CBSE Class 8 Maths Rational Numbers HOTs

Very Short Answer type Questons

Question. Find a rational number between 1/2 and 1/4 such that its denominator is 8.
Answer : 3/8

Question. Find 3(1/7) + (-5/13)
Answer : 39/14

Question. Simplify : 12(2/3) x 2(1/2) x -5/8 x 0/3 x 8/5
Answer : 0

Question. Which number lies between 1/3 and 1/2
(a) 1/4
(b) 23/60
(c) 1/5
(d) 1/6
Answer : 23/60

Question. Find the missing number x, 5/6 + 8/3 = 8/3 - x
Answer : -5/6

Question. Identify the property of multiplication associated with the following statement 1/7 x 7/1 = 1
Answer : Multiplicative inverse.

Question. Simplify
3/8 X -11/5 + 3/5 + 3/8 X 1/5 - 11/4
Answer : -14/4 = -7/2

Question. What is the value of (2/3 + 3/4) + 1/4
Answer : 5/3

Question. Which number should be subtracted from 11/12 so that we obtain 3/4 ?
Answer : 5/3

Question. Simplify : (-6/7 x 4/5) x (-5/8)
Answer : 3/7

Question. How many ribbons of (11/10)m can be cut from a ribbon of length 5(1/2)m?
Answer : 5

Question. Which rational number has no reciprocal?
Answer : 0

Question. Simplify 2/9 x 7/10 - 1 + 2/9 x 1/5 + 4/5
Answer : 0

Question. Which of the following rational number lies between the rational numbers a and b.
(a) a + b / 2
(b) a + b
(c) a + b / 3
(d) (1 / 2) a + b
Answer : a + b / 2

Question. A designer needs 3/5 th of a metre of cloth to make a fancy dress for children taking part in a dance performance. If 200 children are taking part, how much cloth will the designer need?
Answer : 120m

Question. What is the multiplicative inverse of (-4/7) x (-7/4)
Answer : 1

Question. Simplify -8/15 + 4/5 - 4 + 23/15 - 9/5
Answer : –4

Question. Fill up the blank boxes of the magic square such that the sum of the numbers taken vertically, horizontally and diagonally remains the same.'

Answer : 

Question. 11/6 + 3/-4 = .........
Answer : 13/12

Question. What is the value of x 1/3 + 1/3 ÷ 1/3
Answer : 2

Question. Simplify
Answer : 1/3

Question. Find x if 5/8 + x = 25/24
Answer : 3/5

Question. What is the multiplicative inverse of -1(1/7)
Answer : –7/8

Question. The sum of two rational number is 9/13 If one of them is 5/26 then find the other rational number.
Answer : 1/2

Question. 0.3 x 3(1/3 = .........
Answer : 1

Question. Which number lies between 1/10 and 1/100
(a) 2/10
(b) 2/100
(c) 10/2
(d) 100/2
Answer : 2/100

Question. Simplify
Answer : 23.

Question. From a rope 12 m long, two pieces of length 3(3)/5 m and 2,3/10 m are cut off. What is the length of the remaining rope?
Answer : 6(1)/10

Question. A bus is moving at a speed of 40(2)/5 km/hr. How far will it travel in 6(2)/3 hours? If the bus is still at a distance of 10(2)/3 km from the final destination, how far is the final destination from the starting point?
Answer : 269(1)/3 km, 280km

Question. If 3/5 of a number exceeds its 2/7 by 44, find the number.
Answer : 140

Question. Which is longer and by how much, perimeter of a square of side 4(2)/3 m or perimeter of a rectangle with length 5(1)/5 m and breadth 3(2)/5 m?
Answer : Perimeter of square by 22/15 m

Question. Represent : 

CHALLENGES 

1. Simplify : 

2. Let p/q is a rational number such that HCF(p, q)=1. Write a/b = 1/q + 1/p+q . Is a/b in the irreducible form?

4. Find all natural numbers for which 3n+4/5 is an integer.

5. For what values of integer 'a', the rational number a²-5/a+4 is an integer?

6. Express 19/84 in the form 1/m + 1/n for some natural numbers m, n. In how many ways can you express in the given form?

7. Suppose a, b are two rational numbers: a = p/q , b= r/s , where HCF(p, q)=1 and HCF(r, s)=1. When is a/b an integer?

8. Suppose p/q = m/n , where HCF(m, n)=1. Prove that there exists an integer 'u' such that p=mu and q=nu.

9. Write 10 rational numbers between 1.1 and 1.11 in the form p/q. Can you write 100 numbers? Can you write 1000 numbers? What do you predict from your observations?

10. Suppose a, b are positive integers such that 1/a + 1/b is an integer. Prove that a=b. Prove also that 'a' can be 1 or 2, but no other possibility.

SUMMARY

1. A rational number is a number that can be expressed as p/q , where p and q are both integers and q ≠ 0.

2. If the numerator p and the denominator q have the same sign, it is a positive rational number, otherwise, it is a negative rational number.

3. Absolute value of a rational number is its numerical value with no regards to its sign.

4. a. If p/q is a rational number and n is an integer with n ≠ 0, then p/q = pxn/qxn = pn/qn 

b. If q/b is a rational number and n is an integer with n ≠ 0, then p/q = p÷n/q÷n 

5. p/q = r/s only when (p×s)=(q×r)

6. If p/q , r/s and u/v are three rational numbers, then 

7. If p/q and r/s are two rational numbers, then p/q x r/s = pxs/qxs 

8. If p/q, r/s and u/v are three rational numbers, then
a. p/q x r/s is also a rational number (Closure property)
b. p/q x r/s = r/s x p/q (Commutative property of multiplication) 

9. If p/q and r/s are two non-zero rational numbers, then p/q ÷ r/s = p/q x r/s

10. If p/q and r/s are two non-zero rational numbers, then p/q ÷ r/s is also a rational number.

11. For every rational number p/q,p/q ÷ 1 = p/q and p/q ÷ p/q = 1

12. If p/q and r/s are two non-zero rational numbers, then 1/2 x (p/q+r/s) is a rational number between p/q and r/s .

ERROR ANALYSIS

1. While adding two rational numbers, the students add the numerator and the denominator instead of strictly adding the numerators.

2. Some write the incorrect rational number.

3. Some misread the + sign as the x sign.

4. Students forget the division rule.

5. While multiplying rational numbers, students forget to put the negative sign with the answer.

6. Students forget to reduce the answer in the standard form or mixed fraction.

7. While adding rational numbers with different denominators, they forget to find the common denominator.

ACTIVITY I 

Multiplication of Rational Numbers 

ACTIVITY II

Addition / Subtraction of Rational Numbers