Solutions to NCERT Class 7 Math Chapter 8 Rational Numbers

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Solutions to Exercise 8.1 (Page No 133) of NCERT Class 7 Math Chapter 8 Rational Numbers –

1. List five rational numbers between:

(i) -1 and 0

Answer:

Let us write -1 and 0 as rational numbers with denominator 6 (the denominator 6 is chosen so that we are able to easily fit in the required 5 rational numbers in between -1 and 0).

We have, -1 = (-6)/6 and 0 = 0/6

So, (-6)/6 < (-5)/6 < (-4)/6 < (-3)/6 < (-2)/6 < (-1)/6 < 0/6

or, -1 < (-5)/6 < (-4)/6 < (-3)/6 < (-2)/6 < (-1)/6 < 0

Therefore, 5 rational numbers between -1 and 0 are: (-5)/6, (-4)/6, (-3)/6, (-2)/6, (-1)/6

Therefore, 5 rational numbers between -1 and 0 are: , , , ,

(ii) -2 and -1

Answer:

Let us write -2 and 1 as rational numbers with denominator 6 (the denominator 6 is chosen so that we are able to easily fit in the required 5 rational numbers in between -2 and -1).

We have, -2 = (-12)/6 and -1 = (-6)/6

So, (-12)/6 < (-11)/6 < (-10)/6 < (-9)/6 < (-8)/6 < (-7)/6 < (-6)/6

or, -2 < (-11)/6 < (-10)/6 < (-9)/6 < (-8)/6 < (-7)/6 < -1

Therefore, 5 rational numbers between -2 and 1 are: (-11)/6, (-10)/6, (-9)/6, (-8)/6, (-7)/6

(iii) (-4)/5 and (-2)/3

Answer:

We have found by trial and error that in order to be able to fit in 5 rational numbers between (-4)/5 and (-2)/3 we have to make the denominators = 45:

(-4)/5 = (-4 × 9)/(5 × 9) = (-36)/45 and (-2)/3 = (-2 × 15)/(3 × 15) = (-30)/45

So, (-36)/45 < (-35)/45 < (-34)/45 < (-33)/45 < (-32)/45 < (-31)/45 < (-30)/45

or, (-4)/5 < (-35)/45 < (-34)/45 < (-33)/45 < (-32)/45 < (-31)/45 < (-2)/3

Therefore, 5 rational numbers between (-4)/5 and (-2)/3 are: (-35)/45 < (-34)/45 < (-33)/45 < (-32)/45 < (-31)/45

(iv) (-1)/2 and 2/3

Answer:

We know that (-1)/2 lies to the left of 0 on the number line and 2/3 lies to the right of 0 on the number line.

You will find that, (-1)/2 < (-1)/3 < (-1)/4 < 0 < 1/3 < 1/2 < 2/3

Therefore, 5 rational numbers between (-1)/2 and 2/3 are: (-1)/3 < (-1)/4 < 0 < 1/3 < 1/2

2. Write four more rational numbers in each of the following patterns:

(i) (-3)/5, -(-6)/10, (-9)/15, (-12)/20, …..

Answer:

We have,

(-3)/5 = (-3 × 1)/(5 × 1) , (-6)/10 = (-3 × 2)/(5 × 2) , (-9)/15 = (-3 × 3)/(5 × 3) , (-12)/20 = (-3 × 4)/(5 × 4)

or, (-3 × 1)/(5 × 1) = (-3)/5 , (-3 × 2)/(5 × 2) = (-6)/10 , (-3 × 3)/(5 × 3) = (-9)/15 , (-3 × 4)/(5 × 4) = (-12)/20

Thus, we observe a pattern in these numbers. Let us maintain the same pattern to get the numbers.

Let us maintain the same pattern to get the numbers: (-3 × 5)/(5 × 5) = (-15)/25 , (-3 × 6)/(5 × 6) = (-18)/30 , (-3 × 7)/(5 × 7) = (-21)/35 , (-3 × 8)/(5 × 8) = (-24)/40

Therefore, the other numbers would be: (-15)/25 , (-18)/30 , (-21)/35 , (-24)/40

(ii) (-1)/4, (-2)/8, (-3)/12, …..

Answer:

We have,

(-1)/4 = (-1 × 1)/(4 × 1) , (-2)/8 = (-1 × 2)/(4 × 2) , (-3)/12 = (-1 × 3)/(4 × 3)

or, (-1 × 1)/(4 × 1) = (-1)/4 , (-1 × 2)/(4 × 2) = (-2)/8 , (-1 × 3)/(4 × 3) = (-3)/12

Thus, we observe a pattern in these numbers. Let us maintain the same pattern to get the numbers.

Let us maintain the same pattern to get four more numbers: (-1 × 4)/(4 × 4) = (-4)/16 , (-1 × 5)/(4 × 5) = (-5)/20 , (-1 × 6)/(4 × 6) = (-6)/24 , (-1 × 7)/(4 × 7) = (-7)/28

Therefore, the other numbers would be: (-4)/16 , (-5)/20 , (-6)/24 , (-7)/28

(iii) (-1)/6 , 2/(-12) , 3/(-18) , 4/(-24), …..

Answer:

We have,

(-1)/6 = (-1 × (-1))/(6 × (-1)) = 1/(-6)

Therefore, the given pattern can be written as: 1/(-6) , 2/(-12) , 3/(-18) , 4/(-24)

We have,

1/(-6) = (1 × 1)/(-6 × 1) , 2/(-12) = (1 × 2)/(-6 × 2) , 3/(-18) = (1 × 3)/(-6 × 3) = 3/(-18) , 4/(-24) = (1 × 4)/(-6 × 4)

or, (1 × 1)/(-6 × 1) = 1/(-6) , (1 × 2)/(-6 × 2) = 2/(-12) , (1 × 3)/(-6 × 3) = 3/(-18) , (1 × 4)/(-6 × 4) = 4/(-24)

Thus, we observe a pattern in these numbers. Let us maintain the same pattern to get the numbers.

Let us maintain the same pattern to get four more numbers: (1 × 5)/(-6 × 5) = 5/(-30) , (1 × 6)/(-6 × 6) = 6/(-36) , (1 × 7)/(-6 × 7) = 7/(-42) , (1 × 8)/(-6× 8) = 8/(-48)

Therefore, the other numbers would be: 5/(-30) , 6/(-36) , 7/(-42) , 8/(-48)

(iv) (-2)/3 , 2/(-3) , 4/(-6) , 6/(-9), …..

Answer:

We have,

2/(-3) = (-2 × (-1))/(3 × (-1)) , 4/(-6) = (-2 × (-2))/(3 × (-2)) , 6/(-9) = (-2 × (-3))/(3 × (-3))

or, (-2 × (-1))/(3 × (-1)) = 2/(-3) , (-2 × (-2))/(3 × (-2)) = 4/(-6) , (-2 × (-3))/(3 × (-3)) = 6/(-9)

Thus, we observe a pattern in these numbers. Let us maintain the same pattern to get the numbers.

Let us maintain the same pattern to get four more numbers: (-2 × (-4))/(3 × (-4)) = 8/(-12) , (-2 × (-5))/(3 × (-5)) = 10/(-15) , (-2 × (-6))/(3 × (-6)) = 12/(-18) , (-2 × (-7))/(3 × (-7)) = 14/(-21)

Therefore, the other numbers would be: 8/(-12) , 10/(-15) , 12/(-18) , 14/(-21)

3. Give four rational numbers equivalent to:

(i) (-2)/7

Answer:

We have,

(-2)/7 = (-2 × 2)/(7 × 2) = (-4)/14

(-2)/7 = (-2 × 3)/(7 × 3) = (-6)/21

(-2)/7 = (-2 × (-1))/(7 × (-1)) = 2/(-7)

(-2)/7 = (-2 × (-2))/(7 × (-2)) = 4/(-14)

Therefore, the four rational numbers equivalent to (-2)/7 are: (-4)/14 , (-6)/21 , 2/(-7) , 4/(-14)

(ii) 5/(-3)

Answer:

We have,

5/(-3) = (5 × (-2))/(-3 × (-2)) = (-10)/6

5/(-3) = (5 × (2))/(-3 × (2)) = 10/(-6)

5/(-3) = (5 × (4))/(-3 × (4)) = 20/(-12)

5/(-3) = (5 × (5))/(-3 × (5)) = 25/(-15)

(iii) 4/9

Answer:

We have,

4/9 = (4 × 2)/(9 × 2) = 8/18

4/9 = (4 × 3)/(9 × 3) = 12/27

4/9 = (4 × 4)/(9 × 4) = 16/36

4/9 = (4 × 6)/(9 × 6) = 24/54

4. Draw the number line and represent the following rational numbers on it:

(i) 3/4

Answer:

3/4 lies between 0 and 1. The number line between 0 and 1 is divided into 4 equal parts and 3/4 is marked on the number line as shown below:

(ii) (-5)/8

Answer:

(-5)/8 lies between 0 and -1. The number line between 0 and -1 is divided into 8 equal parts and (-5)/8 is marked on the number line as shown below:

(iii) (-7)/4

Answer:

(-7)/4 = (-4-3)/4 = (-4)/4 – 3/4 = -1 – 3/4

Therefore, (-7)/4 lies between -1 and -2. The number line between -1 and -2 is divided into 4 equal parts and (-7)/4 is marked on the number line as shown below:

(iv) 7/8

Answer:

7/8 lies between 0 and 1. The number line between 0 and 1 is divided into 8 equal parts and 7/8 is marked on the number line as shown below:

5. The points P, Q, R, S, T, U, A and B on the number line are such that, TR = RS = SU and AP = PQ = QB. Name the rational numbers represented by P, Q, R and S.

Answer:

P and Q lie on the number line between A and B.

The distance AB = 1 unit.

AP = PQ = QB = 1/3

Therefore,

P = 2 + 1/3 = 7/3

Q = 2 + 2/3 = 8/3

R and S lie between the number line between U and T.

The distance UT = 1 unit.

TR = RS = SU = 1/3

Therefore,

R = -1 – 1/3 = – 4/3

S = -1 – 2/3 = – 5/3

6. Which of the following pairs represents the same rational number?

(i) (-7)/21 and 3/9

Answer:

We convert both into their equivalent forms to compare them.

(-7)/21 = (-7 ÷ 7)/(21 ÷ 7) = (-1)/3

3/9 = (3 ÷ 3)/(9 ÷ 3) = 1/3

Since (-1)/3 ≠ 1/3, we can say that (-7)/21 and 3/9 are different rational numbers.

(ii) (-16)/20 and 20/(-25)

Answer:

We convert both into their equivalent forms to compare them.

(-16)/20 = (-16 ÷ 4)/(20 ÷ 4) = (-4)/5

20/(-25) = (20 ÷ 5)/(-25 ÷ 5) = 4/(-5) = (4 × (-1) )/(-5 × (-1)) = (-4)/5

Since their equivalent forms are equal, we can say that (-16)/20 and 20/(-25) represent the same rational number.

(iii) (-2)/(-3) and 2/3

Answer:

We convert (-2)/(-3) into its equivalent forms to compare them.

(-2)/(-3) = (-2 × (-1) )/(-3 × (-1)) = 2/3

Therefore, we can say that (-2)/(-3) and 2/3 represent the same rational number.

(iv) (-3)/5 and (-12)/20

Answer:

We convert (-12)/20 into its equivalent form.

(-12)/20 = (-12 ÷ 4)/(20 ÷ 4) = (-3)/5

Therefore, we can say that (-3)/5 and (-12)/20 represent the same rational number.

(v) 8/(-5) and (-24)/15

Answer:

We convert both into their equivalent forms to compare them.

8/(-5) = (8 ×(-1))/((-5) × (-1)) = (-8)/5

(-24)/15 = (-24 ÷ 3)/(15 ÷ 3) = (-8)/5

Since their equivalent forms are equal, we can say that 8/(-5) and (-24)/15 represent the same rational number.

(vi) 1/3 and (-1)/9

Answer:

We can write (-1)/9 as – 1/9

Also, we can see that 1/3 lies to the right of 0 on the number line and – 1/9 lies to the left of 0 on the number line.

So, 1/3 > – 1/9 and we say that 1/3 and (-1)/9 are different rational numbers.

(vi) (-5)/(-9) and 5/(-9)

Answer:

We convert both into their equivalent forms to compare them.

(-5)/(-9) = ((-5) × (-1))/((-9) × (-1)) = 5/9

5/(-9) = (5 × (-1))/((-9)× (-1)) = (-5)/9

Also, we can see that 5/9 lies to the right of 0 on the number line and (-5)/9 (which can be written as – 5/9) lies to the left of 0 on the number line.

So, 5/9 > (-5)/9 and we say that (-5)/(-9) and 5/(-9) are different rational numbers.

7. Rewrite the following rational numbers in the simplest form:

(i) (-8)/6

Answer:

We divide both the numerator and the denominator by the HCF of 8, 6 = 2.

So, (-8)/6 = (-8 ÷ 2)/(6 ÷ 2) = (-4)/3 (Reduced to simplest form)

(ii) 25/45

Answer:

We divide both the numerator and the denominator by the HCF of 25, 45 = 5.

So, 25/45 = (25 ÷ 5)/(45 ÷ 5) = 5/9 (Reduced to simplest form)

(iii) (-44)/72

Answer:

We ignore the negative sign and divide both the numerator and the denominator by the HCF of 44, 72 = 4.

So, (-44)/72 = (-44 ÷ 4)/(72 ÷ 4) = (-11)/18 (Reduced to simplest form)

(iv) (-8)/10

Answer:

We ignore the negative sign and divide both the numerator and the denominator by the HCF of 8, 10 = 2.

So, (-8)/10 = (-8 ÷ 2)/(10 ÷ 2) = (-4)/5 (Reduced to simplest form)

8. Fill in the boxes with the correct symbol out of >, <, and =.

(i) (-5)/7 [ ] 2/3

Answer:

To make the comparison easy, we convert both numbers into equivalent forms with the same denominator. The denominator = LCM of 7, 3 = 21.

(-5)/7 = (-5 × 3)/(7 × 3) = (-15)/21

2/3 = (2 × 7)/(3 × 7) = 14/21

Since 14/21 > (-15)/21 we can say that:

(-5)/7 [<] 2/3

(ii) (-4)/5 [ ] -5/7

Answer:

To make the comparison easy, we convert both numbers into equivalent forms with the same denominator. The denominator = LCM of 5, 7 = 35.

(-4)/5 = (-4 × 7)/(5 × 7) = (-28)/35

(-5)/7 = (-5 × 5)/(7 × 5) = (-25)/35

Since -28 < -25, we can say that (-28)/35 < (-25)/35 .

Therefore, (-4)/5 [<] -5/7

(iii) (-7)/8 [ ] 14/(-16)

Answer:

14/(-16) = (14 ÷ 2)/(-16 ÷ 2) = 7/(-8) = (7 × (-1))/(-8 × (-1)) = (-7)/8

So the numbers are both = (-7)/8

Therefore, (-7)/8 [=] 14/(-16)

(iv) (-8)/5 [ ] -7/4

Answer:

To make the comparison easy, we convert both numbers into equivalent forms with the same denominator. The denominator = LCM of 5, 4 = 20.

(-8)/5 = (-8 × 4)/(5 × 4) = (-32)/20

(-7)/4 = (-7 × 5)/(4 × 5) = (-35)/20

Since -35 < -32, we can say that (-35)/20 < (-32)/20.

Therefore, (-8)/5 [>] -7/4.

(v) 1/(-3) [ ] -1/4

Answer:

We can write 1/(-3) as (1 × (-1))/(-3 × (-1)) = (-1)/3

Now, we compare (-1)/3 and (-1)/4

To make the comparison easy, we convert both numbers into equivalent forms with the same denominator. The denominator = LCM of 3, 4 = 12.

(-1)/3 = (-1 × 4)/(3 × 4) = (-4)/12

(-1)/4 = (-1 × 3)/(4 × 3) = (-3)/12

Since -4 < -3, we can say that (-4)/12 < (-3)/12 .

Therefore, 1/(-3) [ <] -1/4

(vi) 5/(-11) [ ] -5/11

Answer:

We can write 5/(-11) = (5 × (-1))/(-11 × (-1)) = (-5)/11

So the numbers are both = (-5)/11

Therefore, 5/(-11) [=] -5/11

(vii) 0 [ ] -7/6

Answer:

We know (-7)/6 (which can be written as – 7/6) lies to the left of 0 on the number line.

Therefore, 0 [ > ] -7/6

9. Which is greater in each of the following:

(i) 2/3 , 5/2

Answer:

To make the comparison easy, we convert both numbers into equivalent forms with the same denominator. The denominator = LCM of 3, 2 = 6.

2/3 = (2 × 2)/(3 × 2) = 4/6

5/2 = (5 × 3)/(2 × 3) = 15/6

Since 15 > 4, we can say that 15/6 > 4/6 or 5/2 > 2/3.

Therefore, 5/2 is greater.

(ii) (-5)/6 , (-4)/3

Answer:

To make the comparison easy, we make the denominators the same. The denominator = LCM of 6, 3 = 6.

(-4)/3 = (-4 × 2)/(3 × 2) = (-8)/6

Since -5 > -8, we can say that (-5)/6 > (-8)/6 or (-5)/6 > (-4)/3.

Therefore, (-5)/6 is greater.

(iii) (-3)/4 , 2/(-3)

Answer:

2/(-3) = (2 × (-1))/(-3 × (-1)) = (-2)/3

Now, we compare (-3)/4 and (-2)/3 .

To make the comparison easy, we convert both numbers into equivalent forms with the same denominator. The denominator = LCM of 4, 3 = 12.

(-3)/4 = (-3 × 3)/(4 × 3) = (-9)/12

(-2)/3 = (-2 × 4)/(3 × 4) = (-8)/12

Since -8 > -9, we can say that (-8)/12 > (-9)/12 or (-2)/3 > (-3)/4 or 2/(-3) > (-3)/4 .

Therefore, 2/(-3) is greater.

(iv) (-1)/4 , 1/4

Answer:

We can see that both have a common denominator = 4.

Since, -1 < 1, we can say that (-1)/4 < 1/4 . Therefore, 1/4 is greater.

(v) -3 2/7 , -3 4/5

Answer:

Converting the mixed fractions into improper fractions:

-3 2/7 = (-23)/7

-3 4/5 = (-19)/5

To make the comparison easy, we convert both numbers into equivalent forms with the same denominator. The denominator = LCM of 7, 5 = 35.

(-23)/7 = (-23 × 5)/(7 × 5) = (-115)/35

(-19)/5 = (-19 × 7)/(5 × 7) = (-133)/35

Since -115 > -133, we can say that (-115)/35 > (-133)/35 or (-23)/7 > (-19)/5 or -3 2/7 > -3 4/5 .

Therefore, -3 2/7 is greater.

10. Write the following rational numbers in ascending order:

(i) (-3)/5 , (-2)/5, (-1)/5

Answer:

We can see that all three have a common denominator = 5.

Since, -3 < -2 < -1, we can say that (-3)/5 < (-2)/5 < (-1)/5 (arranged in ascending order).

(ii) (-1)/3 , (-2)/9, (-4)/3

Answer:

To make the comparison easy, we convert both numbers into equivalent forms with the same denominator. The denominator = LCM of 3, 9, 3 = 9.

(-1)/3 = (-1 × 3)/(3 × 3) = (-3)/9

(-2)/9 = (-2)/9

(-4)/3 = (-4 × 3)/(3 × 3) = (-12)/9

Since, -12 < -3 < -2, we can say that (-12)/9 < (-3)/9 < (-2)/9 or (-4)/3 < (-1)/3 < (-2)/9 (arranged in ascending order).

(iii) (-3)/7, (-3)/2, (-3)/4

Answer:

To make the comparison easy, we convert both numbers into equivalent forms with the same denominator. The denominator = LCM of 7, 2, 4 = 28.

(-3)/7 = (-3 × 4)/(7 × 4) = (-12)/28

(-3)/2 = (-3 × 14)/(2 × 14) = (-42)/28

(-3)/4 = (-3 × 7)/(4 × 7) = (-21)/28

Since, -42 < -21 < -12, we can say that (-42)/28 < (-21)/28 < (-12)/28 or (-3)/2 < (-3)/4 < (-3)/7 (arranged in ascending order).

Solutions to Exercise 8.2 (Page No 141) of NCERT Class 7 Math Chapter 8 Rational Numbers –

1. Find the sum:

(i) 5/4 + ((-11)/4)

Answer:

While adding rational numbers with same denominators, we add the numerators keeping the denominators the same.

5/4 + ((-11)/4) = (5 – 11)/4 = (-6)/4 = (-6 ÷ 2)/(4 ÷ 2) = (-3)/2

(ii) 5/3 + 3/5

Answer:

To make the addition easy, we convert both numbers into equivalent forms with the same denominator. The denominator = LCM of 3, 5 = 15.

5/3 = (5 × 5)/(3 × 5) = 25/15

3/5 = (3 × 3)/(5 × 3) = 9/15

While adding rational numbers with same denominators, we add the numerators keeping the denominators the same.

5/3 + 3/5 = 25/15 + 9/15 = (25 + 9)/15 = 34/15

(iii) (-9)/10 + 22/15

Answer:

To make the addition easy, we convert both numbers into equivalent forms with the same denominator. The denominator = LCM of 10, 15 = 30.

(-9)/10 = (-9 × 3)/(10 × 3) = (-27)/30

22/15 = (22 × 2)/(15 × 2) = 44/30

While adding rational numbers with same denominators, we add the numerators keeping the denominators the same.

(-9)/10 + 22/15 = (-27)/30 + 44/30 = (-27 + 44)/30 = 17/30

(iv) (-3)/(-11) + 5/9

Answer:

(-3)/(-11) = (-3 × (-1))/(-11 × (-1)) = 3/11

To make the addition easy, we convert both numbers into equivalent forms with the same denominator. The denominator = LCM of 11, 9 = 99.

3/11 = (3 × 9)/(11 × 9) = 27/99

5/9 = (5 × 11)/(9 × 11) = 55/99

While adding rational numbers with same denominators, we add the numerators keeping the denominators the same.

(-3)/(-11) + 5/9 = 3/11 + 5/9 = 27/99 + 55/99 = (27 + 55)/99 = 82/99

(v) (-8)/19 + ((-2))/57

Answer:

To make the addition easy, we convert both numbers into equivalent forms with the same denominator. The denominator = LCM of 19, 57 = 57.

(-8)/19 = (-8 × 3)/(19 × 3) = (-24)/57

(-2)/57 = (-2)/57

While adding rational numbers with same denominators, we add the numerators keeping the denominators the same.

(-8)/19 + ((-2))/57 = (-24)/57 + ((-2)/57) = (-24 – 2)/57 = (-26)/57

(vi) (-2)/3 + 0

Answer:

We have,

(-2)/3 + 0 = (-2)/3 + 0/3 = (-2 + 0)/3 = (-2)/3

(vii) -2 1/3 + 4 3/5

Answer:

Converting the mixed fractions into improper fractions:

-2 1/3 = – 7/3

4 3/5 = 23/5

Therefore, -2 1/3 + 4 3/5 = – 7/3 + 23/5

To make the addition easy, we convert both numbers into equivalent forms with the same denominator. The denominator = LCM of 3, 5 = 15.

(-7)/3 = (-7 × 5)/(3 × 5) = (-35)/15

23/5 = (23 × 3)/(5 × 3) = 69/15

While adding rational numbers with same denominators, we add the numerators keeping the denominators the same.

-7/3 + 23/5 = (-35)/15 + 69/15 = (-35 + 69)/15 = 34/15

Therefore, -2 1/3 + 4 3/5 = 34/15

2. Find the value of:

(i) 7/24 – 17/36

Answer:

When subtracting two rational numbers, we add the additive inverse of the rational number that is being subtracted, to the other rational number.

7/24 – 17/36

= 7/24 + additive inverse of 17/36

= 7/24 + ((-17))/36

Let us take the LCM of 24, 36 and make the denominators = LCM.

= (7 × 3)/(24 × 3) + ((-17 × 2)/(36 × 2))

= 21/72 + ((-34)/72)

= (21 – 34 )/72

= – 13/72

Therefore, 7/24 – 17/36 = – 13/72

(ii) 5/63 – ((-6)/21)

Answer:

When subtracting two rational numbers, we add the additive inverse of the rational number that is being subtracted, to the other rational number.

5/63 – ((-6)/21)

= 5/63 + additive inverse of (-6)/21

= 5/63 + 6/21

Let us take the LCM of 63, 21 and make the denominators = LCM.

= 5/63 + ((6 × 3)/(21 × 3))

= 5/63 + (18/63)

= (5 + 18 )/63

= 23/63

Therefore, 5/63 – ((-6)/21) = 23/63

(iii) (-6)/13 – ((-7)/15)

Answer:

When subtracting two rational numbers, we add the additive inverse of the rational number that is being subtracted, to the other rational number.

(-6)/13 – ((-7)/15)

= (-6)/13 + additive inverse of (-7)/15

= (-6)/13 + 7/15

Let us take the LCM of 13, 15 and make the denominators = LCM.

= (-6 × 15)/(13 × 15) + (7 × 13)/(15 × 13)

= (-90)/195 + 91/195

= (-90 + 91 )/195

= 1/195

Therefore, (-6)/13 – ((-7)/15) = 1/195

(iv) (-3)/8 – 7/11

Answer:

When subtracting two rational numbers, we add the additive inverse of the rational number that is being subtracted, to the other rational number.

(-3)/8 – 7/11

= (-3)/8 + additive inverse of 7/11

= (-3)/8 + ((-7))/11

Let us take the LCM of 8, 11 and make the denominators = LCM.

= (-3 × 11)/(8 × 11) + ((-7) × 8)/(11 × 8)

= (-33)/88 + ((-56))/88

= (-33 – 56 )/88

= (-89)/88

Therefore, (-3)/8 – 7/11 = (-89)/88

(v) –2 1/9 – 6

Answer:

When subtracting two rational numbers, we add the additive inverse of the rational number that is being subtracted, to the other rational number.

–2 1/9 – 6

= (-19)/9 – 6/1 (Converting the mixed fraction to an improper fraction)

= (-19)/9 + additive inverse of 6/1

= (-19)/9 + ((-6))/1

Let us take the LCM of 9, 1 and make the denominators = LCM.

= (-19)/9 + ((-6) × 9)/(1 × 9)

= (-19)/9 + ((-54))/9

= (-19 – 54 )/9

= (-73)/9

Therefore, –2 1/9 – 6 = (-73)/9

3. Find the product:

(i) 9/2 × ((-7)/4)

Answer:

Here the steps to follow:

Step 1: Multiply the numerators of the two rational numbers.

Step 2: Multiply the denominators of the two rational numbers.

Step 3: The product = (Result of Step 1 )/(Result of Step 2)

Thus,

9/2 × ((-7)/4) = (9 × (-7))/(2 × 4) = (-63)/8

(ii) 3/10 × (-9)

Answer:

Here the steps to follow:

Step 1: Multiply the numerators of the two rational numbers.

Step 2: Multiply the denominators of the two rational numbers.

Step 3: The product = (Result of Step 1 )/(Result of Step 2)

Thus,

3/10 × (-9) = (3 × (-9))/(10 × 1) = (-27)/10

(iii) (-6)/5 × 9/11

Answer:

Here the steps to follow:

Step 1: Multiply the numerators of the two rational numbers.

Step 2: Multiply the denominators of the two rational numbers.

Step 3: The product = (Result of Step 1 )/(Result of Step 2)

Thus,

(-6)/5 × 9/11 = (-6 × 9)/(5 × 11) = (-54)/55

(iv) 3/7 × ((-2)/5)

Answer:

Here the steps to follow:

Step 1: Multiply the numerators of the two rational numbers.

Step 2: Multiply the denominators of the two rational numbers.

Step 3: The product = (Result of Step 1 )/(Result of Step 2)

Thus,

3/7 × ((-2)/5) = (3 × (-2))/(7 × 5) = (-6)/35

(v) 3/11 × 2/5

Answer:

Here the steps to follow:

Step 1: Multiply the numerators of the two rational numbers.

Step 2: Multiply the denominators of the two rational numbers.

Step 3: The product = (Result of Step 1 )/(Result of Step 2)

Thus,

3/11 × 2/5 = (3 × 2)/(11 × 5) = 6/55

(vi) 3/(-5) × (-5)/3

Answer:

Here the steps to follow:

Step 1: Multiply the numerators of the two rational numbers.

Step 2: Multiply the denominators of the two rational numbers.

Step 3: The product = (Result of Step 1 )/(Result of Step 2)

Thus,

3/(-5) × (-5)/3 = (3 × (-5))/((-5) × 3) = (-15)/(-15) = (-15 ÷ (-15))/(-15 ÷ (-15)) = 1/1 = 1

4. Find the value of:

(i) (-4) ÷ 2/3

Answer:

To divide one rational number by another rational number, we multiply the rational number by the reciprocal of the other.

Therefore,

(-4) ÷ 2/3

= (-4)/1 × (Reciprocal of 2/3)

= (-4)/1 × 3/2

= (-4 × 3)/(1 × 2)

= (-12)/2

= -6

(ii) (-3)/5 ÷ 2

Answer:

To divide one rational number by another rational number, we multiply the rational number by the reciprocal of the other.

Therefore,

(-3)/5 ÷ 2/1

= (-3)/5 × (Reciprocal of 2/1)= (-3)/5 × 1/2

= (-3 × 1)/(5 × 2)

= (-3)/10

(iii) (-4)/5 ÷ (-3)

Answer:

To divide one rational number by another rational number, we multiply the rational number by the reciprocal of the other.

Therefore,

(-4)/5 ÷ ((-3))/1

= (-4)/5 × (Reciprocal of (-3)/1)

= (-4)/5 × 1/((-3))

= (-4 × 1)/(5 × (-3))

= (-4)/(-15)

= (-4 ÷ (-1))/(-15 ÷ (-1))

= 4/15

(iv) (-1)/8 ÷ (3/4)

Answer:

To divide one rational number by another rational number, we multiply the rational number by the reciprocal of the other.

Therefore,

(-1)/8 ÷ 3/4

= (-1)/8 × (Reciprocal of 3/4)

= (-1)/8 × 4/3

= (-1 × 4)/(8 × 3)

= (-4)/24

= (-4 ÷ 4)/(24 ÷ 4)

= (-1)/6

(v) (-2)/13 ÷ 1/7

Answer:

To divide one rational number by another rational number, we multiply the rational number by the reciprocal of the other.

Therefore,

(-2)/13 ÷ 1/7

= (-2)/13 × (Reciprocal of 1/7)

= (-2)/13 × 7/1

= (-2 × 7)/(13 × 1)

= (-14)/13

(vi) (-7)/12 ÷ ((-2)/13)

Answer:

To divide one rational number by another rational number, we multiply the rational number by the reciprocal of the other.

Therefore,

(-7)/12 ÷ ((-2)/13)

= (-7)/12 × (Reciprocal of (-2)/13)

= (-7)/12 × 13/((-2))

= (-7 × 13)/(12 × (-2))

= (-91)/(-24)

= (-91 × (-1))/(-24 × (-1))

= 91/24

(vii) 3/13 ÷ ((-4)/65)

Answer:

To divide one rational number by another rational number, we multiply the rational number by the reciprocal of the other.

Therefore,

3/13 ÷ ((-4)/65)

= 3/13× (Reciprocal of (-4)/65)

= 3/13× 65/((-4))

= (3 × 65)/(13 × (-4))

= 195/(-52)

= (195 ÷ 13)/(-52 ÷ 13 )

= 15/(-4)

= (15 × (-1) )/(-4 × (-1))

= (-15)/4

Extra Questions to Complement Solutions to NCERT Class 7 Mathematics Chapter 8 Rational Numbers:

Very Short Answer Type Questions:

1. Is the integer 5 a rational number?
Answer:
Yes, the integer 5 can be written as 5/1 which is a rational number.

2. Is (-2)/(-11) a negative rational number:
Answer:
No. We have (-2)/(-11) = (-2 × (-1))/(-11 × (-1)) = 2/11 which is a positive rational number.

3. Is 3/(-2) in standard form?
Answer:
No, 3/(-2) is not in standard form because the denominator is negative.

4. How many rational numbers exist between 1 and 2?
Answer:
An unlimited or infinite number of rational numbers exist between 1 and 2.

5. Are the rational numbers 4/9 and (-4)/(-9) equivalent?
Answer:
(-4)/(-9) = (-4 × (-1))/(-9 × (-1)) = 4/9 . Therefore, they are equivalent.

6. Express the product of x and y, where x and y are integers, as a rational number.
Answer:
x × y
= xy
= xy/1 (Expressed as a rational number)

7. In a rational number the numerator is greater than the denominator. What kind of fraction is this?
Answer:
This is an improper fraction.

8. Name a rational number which is neither positive nor negative.
Answer:
The rational number 0 is neither positive nor negative.

9. A car travels x/y miles towards east starting from his home and then from there a/b miles towards west. We are given that x/y > a/b. Where will be now from his home?
Answer:
Since x/y > a/b we will be at a distance of (x/y – a/b) east of his home.

10. The length of the side of a square is 9/4 cm. What is the perimeter?
Answer:
Perimeter of a square = 4 × 9/4 = 9 cm.

Multiple Choice Questions (MCQ):

1. Between which two integers does 5/(-7) lie on the number line?
(a) 0 and 1
(b) 0 and -1
(c) -1 and -2
(d) 1 and 2

Answer: (b) 0 and -1

5/(-7) = (5 × (-1))/(-7 × (-1)) = (-5)/7 . This lies between 0 and -1.

2. In a negative rational number in standard form:
(a) Numerator is negative and denominator is positive
(b) Denominator is negative and numerator is positive
(c) Numerator and denominator are both positive
(d) None of the above

Answer: (a) Numerator is negative and denominator is positive

3. If you multiply a number by its additive inverse, then the product will be:
(a) Negative
(b) Positive
(c) 0
(d) Depending on the number it can be positive or negative

Answer: (a) Negative

4. A rational number is positive. If it is multiplied by -k where k is any rational number, then the product is:
(a) Positive
(b) Negative
(c) Depends on whether the sign of k is positive or negative
(d) Depends on the specific value of k

Answer: (c) Depends on whether the sign of k is positive or negative

If k is positive then x × (-k) = -xk, which is negative
If k is negative then x × (-k) = xk, which is positive

5. Compare (-7)/6 and (-7)/(6 + k) where k is a positive integer.
(a) (-7)/6 is greater than (-7)/(6 + k)
(b) (-7)/(6 + k) is greater than (-7)/6
(c) Depends on the value of k
(d) The number (-7)/(6 + k) does not exist

Answer: (b) (-7)/(6 + k) is greater than (-7)/6

k is a positive integer.
So, (6 + k) is greater than 6.
We have,
7/6 > 7/(6 + k)
or, (-7)/(6 + k) > (-7)/6

Long and Short Answer Type Questions:

1. Reduce to standard form:

(i) 48/(-256) (ii) (-4)/(-64)

Answers:

(i) 48/(-256)

48/(-256)

= (48 ÷ (-16))/(-256 ÷ (-16))

= (-3)/16 (Standard form)

(ii) (-4)/(-64)

(-4)/(-64)

= (-4 ÷ (-4))/(-64 ÷ (-4))

= 1/16 (Standard form)

2. State which is greater among:

(i) (-3)/(-5) and (-4)/(-6) (ii) (-11)/12 and 35/(-36)

Answers:

(i) (-3)/(-5) and (-4)/(-6)

(-3)/(-5)

= (-3 × (-6))/(-5 × (-6))

= 18/30

(-4)/(-6)

= (-4 × (-5))/(-6 × (-5))

= 20/30

Since 20/30 > 18/30 we can say that (-4)/(-6) > (-3)/(-5) .

(ii) (-11)/12 and 35/(-36)

(-11)/12

= (-11 × 3)/(12 × 3)

= (-33)/36

35/(-36)

= (35 × (-1))/(-36 × (-1))

= (-35)/36

Since (-35)/36 > (-33)/36 we can say that 35/(-36) > (-11)/12 .

3. Find the sum:

(i) 3/5 + (-5)/(-3) (ii) (-5)/15 + 6/(-75)

Answers:

(i) 3/5 + (-5)/(-3)

= 3/5 + (-5 × (-1))/(-3 × (-1))

= 3/5 + 5/3

= (3 × 3)/(5 × 3) + (5 × 5)/(3 × 5)

= 9/15 + 25/15

= (9 + 25)/15

= 34/15 (Answer)

(ii) (-5)/15 + 6/(-75)

= (-5)/15 + 6/(-75)

= (-5 × 5)/(15 × 5) + (6 × (-1))/(-75 × (-1))

= (-25)/75 + (-6)/75

= (-25 – 6)/75

= (-31)/75 (Answer)

4. Find:

(i) (-5)/6 – ((-3)/4) (ii) 5 1/5 – 10

Answers:

(i) (-5)/6 – ((-3)/4)

= (-5)/6 + additive inverse of (-3)/4

= (-5)/6 + 3/4

= (-5 × 4)/(6 × 4) + (3 × 6)/(4 × 6)

= (-20)/24 + 18/24

= (-20 +18)/24

= (-2)/24

= (-2 ÷ 2)/(24 ÷ 2)

= (-1)/12 (Answer)

(ii) 5 1/5 – 10

= 26/5 – 10/1

= 26/5 – (10 × 5)/(1 × 5)

= 26/5 – 50/5

= (26 – 50)/5

= – 24/5 (Answer)

5. List 5 rational numbers lie between 1/5 and 1/2 ?

Answer:

We have,

1/5 = (1 × 4)/(5 × 4) = 4/20 and 1/2 = (1 × 10)/(2 × 10) = 10/20

Therefore,

4/20 < 5/20 < 6/20 < 7/20 < 8/20 < 9/20 < 10/20

Therefore,

5 rational numbers between 1/5 and 1/2 are 5/20 , 6/20 , 7/20 , 8/20 , 9/20 .

6. Write the next 2 numbers in the pattern:

2/3 , -4/9 , 8/27 , -16/81 ,….

Answer:

We have,

-4/9 = (2 × (-2))/(3 × 3)

8/27 = (2 × (-2) × (-2))/(3 × 3 × 3)-16/81 = (2 × (-2) × (-2) × (-2))/(3 × 3 × 3 × 3)

Next two numbers are:

(2 × (-2) × (-2) × (-2) × (-2))/(3 × 3 × 3 × 3 × 3) = 32/243 and

(2 × (-2) × (-2) × (-2) × (-2) × (-2) )/(3 × 3 × 3 × 3 × 3 × 3) = (-64)/729

7. Find the product as a rational number:

(i) 4 × 6 1/5 (ii) -2 1/5 × ((-7)/6)

Answer:

(i) 4 × 6 1/5

= 4 × 31/5

= 124/5 (Answer)

(ii) -2 1/5 × ((-7)/6)

= (-11)/5 × ((-7)/6)

= (-11 × (-7))/(5 × 6)

= 77/30 (Answer)

8. Find the value of:

(i) 4 2/5 ÷ (13/(-5)) (ii) 1 ÷ 1/13

Answers:

(i) 4 2/5 ÷ (13/(-5))

= 22/5 ÷ (13/(-5))

= 22/5 × (Reciprocal of 13/(-5))

= 22/5 × (-5)/13

= (22 × (-5))/(5 × 13)

= (-22)/13 (Answer)

(ii) 1 ÷ 1/13

= 1/1 × (Reciprocal of 1/13)

= 1/1 × 13/1

= (1 × 13)/(1 × 1)

= 13 (Answer)

9. Find the value of:

9/2 × 3/5 + 3/5 × 1/6

Answer:

9/2 × 3/5 + 3/5 × 1/6

= (9 × 3)/(2 × 5) + (3 × 1)/(5 × 6)

= 27/10 + 3/30

= (27 × 3)/(10 × 3) + 3/30

= 81/30 + 3/30

= (81 + 3)/30

= 84/30

= (84 ÷ 6)/(30 ÷ 6)

= 14/5 (Answer)

10. Find the value of:

15/4 ÷ 9/28 + 3 1/14 × 1/3

Answer:

15/4 ÷ 9/28 + 3 1/14 × 1/3

= 15/4 × (Reciprocal of 9/28) + 43/14 × 1/3

= 15/4 × 28/9 + 43/14 × 1/3

= (15 × 28)/(4 × 9) + (43 × 1)/(14 × 3)

= 35/3 + 43/42

= (35 × 14)/(3 × 14) + 43/42

= 490/42 + 43/42

= (490 + 43)/42

= 533/42 (Answer)

Fill in the Blanks:

(a) -1/3  (  ) -1/6

(b) When an integer is divided by another integer we get a _________ number.

(c) When the numerator and denominator are both negative integers, the rational number is _________.

(d) A rational number multiplied by negative of its reciprocal is equal to ________.

(e) All integers are _________ numbers but all rational numbers are not _________.

Answers:

(a) -1/3  ( < ) -1/6

(b) When an integer is divided by another integer we get a rational number.

(c) When the numerator and denominator are both negative integers, the rational number is positive.

(d) A rational number multiplied by negative of its reciprocal is equal to -1.

(e) All integers are rational numbers but all rational numbers are not integers.

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Frequently Asked Questions (FAQs) on NCERT Solutions to Class 7 Maths Chapter 8 Rational Numbers: