Rational Numbers Test

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Rational Numbers Test - 20

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Weekly Quiz Competition

Question 1
1 / -0

Fill the blank spaces: $$....... + \dfrac {3}{7} = ...... + \dfrac {3}{8}$$

A
$$\dfrac {3}{7}, \dfrac {3}{8}$$

B
$$\dfrac {3}{8}, \dfrac {3}{7}$$

C
$$\dfrac {8}{3}, \dfrac {8}{7}$$

D
$$\dfrac {8}{3}, \dfrac {7}{3}$$

Solution

Addition is commutative for rational numbers. i.e. $$a + b = b + a$$
$$\therefore \dfrac {3}{8} + \dfrac {3}{7} = \dfrac {3}{7} + \dfrac {3}{8}$$
$$\therefore$$ The missing terms are $$\dfrac {3}{8}, \dfrac {3}{7}$$
Question 2
1 / -0

The missing value in $$........ + \dfrac {2}{7} = \dfrac {2}{7} + \dfrac {-11}{13}$$ is

A
$$\dfrac {2}{7}$$

B
$$\dfrac {-11}{13}$$

C
$$\dfrac {-2}{7}$$

D
$$\dfrac {11}{13}$$

Solution

Addition is commutative for rational numbers. i.e. $$a + b = b + a$$
$$\therefore \dfrac {-11}{13} + \dfrac {2}{7} = \dfrac {2}{7} + \dfrac {-11}{13}$$
Question 3
1 / -0

Fill in the blank: $$\left (-\dfrac {1}{3}\right ) + \left [\left (\dfrac {-4}{3}\right ) + \dfrac {3}{7}\right ] = \left [\left (\dfrac {-1}{3}\right ) + ..... \right ] + \dfrac {3}{7}$$

A
$$\dfrac {-1}{3}$$

B
$$\dfrac {-4}{3}$$

C
$$\dfrac {3}{7}$$

D
$$\dfrac {-3}{7}$$

Solution

Addition is associative for rational numbers
i.e. $$a + (b + c) = (a + b) + c$$
$$\therefore \left (-\dfrac {1}{3}\right ) + \left [\left (\dfrac {-4}{3}\right ) + \dfrac {3}{7}\right ] = \left [\left (\dfrac {-1}{3}\right ) + \left (\dfrac {-4}{3}\right ) \right ] + \dfrac {3}{7}$$
Question 4
1 / -0

......... are associative for rational numbers.

A
Multiplication and addition

B
Multiplication and subtraction

C
Division and subtraction

D
Division and addition

Solution

Multiplication and addition are associative for rational numbers because when any three rational numbers are added or multiplied, the result so obtained will remain the same irrespective of the change in their order. i.e. $$a + (b + c) = (a + b) + c$$
$$a\times (b\times c) = (a\times b)\times c$$.
Question 5
1 / -0

Which property is depicted by $$\dfrac {1}{2} \times \left (6\times \dfrac {4}{3}\right ) = \left (\dfrac {1}{2} \times 6 \right )\times \dfrac {4}{3}$$?

A
Commutative

B
Closure

C
Associative

D
Distributive

Solution

For any three rational numbers, their product remains the same irrespective of their orders. i.e. $$a\times (b\times c) = (a\times b)\times c$$.
$$\therefore$$ Here, associative property is depicted.
Question 6
1 / -0

Find the missing value: $$-61 + ....... = \dfrac {2}{3} + (-61)$$

A
$$61$$

B
$$-6$$

C
$$\dfrac {2}{3}$$

D
$$\dfrac {-2}{3}$$

Solution

Addition is commutative for rational numbers. i.e. $$a + b = b + a.$$
$$\therefore -61 + \dfrac {2}{3} = \dfrac {2}{3} + (-61)$$
Question 7
1 / -0

................. are not associative for rational numbers.

A
Addition and multiplication

B
Subtraction and multiplication

C
Subtraction and division

D
Addition and division

Solution

Let us add three rational numbers $$\dfrac{-7}{5},\ \dfrac{2}{-11}$$ and $$\dfrac{-13}{25}$$

Then we know $$ \displaystyle \frac{-7}{5} + \left(\displaystyle \frac{2}{-11} + \frac{-13}{25} \right) = \left(\displaystyle \dfrac{-7}{5} + \frac{2}{-11} \right) + \frac{-13}{25}$$

So we can say that addition is associative for rational numbers.

Now let us multiply three rational numbers $$\dfrac 16,\ 6, \ \dfrac 43$$

Then we get $$\displaystyle \frac{1}{3}\times \left ( 6 \times\frac{4}{3} \right )= \left ( \frac{1}{3} \times 6\right )\times \frac{4}{3}$$

So we can say that multiplication is associative for rational numbers.

But when we divide three rational numbers in an order, the result so obtained will change if the order is changed.

For example: $$(81\div 9)\div 3\ne $$ $$81\div(9\div 3)$$

And when we subtract three rational numbers, the result changes when we change the order of subtraction

For example: $$5-(3-9)\ne (5-3)-2$$

Therefore, subtraction and division are not associative for rational numbers.
Question 8
1 / -0

Simplify using commutative and associative property :
$$\dfrac {2}{9} + \dfrac {-3}{5} + \dfrac {1}{3}$$

A
$$\dfrac {10}{18}$$

B
$$\dfrac {-2}{45}$$

C
$$\dfrac {2}{45}$$

D
$$\dfrac {2}{27}$$

Solution

$$\dfrac {2}{9} + \dfrac {-3}{5} + \dfrac {1}{3} = \dfrac {2}{9} + \left (\dfrac {-3}{5} + \dfrac {1}{3}\right )$$

$$= \dfrac {2}{9} + \left (\dfrac {1}{3} + \dfrac {-3}{5}\right )$$ (commutative property)

$$=\left (\dfrac {2}{9} + \dfrac {1}{3}\right ) + \dfrac {-3}{5}$$ (associative property)

$$= \dfrac {2 + 3}{9} + \dfrac {-3}{5}$$

$$= \dfrac {5}{9} + \dfrac {-3}{5}$$

$$= \dfrac {25 - 27}{45}$$

$$= \dfrac {-2}{45}$$
Question 9
1 / -0

................. is the additive inverse of $$\dfrac {-p}{q}$$, where $$\dfrac {-p}{q}$$ is a rational number.

A
$$\dfrac {p}{q}$$

B
$$\dfrac {p}{-q}$$

C
$$\dfrac {-p}{q}$$

D
$$\dfrac {q}{p}$$

Solution

If $$\left (\dfrac {-p}{q}\right )$$ is a rational number, then its negative $$\left [ - \left (\dfrac {-p}{q}\right ) = \dfrac {p}{q}\right ]$$ is called the additive inverse of $$\dfrac {-p}{q}$$.
Also addition of $$\left (\dfrac {-p}{q}\right )$$ and $$\left (\dfrac {p}{q}\right )$$ results zero.
Question 10
1 / -0

For any integer $$a$$, its additive inverse is ..........

A
$$-a$$

B
$$a$$

C
$$\dfrac {1}{a}$$

D
$$\dfrac {-1}{a}$$

Solution

The additive inverse of a number $$a$$ is the number that, when added to $$a$$ yields zero.

This number is also known as the negative, or negation.

If $$a$$ is an integer, then its negative or additive inverse is $$(-a)$$

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Rational Numbers Test - 20

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Rational Numbers Test - 20

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Question 1

$$\dfrac {3}{7}, \dfrac {3}{8}$$

$$\dfrac {3}{8}, \dfrac {3}{7}$$

$$\dfrac {8}{3}, \dfrac {8}{7}$$

$$\dfrac {8}{3}, \dfrac {7}{3}$$

Solution

Question 2

$$\dfrac {2}{7}$$

$$\dfrac {-11}{13}$$

$$\dfrac {-2}{7}$$

$$\dfrac {11}{13}$$

Solution

Question 3

$$\dfrac {-1}{3}$$

$$\dfrac {-4}{3}$$

$$\dfrac {3}{7}$$

$$\dfrac {-3}{7}$$

Solution

Question 4

Multiplication and addition

Multiplication and subtraction

Division and subtraction

Division and addition

Solution

Question 5

Commutative

Closure

Associative

Distributive

Solution

Question 6

$$61$$

$$-6$$

$$\dfrac {2}{3}$$

$$\dfrac {-2}{3}$$

Solution

Question 7

Addition and multiplication

Subtraction and multiplication

Subtraction and division

Addition and division

Solution

Question 8

$$\dfrac {10}{18}$$

$$\dfrac {-2}{45}$$

$$\dfrac {2}{45}$$

$$\dfrac {2}{27}$$

Solution

Question 9

$$\dfrac {p}{q}$$

$$\dfrac {p}{-q}$$

$$\dfrac {-p}{q}$$

$$\dfrac {q}{p}$$

Solution

Question 10

$$-a$$

$$a$$

$$\dfrac {1}{a}$$

$$\dfrac {-1}{a}$$

Solution

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