# Grade 8 Operations on Rational Numbers Worksheet (For CBSE, ICSE, IAS, NET, NRA 2022)

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## (1) Evaluate. Express Your Answer as a Rational Number in Its Lowest Form

(a)

(b)

(c)

## (2) Evaluate. Express Your Answer as a Rational Number in Its Lowest Form

(a)

(b)

(c)

## (3) Evaluate. Express Your Answer as a Rational Number in Its Standard Form

(a)

(b)

(c)

## (4) Evaluate. Express Your Answer as a Rational Number in Its Standard Form

(a)

(b)

(c)

## (5) Evaluate. Express Your Answer as a Rational Number in Its Lowest Form

(a)

(b)

(c)

## (6) Find the Value of X. Express Your Answer as a Rational Number in Its Lowest Form

(a)

(b)

## (7) Find the Value of X. Express Your Answer as a Rational Number in Its Standard Form

(a)

(b)

## (8) Find the Value of X. Express Your Answer as a Rational Number in Its Lowest Form

(a)

(b)

## (9) Find the Value of X. Express Your Answer as a Rational Number in Its Standard Form

(a)

(b)

## (10) P, Q, R, and S Are Rational Numbers, Where P < 0, Q > 0, R < 0, and S > 0. Find the Sign of the Given Expressions. Write + for Positive, – for Negative, and X to Indicate That the Sign Cannot be Determined

(a)

(b)

(C)

## (11) a, B, C, and D Are Rational Numbers, Where a > 0, B < 0, C > 0, and D < 0. Find the Sign of the Given Expressions. Write + for Positive, – for Negative, and X to Indicate That the Sign Cannot be Determined

(a)

(b)

(C)

## (12) T, U, V, and W Are Rational Numbers, Where T < 0, U > 0, V < 0, and W > 0. Find the Sign of the Given Expressions. Write + for Positive, – for Negative, and X to Indicate That the Sign Cannot be Determined

(a)

(b)

(C)

## (13) W, X, Y, and Z Are Rational Numbers, Where W < 0, X > 0, Y < 0, and Z > 0. Find the Sign of the Given Expressions. Write + for Positive, – for Negative, and X to Indicate That the Sign Cannot be Determined

(a)

(b)

(C)

## (14) a Crane Lifts the Material According Signal It Receives from Signal Man. When Man Given Green Signal Operator Lift Material by 1/4 Meter and Man Give Red Signal Operator down Material by 1/5 Meter

Now write a sequence of signal that moves the material accordingly.

(a) Material lifted up to 6m.

(b) Material down by 4m.

## (15) a Crane Lift the Material According Signal It Receive from Signal Man. When Man Given Green Signal Operator Lift Material by 1/4 Meter and Man Give Red Signal Operator down Material by 1/5 Meter

Now write a sequence of signal that moves the material accordingly.

(a) Material lifted up to 8m

(b) Material down by 1m

## (16) a Crane Lift the Material According Signal It Receive from Signal Man. When Man Given Green Signal Operator Lift Material by 1/4 Meter and Man Give Red Signal Operator down Material by 1/5 Meter

(a) Due to mistake of signal man material was at same place hanged from when man start giving signal, so write a sequence of that signal.

(b) Can a sequence of signals lift or down material by meter? If yes, write that sequence. Otherwise, explain why it is not possible.

(17) Find rational number that lies between and .

(18) Find three rational number that lies between and .

## (19) Evaluate. Write the Answer in the Decimal Form

(a)

(b)

## (20) Evaluate. Write the Answer in the Decimal Form

(a)

## (21) Evaluate. Write the Answer in the Decimal Form

(a)

(22) If , evaluate the following.

(23) If , evaluate the following.

## Answers and Explanations

### Answer 1 (A)

- So,
- And lowest form of evaluated rational number is

### Answer 1 (B)

- So,
- And lowest form of evaluated rational number is

### Answer 1 (C)

- So,
- And lowest form of evaluated rational number is

### Answer 2 (A)

- So,
- And lowest form of evaluated rational number is

### Answer 2 (B)

- So,
- And for lowest form

- So, lowest form of derived rational number

### Answer 2 (C)

- So,
- For lowest form,

- So, lowest form of evaluated rational number is

### Answer 3 (A)

- So,
- For Standard form,

- So, standard form of evaluated rational number

### Answer 3 (B)

- So,
- So, standard form of evaluated rational number is

### Answer 3 (C)

- So,
- For standard form,

)

- So, standard form of evaluated rational number

### Answer 4 (A)

- So,
- For Standard form

- So, standard form of evaluated rational number

### Answer 4 (B)

- So,
- So, Standard form of evaluated rational number is

### Answer 4 (C)

- So,
- So, Standard form of evaluated rational number is

### Answer 5 (A)

- So,
- So, lowest form of evaluated rational number is

### Answer 5 (B)

- So,
- So, lowest form of evaluated rational number is

### Answer 5 (C)

- So,
- So, lowest form of evaluated rational number is

### Answer 6 (A)

- Now, for the lowest form of evaluated rational number:

- So, lowest form of rational number is

### Answer 6 (B)

- So, lowest form of evaluated rational number is

### Answer 7 (A)

- For, Standard form of evaluated rational number;

- So, standard form of evaluated rational number

### Answer 7 (B)

- Now, for Standard form of evaluated rational number;

- So, standard form of evaluated rational number

### Answer 8 (A)

- For, lowest form of evaluated x value;

- So, lowest form of evaluated x value is

### Answer 8 (B)

- For, lowest form of evaluated x value;

- So, lowest form of evaluated x value

### Answer 9 (A)

- Now, for standard form of evaluated x value,

- So, standard form of evaluated x value is

### Answer 9 (B)

- Now, for standard form of evaluated x value,

- So, standard form of evaluated x value

### Answer 10 (A)

- And given.
- Here P and R both are minus rational number.
- So, their summations also have minus rational number.
- Therefore,

### Answer 10 (B)

- And given.
- Here Q and S both are Positive rational number.
- So, its Multiplication also has positive rational number.
- Therefore,

### Answer 10 (C)

- Here given
- So, Q is positive rational number and R is negative rational number.
- So, we cannot exactly know the sign of its summation until we know its value.
- Therefore,

### Answer 11 (A)

- Here, A > 0 and B < 0 given.
- So, A is positive rational number and B is negative rational number.
- So, Division of one positive rational by another negative rational number is a negative rational number.
- Therefore,

### Answer 11 (B)

- Here, A > 0 and C > 0 given.
- So, A is positive rational number and C is also a positive rational number.
- So, multiplication of one positive rational by another positive rational number is a positive rational number.
- So,

### Answer 11 (C)

- Here, B < 0 and D < 0 given.
- So, B is negative rational number and D is also a negative rational number.
- So, Summation of one negative rational to another negative rational number is a negative rational number.
- So,

### Answer 12 (A)

- Here, T < 0 and U > 0 given.
- So, T is negative rational number and U is Positive rational number.
- So, Subtraction from one negative rational to another positive rational number is a negative rational number.
- So,

### Answer 12 (B)

- Here, V < 0 and W > 0 given.
- So, V is negative rational number and W is Positive rational number.
- So, for summation of one negative rational to another positive rational number one cannot judge the answer՚s sign it may be a positive rational number or may be a negative rational number.
- So,

### Answer 12 (C)

- Here, T < 0 and W > 0 given.
- So, T is negative rational number and W is Positive rational number.
- So, Division of one negative rational by another positive rational number is a negative rational number.
- So,

### Answer 13 (A)

- Here, W < 0 and X > 0 given.
- So, W is negative rational number and X is Positive rational number.
- So, Multiplication of one negative rational with another positive rational number is a negative rational number.
- So,

### Answer 13 (B)

- Here, X > 0 and Z > 0 given.
- So, X is Positive rational number and Z is also a Positive rational number.
- So, Division of one Positive rational with another positive rational number is also a positive rational number.
- So,

### Answer 13 (C)

- Here, Y < 0 and Z > 0 given.
- So, Y is negative rational number and Z is also a Positive rational number.
- So, subtraction of one negative rational by another positive rational number is also a negative rational number.
- So,

### Answer 14 (A)

- Material lifted up to 6m.
- Now, when operator receive green signal from signal man than he lift material only by meter.
- So,

- Operator receive green signal number of times

- So, sequence of signal that will lift material by 6 meter is
**24 times green signal**.

### Answer 14 (B)

- Material down by 4m.
- When operator receive red signal from signal man than he down the material by meter.

So,

- So, operator receives red signal number of times.

- So, sequence of signal that will Down material by 4 meter is
**20 times red signal**.

### Answer 15 (A)

- Material lifted up to 8m.
- Now, when operator receive green signal from signal man than he lift material only by meter.
- So,

- Operator receive green signal number of times

- So, sequence of signal that will lift material by 6 meter is 3
**2 times green signal**.

### Answer 15 (B)

- Material down by 1m.
- When operator receive red signal from signal man than he down the material by meter.

So,

- So, operator receives red signal number of times.

- So, sequence of signal that will Down material by 1 meter is
**5 times red signal**.

### Answer 16 (A)

- When operator receive green signal he lift material by meter and when receive red signal he down material by meter.
- So, to lift the material by 1 meter operator receive 4 green signals and to down material 1 meter operator should receive 5 red signals.
- So, in another way one can say that operator received 4 green signal and 5 red signal from signal man that՚s why material as hanged at same place since starting of activity.
- So sequence of signal is,
**4 green signals and 5 red signals**.

### Answer 16 (B)

- When operator receive green signal he lift material by meter and when receive red signal he down material by meter.
- So, material was lifted by number of meter is in multiplication of meter, because one cannot give half signal.
- Same way material was down by number of meter is in multiplication of meter, because one cannot give half signal.
- So, by summation of green and red signal material was lifted/down by,

- here we give minus sign to red signal to identify actual movement direction of material

- Here positive sign indicate lift of material.
- So, by one green and one red signal will lift the material by meter.
- Here 20 is H. C. F of 5 and 4 and is not dividable by 3 (here we consider 3 because we have to movement the material by meter) so one cannot lift or down the material by meter by giving any sequence of red or green signal.

### Answer (17)

- Here given two rational number are and .
- So, the rational number that lies between this given two rational number is;

- So, the rational number that lies between
**and****is**

### Answer (18)

- The number lies between and ;

- Now, Number lies between and ;

- Now, Number lies between and ;

- So, three number lies between and are

### Answer 19 (A)

- Here in , 45 is given in the bar form that means 45 is repetitive number after decimal point,
- So in another way we can write this decimal number as below.

- Same way for decimal number we can write as below.

- Now for summation of this two decimal bar number we can write as below,

- Now, for , here 81 is repetitive number after decimal point
- So, in another way we can write this decimal number as below,

- So,

### Answer 19 (B)

- Here in , 67 is given in the bar form that means 67 is repetitive number after decimal point,
- So in another way we can write this decimal number as below.

- Same way for decimal number we can write as below.

- Now for Subtraction of this two decimal bar number we can write as below,

- Now, for , here 4 is repetitive number after decimal point
- So, in another way we can write this decimal number as below,

- So,

### Answer 20 (A)

- For multiplication of bar number; we cannot directly multiply this bar number with given number 2.
- For that first we have to convert this bar number in to rational number than multiply this rational number with given number to get answer.
- So,

- Now, for multiplication;

- Now, for , here 90 is repetitive number after decimal point
- So, in another way we can write this decimal number as below,

- So,

### Answer 21 (A)

- For Division of bar number; we cannot directly divide bar number by given number 3.
- For that first we have to convert this bar number in to rational number than Divide this rational number by given number to get answer.
- So,

- Now for division,

- Now, for , here 23 is repetitive number after decimal point
- So, in another way we can write this decimal number as below,

- So,

### Answer (22)

- Here, given
- Now,

- So,

### Answer (23)

- So,