"
Answer: Multiplicative Inverse