Maths Chapter-01: Rational Numbers

  1. Using appropriate properties find the value of \(\frac{-2}{3}\times \frac{3}{5}+\frac{5}{2}-\frac{3}{5}\times \frac{1}{6}\)

Sol. \(\frac{-2}{3}\times \frac{3}{5}+\frac{5}{2}-\frac{3}{5}\times \frac{1}{6}\) – \(\frac{-2}{3}\times \frac{3}{5}-\frac{3}{5}\times \frac{1}{6}+\frac{5}{2} \)

=\(\frac{-3}{5}(\frac{2}{3}+\frac{1}{6})+\frac{5}{2}\)

=\(\frac{-3}{5}(\frac{4+1}{6})+\frac{5}{2}\)

=\(\frac{-3}{5}(\frac{5}{6})+\frac{5}{2}\)

=\(\frac{-15}{30}+\frac{5}{2}\)

=\(\frac{-1}{2}+\frac{5}{2}\)

=\(\frac{4}{2}\)

=\(2\)

\(\ therefore\frac{-2}{3}\times \frac{3}{5}+\frac{5}{2}-\frac{3}{5}\times \frac{1}{6}\) =2

 

  1. Using appropriate properties find the value of \(\frac{2}{5}\times \frac{-3}{7}-\frac{1}{6}\times \frac{3}{2}+\frac{1}{14}\times \frac{2}{5}\)

Sol. \(\frac{2}{5}\times \frac{-3}{7}-\frac{1}{6}\times \frac{3}{2}+\frac{1}{14}\times \frac{2}{5}\)

= \(\frac{-2}{5}\times \frac{3}{7}-\frac{1}{2}\times \frac{1}{2}+\frac{1}{7}\times \frac{1}{5}\)

=\(\frac{-6}{35}-\frac{1}{4}+\frac{1}{35}\)

=\(\frac{-5}{35}-\frac{1}{4}\)

=\(\frac{-1}{7}-\frac{1}{4}\)

=\(\frac{-11}{28}\)

=\(\frac{2}{5}\times \frac{-3}{7}-\frac{1}{6}\times \frac{3}{2}+\frac{1}{14}\times \frac{2}{5}\)

=\(\frac{-11}{28}\)

 

  1. Solving the additive inverse of \(\frac{2}{8}\)

Sol. Additive inverse of \(\frac{2}{8} is \frac{-2}{8}\)

 

  1. Solving the additive inverse of \(\frac{-5}{9}\)

Sol. Additive inverse of \(\frac{-5}{9} is\frac{5}{9}\)

 

  1. Solving the additive inverse of \(\frac{-6}{-5}\)

Sol. \(\frac{-6}{-5}\frac{-6}{-5} \)

Additive inverse of \(\frac{6}{5} is\frac{-6}{5} \)

Additive inverse of \(\frac{-6}{-5}\frac{-6}{5} \)

 

  1. Solving the additive inverse of \(\frac{2}{-9}\)

Sol. Additive inverse of \(\frac{2}{-9} is\frac{2}{9} \)

 

  1. Verify that –(-x )=x for \( x=\frac{11}{15}\)

Sol. \(x = \frac{11}{15}\)

\(– x =\frac{-11}{15}\) \(-(-x)= -(\frac{-11}{15})\)

=\(\frac{11}{15}=x\)

\(\ therefore -(-x)=x\)

 

  1. Verify that –(-x )=x for \( x=\frac{-13}{17}\)

Sol. \(x = \frac{-13}{17}\)

\(– x =(-\frac{-13}{17})=\frac{13}{17}\) \(-(-x)= -(\frac{-13}{17})\)

=\(\frac{-13}{17}=x\)

\(\ therefore -(-x)=x\)

 

  1. Solve that multiplicative inverse of -13

Sol. Given multiplicative inverse -13 is\(\frac{-1}{13}\)

 

  1. Solve that multiplicative inverse of \(\frac{-13}{19}\)

Sol. Given multiplicative inverse \(\frac{-13}{19} is\frac{-19}{13} \)

 

  1. Solve that multiplicative inverse of \(\frac{-5}{8}\times \frac{-3}{7}\)

Sol. Given multiplicative inverse \(\frac{-5}{8}\times \frac{-3}{7}\)is \(\frac{8}{5}\times \frac{7}{3}\) or \(\frac{-8}{5}\times \frac{-7}{3}\)

 

  1. What is the multiplicative inverse of -1.

Sol. The multiplicative inverse of -1 is -1.

 

  1. Name the property under multiplicative used in each of the following.

(i) \(\frac{-4}{5}\times 1=1\times \frac{-4}{5}=\frac{-4}{5}\)

(ii) \(\frac{-13}{17}\times \frac{-2}{7}=\frac{-2}{7}\times \frac{-13}{17}\)

(iii) \(\frac{-19}{29}\times \frac{29}{-19}=1\)

Sol. (i) ROLE OF 1

(ii) COMMUTATIVITY

(iii) MULTIPLICATIVE INVERSE

 

  1. Multiply \(\frac{6}{13}\) by the reciprocal of \(\frac{-7}{6}\)

Sol. Reciprocal of \(\frac{-7}{6}\) is \(\frac{6}{-7}\)

\(\ therefore \frac{6}{13}\times \frac{6}{-7}\)

=\(\frac{-36}{91}\)

 

  1. what property allows you to compute \(\frac{1}{3}\times (6\times \frac{4}{3})(\frac{1}{3}\times 6)\times \frac{4}{5}\)

Sol. Associativity

 

  1. Is \(\frac{8}{9} \) the multiplicative inverse of \(-1\frac{1}{8}\) ? why or why not.

Sol.

\(-1\frac{1}{8} = \frac{-9}{8}\)

=\( \frac{8}{9}\times \frac{-9}{8}\)

=\(-1 \neq 1\)

\(\ therefore\frac{8}{9}\) is not the multiplicative inverse of \( -1\frac{1}{8}\)

 

  1. Is 0.3 the multiplicative inverse of \(3\frac{1}{3}\) ? why or why not.

Sol. \(3\frac{1}{3} = \frac{10}{3} =3.3\)

\( 3.3\times 0.3= 0.99\neq 1\)

\(\ therefore \, 0.3\, is\, not\, the\, multiplicative\, inverse\, of\, 3\frac{1}{3}\)

 

Exercise 1.1:

Question 1:

Solve the following using appropriate properties:

  1. ii)\(\frac{2}{5}* \frac{-3}{7} – \frac{1}{6} *\frac{3}{2}+\frac{1}{14}*\frac{2}{5}\)

Answers:

  1. ii) Use commutativity of rational numbers

\(\frac{2}{5}* \frac{-3}{7} – \frac{1}{6} *\frac{3}{2}+\frac{1}{14}*\frac{2}{5}\)

Using distributive property

\(\frac{2}{5} * [\frac{-3}{7} + \frac{1}{14}] – \frac{1}{4}\)

= \( \frac{2}{5} * \frac{-3*2+1}{14} – \frac{1}{4}\)

=\( \frac{2}{5} * \frac{-5}{14} – \frac{1}{4}\)

=\( \frac{-4-7}{28} = \frac{-11}{28}\)

 

Question 2:

(i) \( \frac{-5}{-7}\)

(ii) \( \frac{-4}{-7}\)

(iii)\( \frac{2}{-5}\)

(iv)\( \frac{5}{-9}\)

Solution:

(i) \( \frac{-5}{-7}\)

additive inverse = \( \frac{5}{-7}\)

(ii) \( \frac{-4}{-7}\) = \( \frac{4}{7}\)

Additive inverse = \( \frac{-4}{7}\)

(iii) \( \frac{2}{-5}\)

Additive inverse = \( \frac{2}{5}\)

(iv) \( \frac{5}{-9}\)

Additive inverse = \( \frac{5}{9}\)
Question 3:

(i) x = \( \frac{11}{15}\)

(ii) x= \( \frac{-13}{17}\)

Solution:

x = \( \frac{11}{15}\)

The additive inverse of is x = \( \frac{-11}{15}\)

This equality

x = \( \frac{11}{15}+frac{-11}{15}= 0\)

=\( -(-x) = x \)

x= \( \frac{-13}{17}\)

The additive inverse of x= \( \frac{-13}{17}\) = x= \( \frac{13}{17}\)

This equaliity

x = \( \frac{-13}{17}+frac{13}{17}= 0 \)

=\(-(-x) = x \)

 

Question 4:

(i) -12

(ii) \( \frac{-19}{13}\)

(iii) \( \frac{1}{3}\)

(iv) \( \frac{-3}{8} * \frac{-3}{8}\)

(v) \( -1 * \frac{-3}{5}\)

Solutions:

  1. The multiplicative inverse of -12 is \( \frac{1}{12}\)
  2. The multiplicative inverse of \( \frac{-19}{13}\) is\( \frac{-13}{19}\)
  3. The multiplicative inverse of \( \frac{1}{3}\) is \( \frac{3}{1}\) = 3
  4. The multiplicative inverse of \( -1 * \frac{-3}{5}\) is \( \frac{5}{3}\)

 

Question 5:

Find out the multiplication property that is used in the following sums:

(i) \( \frac{-3}{7} * 1 = 1 * \frac{-3}{7} = \frac{-3}{7}\)

(ii) \( \frac{-14}{17}* \frac{-3}{4} = \frac{-3}{4} * \frac{-14}{17}\)

(iii) \( \frac{-17}{37} * \frac{37}{-17} = 1 \)

Solutions:

  1. This use the multiplicative identity property and the multiplicative identity is 1
  2. This uses the commutativity property
  3. This uses the multiplicative inverse property

Question 6:

Find the product of \( \frac{7}{15}\) and the reciprocal of \( \frac{-4}{17}\)

Solution:

Reciprocal of \( \frac{-4}{17}\) = \( \frac{-17}{14}\)

\( \frac{7}{15}* \frac{-17}{14} = \frac{119}{60}\)

 

Question 7:

  1. Which rational number does not have a reciprocal
  2. Which rational number is equal to its own reciprocal
  3. Which rational number is equal to its own negative

Solution:

  1. Zero is the rational number that has no reciprocal
  2. One and negative one are rational numbers that are equal to their own reciprocal
  3. Zero is a rational number that is also equal to its own negative.

 

Question 8: Answer or complete the following sentences:

  1. Does zero have a reciprocal?
  2. The two numbers _____ and _________ are their own reciprocals.
  3. What is the reciprocal of -6?
  4. What is the reciprocal of ?
  5. The product of two rational numbers is always a __________.
  6. The reciprocal of a positive rational number is ___________.

 

Solutions:

  1. No it doesn’t
  2. 1, -1
  3. -1/6
  4. X
  5. Rational number
  6. Positive rational number

 

Question 9:

Find the multiplicative inverse of the following:

  1. -12
  2. \( \frac{-12}{18}\)
  3. \( \frac{2}{7}\)
  4. -1
  5. \( \frac{-3}{8} * \frac{-5}{7}\)
  6. \( -2 * \frac{-2}{7}\)

Solutions:

  1. \( \frac{-1}{12}\)
  2. \( \frac{-12}{18}\)
  3. \( \frac{7}{2}\)
  4. -1
  5. \( \frac{56}{15}\)
  6. \( \frac{7}{2}\)

 

Question 10:

Is 0.3 the multiplicative inverse of \( 3 \frac{1}{3}\)? Why or why not?

Answer:

\( 3 \frac{1}{3}= \frac{10}{3}\) \( 0.3*3 \frac{1}{3} = 0.3 * \frac{10}{3} = \frac{3}{10} * \frac{10}{3} = 1 \)

As proved above the product is one.

Therefore, 0.3 is the multiplicative inverse of \( 3 \frac{1}{3}\)