Looking for complete, accurate, and step-by-step solutions for DAV Class 7 Secondary Mathematics Chapter 1: Rational Numbers (Brain Teasers, HOTS & Value Based Questions)? You are in the right place! Below you will find line-by-line handwritten-style explanations for all questions of Brain Teasers (Q.1 to Q.6), Higher Order Thinking Skills (HOTS), and Value-Based Questions (VBQs) strictly following the DAV Education Board curriculum.
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Q.1. (A) Tick (✓) the correct option
(i) The value of x such that −3/8 and x/−24 are equivalent rational numbers is:
- (a) 64
- (b) −64
- (c) −9
- (d) 9
Two rational numbers a/b and c/d are equivalent if a × d = b × c.
Here, −3/8 = x/−24
⇒ (−3) × (−24) = 8 × x
⇒ 72 = 8x
⇒ x = 72/8 = 9
Correct Option: (d) 9
(ii) Which of the following is a negative rational number?
- (a) −2/−3
- (b) 0
- (c) 5/−7
- (d) −4/−9
• In (a) −2/−3 = 2/3 (Both signs cancel out → Positive).
• In (b) 0 is neither positive nor negative.
• In (c) 5/−7 = −5/7 (Only one negative sign → Negative Rational Number).
• In (d) −4/−9 = 4/9 (Both signs cancel out → Positive).
Correct Option: (c) 5/−7
(iii) In a number line, if a point M represents −12/20, its standard form is:
- (a) −6/10
- (b) −3/5
- (c) 3/−5
- (d) −4/5
To find the standard form, divide both the numerator and denominator by their Highest Common Factor (HCF).
• HCF of 12 and 20 is 4.
⇒ (−12 ÷ 4)/(20 ÷ 4) = −3/5
Correct Option: (b) −3/5
(iv) Which of the following is the smallest rational number out of −5/7, −3/7, −1/7, and −6/7?
- (a) −6/7
- (b) −1/7
- (c) −3/7
- (d) −5/7
Since all denominators are identical and positive (7), compare the numerators directly: −6, −5, −3, −1.
On the number line, numbers positioned furthest to the left are smallest.
Since −6 < −5 < −3 < −1, the smallest rational number is −6/7.
Correct Option: (a) −6/7
(v) The standard form of 18/−24 is:
- (a) −9/12
- (b) 3/−4
- (c) −3/4
- (d) −6/8
A rational number is in standard form when its denominator is a positive integer and HCF(numerator, denominator) = 1.
1. Shift the negative sign to numerator: 18/−24 = −18/24
2. HCF of 18 and 24 is 6.
⇒ (−18 ÷ 6)/(24 ÷ 6) = −3/4
Correct Option: (c) −3/4
Q.1. (B) Answer the following questions
(a) Find the average of the rational numbers 4/5, 2/3, and 5/6.
Formula: Average = (Sum of given rational numbers) ÷ (Total count of numbers)
Here, total count of numbers = 3.
Step 1: Find the sum of the rational numbers:
Sum = 4/5 + 2/3 + 5/6
LCM of denominators (5, 3, 6):
• 5 = 5
• 3 = 3
• 6 = 2 × 3
⇒ LCM = 2 × 3 × 5 = 30.
Convert each fraction to denominator 30:
• 4/5 = (4 × 6)/(5 × 6) = 24/30
• 2/3 = (2 × 10)/(3 × 10) = 20/30
• 5/6 = (5 × 5)/(6 × 5) = 25/30
Sum = (24 + 20 + 25)/30 = 69/30 = 23/10 (dividing numerator and denominator by 3).
Step 2: Compute Average:
Average = 23/10 ÷ 3 = 23/10 × 1/3 = 23/30.
Ans: 23/30
(b) How will you write −12/−18 in standard form?
1. In −12/−18, both negative signs cancel out: −12/−18 = 12/18.
2. Find the HCF of 12 and 18: Prime factors of 12 = 2 × 2 × 3; Prime factors of 18 = 2 × 3 × 3 ⇒ HCF = 6.
3. Divide numerator and denominator by 6:
(12 ÷ 6)/(18 ÷ 6) = 2/3.
(Note: If the question in your edition is 12/−18, the standard form is −2/3).
Ans: 2/3
(c) How many rational numbers are there between any two rational numbers?
Between any two distinct rational numbers, there are infinitely many (uncountable) rational numbers. This property is known as the dense property of rational numbers.
Ans: Infinitely many rational numbers
(d) On the number line, does the rational number −5/−7 lie on the right or left of zero?
Simplifying −5/−7:
−5/−7 = 5/7.
Since 5/7 > 0 (positive rational number), it lies to the right side of zero on the number line.
Ans: Right of zero
(e) Express 7/−8 as a rational number with denominator 40.
Given rational number: 7/−8.
We require the new denominator to be 40.
To find the multiplier: 40 ÷ (−8) = −5.
Multiply both the numerator and the denominator by −5:
7/−8 = (7 × −5)/(−8 × −5) = −35/40.
Ans: −35/40
Q.2. Answer the following questions with reasons
(i) Is every whole number a rational number?
Yes. Any whole number w (such as 0, 1, 2, 3…) can be expressed as w/1, where both w and 1 are integers and the denominator 1 ≠ 0. Thus, every whole number is a rational number.
(ii) Is every integer a rational number?
Yes. Any integer z (positive, zero, or negative) can be written in the form z/1. Since z and 1 are integers and the denominator 1 ≠ 0, all integers are rational numbers.
(iii) Is every rational number a fraction?
No. A fraction is defined as a ratio of two whole numbers where the numerator and denominator are non-negative. However, rational numbers can have negative integers in the numerator or denominator (e.g., −3/4 or −7/5). Therefore, negative rational numbers are not fractions.
(iv) Is every fraction a rational number?
Yes. Every fraction a/b consists of positive whole numbers where b ≠ 0. Because all whole numbers are integers, every fraction strictly satisfies the definition of a rational number.
Q.3. Arrange the following rational numbers in ascending order
Numbers to arrange: −5/6, −7/12, −2/3, −9/14
All denominators are positive: 6, 12, 3, 14.
• Prime factorization of 6 = 2 × 3
• Prime factorization of 12 = 22 × 3 = 4 × 3
• Prime factorization of 3 = 3
• Prime factorization of 14 = 2 × 7
⇒ LCM(6, 12, 3, 14) = 22 × 3 × 7 = 4 × 21 = 84.
Step 2: Convert each rational number to common denominator 84:
• −5/6 = (−5 × 14)/(6 × 14) = −70/84
• −7/12 = (−7 × 7)/(12 × 7) = −49/84
• −2/3 = (−2 × 28)/(3 × 28) = −56/84
• −9/14 = (−9 × 6)/(14 × 6) = −54/84
Step 3: Compare numerators:
Numerators are: −70, −49, −56, −54.
Arranging numerators from smallest to greatest:
−70 < −56 < −54 < −49
Step 4: Write in ascending order:
−70/84 < −56/84 < −54/84 < −49/84
⇒ −5/6 < −2/3 < −9/14 < −7/12
Ans: −5/6, −2/3, −9/14, −7/12
Q.4. Arrange the following rational numbers in descending order
Numbers to arrange: 3/−4, −5/12, −7/18, 2/−3
• 3/−4 = −3/4
• −5/12 (already positive)
• −7/18 (already positive)
• 2/−3 = −2/3
Step 2: Find LCM of denominators (4, 12, 18, 3):
• 4 = 22
• 12 = 22 × 3
• 18 = 2 × 32
• 3 = 3
⇒ LCM = 22 × 32 = 4 × 9 = 36.
Step 3: Convert to common denominator 36:
• −3/4 = (−3 × 9)/(4 × 9) = −27/36
• −5/12 = (−5 × 3)/(12 × 3) = −15/36
• −7/18 = (−7 × 2)/(18 × 2) = −14/36
• −2/3 = (−2 × 12)/(3 × 12) = −24/36
Step 4: Compare numerators in descending order (greatest to smallest):
−14 > −15 > −24 > −27
Step 5: Write in descending order:
−14/36 > −15/36 > −24/36 > −27/36
⇒ −7/18 > −5/12 > 2/−3 > 3/−4
Ans: −7/18, −5/12, 2/−3, 3/−4
Q.5. On a number line, find the length of the line segment joining:
Key Concept: The length of a line segment between any two points a and b on a number line is given by the positive absolute distance: Length = |Larger Number − Smaller Number|.
(i) 3 and −3
Ans: 6 units
(ii) 3/4 and −1/4
Ans: 1 unit
(iii) −2½ and 1½
• −2½ = −5/2
• 1½ = 3/2
Length = 3/2 − (−5/2) = 3/2 + 5/2 = (3 + 5)/2 = 8/2 = 4 units.
Ans: 4 units
Q.6. Find three rational numbers between −1/3 and 1/2
LCM(3, 2) = 6.
Step 2: Convert to common denominator 6:
• −1/3 = (−1 × 2)/(3 × 2) = −2/6
• 1/2 = (1 × 3)/(2 × 3) = 3/6
Step 3: List rational numbers lying between −2/6 and 3/6:
Integers between −2 and 3 are: −1, 0, 1, 2.
Therefore, rational numbers between −1/3 and 1/2 include:
−1/6, 0, 1/6, 2/6.
Ans: Any three numbers: −1/6, 0, 1/6
🔥 Higher Order Thinking Skills (HOTS)
HOTS Q1: Find five rational numbers between −2/3 and 1/4.
LCM(3, 4) = 12.
Step 2: Convert to common denominator:
• −2/3 = (−2 × 4)/(3 × 4) = −8/12
• 1/4 = (1 × 3)/(4 × 3) = 3/12
Step 3: Identify integers between −8 and 3:
Integers are: −7, −6, −5, −4, −3, −2, −1, 0, 1, 2.
Thus, any five rational numbers are:
−7/12, −5/12, −1/12, 0, 1/12.
Ans: −7/12, −5/12, −1/12, 0, 1/12
HOTS Q2: Express the rational number −192/168 in standard form using the HCF method.
• 192 = 2 × 2 × 2 × 2 × 2 × 2 × 3 = 26 × 3
• 168 = 2 × 2 × 2 × 3 × 7 = 23 × 3 × 7
Step 2: Find the HCF:
HCF(192, 168) = 23 × 3 = 8 × 3 = 24.
Step 3: Divide numerator and denominator by 24:
(−192 ÷ 24)/(168 ÷ 24) = −8/7.
Ans: −8/7
HOTS Q3: If the product of two rational numbers is −16/9 and one of the numbers is −4/3, find the other number.
Let the required rational number be x.
According to the problem:
x × (−4/3) = −16/9
⇒ x = −16/9 ÷ (−4/3)
⇒ x = −16/9 × (3/−4)
⇒ x = (−16 × 3)/(9 × −4) = −48/−36
Dividing numerator and denominator by −12:
x = 4/3 (or 11/3).
Ans: 4/3
🌱 Value Based Questions (VBQ)
VBQ 1: A community kitchen distributed 5/8 quintal of food grains equally among 5 needy families affected by floods.
(a) Find the share of food grains received by each family.
Share per family = (Total food grain) ÷ (Number of families)= 5/8 ÷ 5
= 5/8 × 1/5
= 1/8 quintal.
(b) What core value is exhibited by the community members?
Core Values Depicted: Empathy, social responsibility, generosity, and compassion toward suffering fellow human beings in times of natural crisis.Ans: (a) 1/8 quintal | (b) Empathy, Social Responsibility
VBQ 2: Rohan actively participates in ‘Van Mahotsav’. He watered 3/4 part of the school herbal garden in the morning and 1/6 part in the afternoon.
(a) What total part of the garden did Rohan water during the day?
Total part = 3/4 + 1/6LCM(4, 6) = 12.
= (3 × 3)/12 + (1 × 2)/12
= 9/12 + 2/12
= 11/12 part.
(b) What moral value is reflected in Rohan’s dedication?
Core Values Depicted: Environmental stewardship, love for nature, responsibility toward green preservation, and active civic participation.Ans: (a) 11/12 part | (b) Environmental Consciousness, Care for Nature
📚 DAV Class 7 Maths Chapter 1: Rational Numbers – Complete Solutions Index
| Worksheet / Section | Key Topics Covered | Solution Link |
|---|---|---|
| Worksheet 1 | Identification, Positive & Negative Rational Numbers | View Worksheet 1 → |
| Worksheet 2 | Equivalent Rational Numbers & Finding x | View Worksheet 2 → |
| Worksheet 3 | Standard Form of Rational Numbers | View Worksheet 3 → |
| Worksheet 4 | Representation on Number Line | View Worksheet 4 → |
| Worksheet 5 | Comparison of Rational Numbers | View Worksheet 5 → |
| Worksheet 6 | Ascending & Descending Order | View Worksheet 6 → |
| Brain Teasers, HOTS & VBQs | Complete Revision, Tough Problems, Value Questions | Current Page |