# Chapter 1: Rational Numbers

Exercise – 1.1

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 1.2:

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  $$3 \frac{1}{3}$$