Comprehensive, line-by-line step-by-step solutions for DAV Class 7 Secondary Mathematics Chapter 2: Operations on Rational Numbers (Brain Teasers, HOTS & Value Based Questions). This guide includes handwritten-style workings for all questions of Brain Teasers (Q.1 to Q.5), Higher Order Thinking Skills (HOTS), and Value-Based Questions (VBQs) aligned strictly with the DAV Board curriculum.
DAV Class 7 Maths Chapter 2: all worksheets All Class 7 Maths chapters →
📑 Quick Navigation & Worksheet Directory
Q.1. (A) Tick (✓) the correct option
(a) The sum of a rational number and its additive inverse is always:
- (i) 1
- (ii) 0
- (iii) −1
- (iv) Same rational number
By definition of additive inverse, for any rational number a/b, its additive inverse is −a/b.
a/b + (−a/b) = 0.
Correct Option: (ii) 0
(b) The reciprocal of −1 is:
- (i) 1
- (ii) −1
- (iii) 0
- (iv) Does not exist
Reciprocal of x is 1/x.
Reciprocal of −1 = 1/(−1) = −1.
Correct Option: (ii) −1
(c) Which rational number does not have a reciprocal?
- (i) 1
- (ii) −1
- (iii) 0
- (iv) 1/2
0 can be written as 0/1. Its reciprocal would be 1/0, which is undefined (division by zero is not permissible in mathematics).
Correct Option: (iii) 0
(d) If x = 2/3 and y = −1/4, then the value of (x + y) ÷ (x − y) is:
- (i) 5/11
- (ii) 11/5
- (iii) −5/11
- (iv) 1
1. x + y = 2/3 + (−1/4) = (8 − 3)/12 = 5/12.
2. x − y = 2/3 − (−1/4) = 2/3 + 1/4 = (8 + 3)/12 = 11/12.
3. (x + y) ÷ (x − y) = 5/12 ÷ 11/12 = 5/12 × 12/11 = 5/11.
Correct Option: (i) 5/11
(e) Multiplicative inverse of a negative rational number is:
- (i) A positive rational number
- (ii) A negative rational number
- (iii) 0
- (iv) 1
Reciprocal of −a/b is −b/a. The sign does not change when taking the multiplicative inverse.
Correct Option: (ii) A negative rational number
Q.1. (B) Answer the following questions
(a) Write the additive inverse of −5/9.
The additive inverse of a rational number x is −x.
Additive inverse of −5/9 = −(−5/9) = 5/9.
Ans: 5/9
(b) Find the reciprocal of (−3/8 × −4/7).
Step 1: Simplify the product:
Product = (−3 × −4)/(8 × 7) = 12/56.
Dividing numerator and denominator by 4: 12 ÷ 4/56 ÷ 4 = 3/14.
Step 2: Take the reciprocal:
Reciprocal of 3/14 = 14/3 (or 42/3).
Ans: 14/3
(c) Is division commutative for rational numbers? Justify with an example.
No, division is not commutative for rational numbers.
Justification / Example:
Let a = 1/2 and b = 1/4.
• a ÷ b = 1/2 ÷ 1/4 = 1/2 × 4/1 = 2.
• b ÷ a = 1/4 ÷ 1/2 = 1/4 × 2/1 = 1/2.
Since 2 ≠ 1/2, a ÷ b ≠ b ÷ a.
Ans: No
(d) What should be added to −7/12 to get 5/8?
Let the required number to be added be x.
According to the question: −7/12 + x = 5/8
⇒ x = 5/8 − (−7/12) = 5/8 + 7/12
LCM of denominators 8 and 12:
• 8 = 23; 12 = 22 × 3 ⇒ LCM = 24.
⇒ x = (5 × 3 + 7 × 2)/24 = (15 + 14)/24 = 29/24 (or 15/24).
Ans: 29/24
(e) What is the product of a non-zero rational number and its multiplicative inverse?
For any non-zero rational number a/b, its multiplicative inverse (reciprocal) is b/a.
Product = a/b × b/a = 1.
Ans: 1
Q.2. Evaluate the following expressions
(i) Evaluate: (−3/5 × 5/6) + (9/12 ÷ 3/−4)
(−3/5 × 5/6) = (−3 × 5)/(5 × 6) = −15/30 = −1/2.
Step 2: Simplify the second bracket:
(9/12 ÷ 3/−4) = 9/12 × (−4/3) = (9 × −4)/(12 × 3) = −36/36 = −1.
Step 3: Combine both terms:
−1/2 + (−1) = −1/2 − 1/1 = (−1 − 2)/2 = −3/2 (or −1½).
Ans: −3/2
(ii) Evaluate: (−2/3 + 5/6) − (−1/4 × 8/3)
−2/3 + 5/6 = (−4 + 5)/6 = 1/6.
Step 2: Simplify second bracket:
−1/4 × 8/3 = (−1 × 8)/(4 × 3) = −8/12 = −2/3.
Step 3: Subtract:
1/6 − (−2/3) = 1/6 + 2/3 = (1 + 4)/6 = 5/6.
Ans: 5/6
Q.3. Simplify using suitable properties (Distributive Property)
(i) 7/8 × (−3/12) + 7/8 × (5/12)
Formula: a × b + a × c = a × (b + c)
Here, a = 7/8, b = −3/12, c = 5/12.
= 7/8 × [−3/12 + 5/12]
= 7/8 × [(−3 + 5)/12]
= 7/8 × [2/12]
= 7/8 × 1/6
= (7 × 1)/(8 × 6) = 7/48.
Ans: 7/48
Q.4. Word Problems on Operations on Rational Numbers
(i) The sum of two rational numbers is −5. If one of the numbers is 2/3, find the other number.
Given: First Number + x = Sum
⇒ 2/3 + x = −5
⇒ x = −5 − 2/3
⇒ x = (−5 × 3)/3 − 2/3
⇒ x = (−15 − 2)/3 = −17/3 (or −52/3).
Ans: −17/3
(ii) From a rope of length 40 metres, a man cuts equal-sized pieces. How many pieces can be cut if each piece is of length 4/9 metre?
Length of each piece = 4/9 m
Number of pieces = (Total length) ÷ (Length of one piece)
= 40 ÷ 4/9
= 40 × 9/4
= 10 × 9 = 90 pieces.
Ans: 90 pieces
(iii) The product of two rational numbers is −8/9. If one of the numbers is −4/15, find the other.
Given: y × (−4/15) = −8/9
⇒ y = −8/9 ÷ (−4/15)
⇒ y = −8/9 × (15/−4)
⇒ y = (−8 × 15)/(9 × −4) = −120/−36
Dividing numerator and denominator by −12:
⇒ y = 10/3 (or 31/3).
Ans: 10/3
🔥 Higher Order Thinking Skills (HOTS)
HOTS Q1: If x = 3/4 and y = −2/5, evaluate (x − y) ÷ (x + y).
x − y = 3/4 − (−2/5) = 3/4 + 2/5 = (15 + 8)/20 = 23/20.
Step 2: Calculate (x + y):
x + y = 3/4 + (−2/5) = 3/4 − 2/5 = (15 − 8)/20 = 7/20.
Step 3: Perform division:
(x − y) ÷ (x + y) = 23/20 ÷ 7/20 = 23/20 × 20/7 = 23/7 (or 32/7).
Ans: 23/7
HOTS Q2: Divide the sum of −13/5 and 12/7 by the product of −31/7 and −1/2.
−13/5 + 12/7 = (−91 + 60)/35 = −31/35.
Step 2: Find the product:
−31/7 × −1/2 = 31/14.
Step 3: Divide the sum by the product:
(−31/35) ÷ (31/14) = −31/35 × 14/31 = −14/35.
Dividing numerator and denominator by 7:
= −2/5.
Ans: −2/5
🌱 Value Based Questions (VBQ)
VBQ 1: During a health relief initiative, a self-help group stitched cotton face masks for orphanages. Each mask required 3/5 metre of cotton cloth. If the group had 30 metres of cloth in total:
(a) How many masks could the group produce from the available cloth?
Number of masks = 30 ÷ 3/5 = 30 × 5/3 = 10 × 5 = 50 masks.(b) What social value does this initiative promote?
Core Values Depicted: Community empathy, service to society, health consciousness, and altruism towards vulnerable sections of the community.Ans: (a) 50 masks | (b) Community Care, Compassion
📚 DAV Class 7 Maths Chapter 2: Operations on Rational Numbers – Complete Solutions Index
| Worksheet / Section | Key Operations Covered | Solution Link |
|---|---|---|
| Worksheet 1 | Addition of Rational Numbers & Properties | View Worksheet 1 → |
| Worksheet 2 | Subtraction & Additive Inverse | View Worksheet 2 → |
| Worksheet 3 | Multiplication & Reciprocals | View Worksheet 3 → |
| Worksheet 4 | Division of Rational Numbers | View Worksheet 4 → |
| Worksheet 5 | Finding Rational Numbers Between Two Numbers | View Worksheet 5 → |
| Worksheet 6 | Practical Word Problems | View Worksheet 6 → |
| Brain Teasers, HOTS & VBQs | Mixed Evaluation, Distributive Laws, Complex Fractions | Current Page |