Find three rational numbers between, 5 7 and 6 7

We know, The numbers which can be expressed in the form of p q , where p and q are integers and q ≠ 0 are called rational numbers. Convert given fractions into equivalent fractions 5 7 = 5 × 10 7 × 10 = 50 70 6 7 = 6 × 10 7 × 10 = 60 70 Therefore, three rational numbers between 5 7 and 6 7 are 51 70 , 52 70 and 53 70

Find three rational numbers between, 5 7 and 6 7

We know, The numbers which can be expressed in the form of p q , where p and q are integers and q ≠ 0 are called rational numbers. Convert given fractions into equivalent fractions 5 7 = 5 × 10 7 × 10 = 50 70 6 7 = 6 × 10 7 × 10 = 60 70 Therefore, three rational numbers between 5 7 and 6 7 are 51 70 , 52 70 and 53 70

Find three rational numbers between, 1 4 and 1 5

We know, The numbers which can be expressed in the form of p q , where p and q are integers and q ≠ 0 are called rational numbers. Given: 1 4 and 1 5 L.C.M of 4 , 5 is 20 . 1 4 = 1 × 5 4 × 5 = 5 20 1 5 = 1 × 4 5 × 4 = 4 20 Convert into equivalent fractions 5 20 = 5 × 4 20 × 4 = 20 80 4 20 = 4 × 4 20 × 4 = 16 80 Therefore, three rational numbers between 1 4 and 1 5 are 17 80 , 18 80 and 19 80

Represent 7 on number line.

Steps to represent 7 on number line. (i) Draw a line segment A B = 7 and extend it to C such that B C = 1 . (ii) Draw the perpendicular bisector of A C and mark the mid point of A C as O (iii) With O as centre and O A as radius, draw the semicircle. (iv) At point B , draw a perpendicular B D which intersect the semicircle at D . (v) With B as centre and B D as radius, draw an arc which intersect A C produced at E . (vi) B D = B E = 7 and point E represent 7 Proof: A B = 7 c m , B C = 1 c m , A C = 7 + 1 = 8 c m O A = O B = 8 2 = 4 c m O B = O C - B C = 4 - 1 = 3 In △ O B D ⇒ B D 2 = O D 2 - O B 2 = 4 2 - 3 2 = 16 - 9 ⇒ B D 2 = 7 ⇒ B D = 7 ∴ B D = B E = 7 c m point represents 7

Represent 17 on number line.

(i) Draw a line segment A B = 17 and extend it to C such that BC = 1 unit . (ii) Draw the perpendicular bisector of A C and mark the mid point of A C as O (iii) With O as centre and O A as radius, draw the semicircle. (iv) At point B , draw a perpendicular B D which intersect the semicircle at D . (v) With B as centre and B D as radius, draw an arc which intersect A C produced at E . (vi) ∴ B D = B E = 17 and point E represent 17 Proof: A B = 17 c m , B C = 1 c m , A C = 17 + 1 = 18 c m O A = O B = 18 2 = 9 c m O B = O C - B C = 9 - 1 = 8 In △ O B D ⇒ B D 2 = O D 2 - O B 2 = 9 2 - 8 2 = 81 - 64 ⇒ B D 2 = 17 ⇒ B D = 17 ∴ B D = B E = 17 c m and point E represents 17

Represent 26 on number line.

Steps to represent 26 on number line. (i) Draw a line segment A B = 26 and extend it to C such that B C = 1 c m . (ii) Draw the perpendicular bisector of A C and mark the mid point of A C as O (iii) With O as centre and O A as radius, draw the semicircle. (iv) At point B , draw a perpendicular B D which intersect the semicircle at D . (v) With B as centre and B D as radius, draw an arc which intersect A C produced at E . (vi) ∴ B D = B E = 26 and point E represent 26 Proof: A B = 26 c m , B C = 1 c m , A C = 26 + 1 = 27 c m O A = O B = 27 2 = 13.5 c m O B = O C - B C = 13.5 - 1 = 12.5 In △ O B D ⇒ B D 2 = O D 2 - O B 2 = 13 .5 2 - 12.5 2 = 182.25 - 156.25 = 26 ⇒ B D = 26

Represent 7.2 on number line.

Steps to represent 7 . 2 on number line. (i) Draw a line segment A B = 7.2 c m and extend it to (such that B C = 1 c m ) (ii) Draw the perpendicular bisector of A C and mark the mid point of A C as O (iii) At point B , draw a perpendicular B D which intersect the semicircle at D . (iv) With B as centre and B D as radius, draw an arc which intersect A C produced at E . Therefore B D = 7.2 and point E represent 7.2 . Proof: A B = 7.2 c m , B C = 1 c m , A C = 7.2 + 1 = 8.2 c m O A = O B = 8.2 2 = 4.1 c m O B = O C - B C = 4.1 - 1 = 3.1 In △ O B D ⇒ B D 2 = O D 2 - O B 2 = 4.1 2 - 3.1 2 = 16.81 - 9.61 ⇒ B D 2 = 7.2 ⇒ B D = 7.2 ∴ B D = B E = 7.2 c m and point E represents 7.2 .

Represent the following on number line. 6.8

Steps to represent 6 . 8 on number line. (i) Draw a line segment A B = 6 . 8 and extend it to C such that B C = 1 c m . (ii) Draw the perpendicular bisector of A C and mark the mid point of A C as O (iii) With O as centre and O A as radius, draw the semicircle. (iv) At point B , draw a perpendicular B D which intersect the semicircle at D . (v) With B as centre and B D as radius, draw an arc which intersect A C produced at E . (vi) ∴ B D = B E = 6 . 8 and point E represent 6 . 8 Proof: A B = 6.8 cm , B C = 1 cm , A C = 6.8 + 1 = 7.8 cm O A = O C = 7.8 2 = 3.9 cm O B = O C - B C = 3.9 - 1 = 2.9 cm In △ O B D ⇒ B D 2 = O D 2 - O B 2 = 3 .9 2 - 2.9 2 = 15.21 - 8.41 ⇒ B D 2 = 6.8 ⇒ B D = 6.8 ∴ B D = B E = 6.8 c m and point in E represents 6.8 .

Represent the following on number line. 9.8

Steps to represent 9 . 8 on number line. (i) Draw a line segment A B = 9 . 8 and extend it to C such that B C = 1 c m . (ii) Draw the perpendicular bisector of A C and mark the mid point of A C as O (iii) With O as centre and O A as radius, draw the semicircle. (iv) At point B , draw a perpendicular B D which intersect the semicircle at D . (v) With B as centre and B D as radius, draw an arc which intersect A C produced at E . (vi) ∴ B D = B E = 9 . 8 and point E represent 9 . 8 Proof: A B = 9.8 c m B C = 1 c m A C = 9.8 + 1 = 10.8 c m O A = O B = 10.8 2 = 5.4 c m O B = O C - B C = 5.4 - 1 = 4.4 c m In △ O B D ⇒ B D 2 = O D 2 - O B 2 = 5.4 2 - 4.4 2 = 29.16 - 19.36 ⇒ B D 2 = 9.8 ⇒ B D = 9.8 ∴ B D = B E = 9.8 c m and point E represents 9.8 .

Represent the following on number line. 2.3

Steps to represent 2 . 3 on number line. (i) Draw a line segment A B = 2 . 3 and extend it to C such that B C = 1 c m . (ii) Draw the perpendicular bisector of A C and mark the mid point of A C as O (iii) With O as centre and O A as radius, draw the semicircle. (iv) At point B , draw a perpendicular B D which intersect the semicircle at D . (v) With B as centre and B D as radius, draw an arc which intersect A C produced at E . (vi) ∴ B D = B E = 2 . 3 and point E represent 2 . 3 Proof: A B = 2.3 c m B C = 1 c m A C = 2.3 + 1 = 3.3 c m O A = O B = 3.3 2 = 1.65 c m O B = O C - B C = 1.65 - 1 = 0.65 c m In △ O B D ⇒ B D 2 = O D 2 - O B 2 = 1.65 2 - 0.65 2 = 2.7225 - 0.4225 ⇒ B D 2 = 2.3 ⇒ B D = 2.3 ∴ B D = B E = 2.3 c m and point E represents 2.3