CBSE Class 8 Mathematics Rational Numbers Assignment Set 08

Question. Insert 5 rational numbers between \(-\frac{1}{3}\) and \(\frac{1}{2}\)
Answer: \(-\frac{3}{12}, -\frac{2}{12}, -\frac{1}{12}, \frac{0}{12}, \frac{1}{12}\) (This is one answer. There may be different other answers.)

Question. Insert 3 rational numbers between \(\frac{1}{6}\) and \(\frac{1}{3}\)
Answer: \(\frac{9}{48}, \frac{13}{48}, \frac{7}{24}\)

Question. Prove that \(\frac{2}{3} \times \left(\frac{11}{12} \times \frac{-15}{22}\right) = \left(\frac{2}{3} \times \frac{11}{12}\right) \times \frac{-15}{22}\)
Answer:
L.H.S. \(= \frac{2}{3} \times \left(\frac{11}{12} \times \frac{-15}{22}\right) = \frac{2}{3} \times \left(\frac{1}{12} \times \frac{-15}{2}\right) = \frac{2}{3} \times \left(-\frac{5}{8}\right) = -\frac{10}{24} = -\frac{5}{12}\)
R.H.S. \(= \left(\frac{2}{3} \times \frac{11}{12}\right) \times \frac{-15}{22} = \frac{11}{18} \times \frac{-15}{22} = \frac{1}{6} \times \frac{-5}{2} = -\frac{5}{12}\)
Since L.H.S. = R.H.S., the property is proved.

Question. Is \(\frac{8}{9}\) the multiplicative inverse of \(-1\frac{1}{8}\)? Why or why not.
Answer: No. Because \(-1\frac{1}{8} = -\frac{9}{8}\) and its product with \(\frac{8}{9}\) is \(\frac{8}{9} \times \left(-\frac{9}{8}\right) = -1\). For \(\frac{8}{9}\) to be the multiplicative inverse, the product must be \(1\).

Question. Verify \(-(x + y) = (-x) + (-y)\) if \(x = -\frac{2}{3}\) and \(y = -\frac{5}{7}\)
Answer:
L.H.S. \(= -(x + y) = -\left(-\frac{2}{3} + \left(-\frac{5}{7}\right)\right) = -\left(\frac{-14 - 15}{21}\right) = -\left(-\frac{29}{21}\right) = \frac{29}{21}\)
R.H.S. \(= (-x) + (-y) = -\left(-\frac{2}{3}\right) + \left(-\left(-\frac{5}{7}\right)\right) = \frac{2}{3} + \frac{5}{7} = \frac{14 + 15}{21} = \frac{29}{21}\)
Since L.H.S. = R.H.S., the property is verified.

Question. What should be added to \(\frac{19}{27}\) to make it \(-\frac{13}{15}\)
Answer: \(-\frac{212}{135}\)

Question. Simplify the following
(i) \(\left(-\frac{2}{3}\right) + \frac{5}{9} + \left(-\frac{7}{6}\right)\)
(ii) \(\frac{1}{12} + \left(-\frac{5}{18}\right) + \left(-\frac{7}{24}\right)\)
Answer:
(i) \(-\frac{23}{18}\)
(ii) \(-\frac{35}{72}\)

Question. Verify the property:- \(x \times (y \times z) = (x \times y) \times z\) if \(x = \frac{7}{4}\), \(y = -\frac{11}{3}\), \(z = \frac{1}{2}\)
Answer:
L.H.S. \(= x \times (y \times z) = \frac{7}{4} \times \left(-\frac{11}{3} \times \frac{1}{2}\right) = \frac{7}{4} \times \left(-\frac{11}{6}\right) = -\frac{77}{24}\)
R.H.S. \(= (x \times y) \times z = \left(\frac{7}{4} \times \left(-\frac{11}{3}\right)\right) \times \frac{1}{2} = -\frac{77}{12} \times \frac{1}{2} = -\frac{77}{24}\)
Since L.H.S. = R.H.S., the property is verified.

Question. Verify the property:- \(x \times (y + z) = (x \times y) + (x \times z)\) if \(x = -\frac{3}{4}\); \(y = \frac{5}{2}\); \(z = \frac{7}{6}\)
Answer:
L.H.S. \(= x \times (y + z) = -\frac{3}{4} \times \left(\frac{5}{2} + \frac{7}{6}\right) = -\frac{3}{4} \times \left(\frac{15 + 7}{6}\right) = -\frac{3}{4} \times \frac{22}{6} = -\frac{3}{4} \times \frac{11}{3} = -\frac{11}{4}\)
R.H.S. \(= (x \times y) + (x \times z) = \left(-\frac{3}{4} \times \frac{5}{2}\right) + \left(-\frac{3}{4} \times \frac{7}{6}\right) = -\frac{15}{8} - \frac{7}{8} = -\frac{22}{8} = -\frac{11}{4}\)
Since L.H.S. = R.H.S., the property is verified.

Question. Verify that \((x \div y) \times z \neq x \div (y \times z)\) if \(x = \frac{8}{15}\), \(y = \frac{2}{3}\), \(z = \frac{4}{10}\)
Answer:
L.H.S. \(= (x \div y) \times z = \left(\frac{8}{15} \div \frac{2}{3}\right) \times \frac{4}{10} = \left(\frac{8}{15} \times \frac{3}{2}\right) \times \frac{2}{5} = \frac{4}{5} \times \frac{2}{5} = \frac{8}{25}\)
R.H.S. \(= x \div (y \times z) = \frac{8}{15} \div \left(\frac{2}{3} \times \frac{4}{10}\right) = \frac{8}{15} \div \left(\frac{2}{3} \times \frac{2}{5}\right) = \frac{8}{15} \div \frac{4}{15} = \frac{8}{15} \times \frac{15}{4} = 2\)
Since L.H.S. \(\neq\) R.H.S., the inequality is verified.