Finding Common Ground — Class 7 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 7 Mathematics chapter "Finding Common Ground" — 8 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 7 Mathematics chapter "Finding Common Ground" — 8 important questions with detailed answers for CBSE board…
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NCERT Solutions for Class 7 Mathematics — Finding Common Ground. Step-by-step answers to all 8 textbook questions from this chapter, written for CBSE board preparation and free to use.
Finding Common Ground — All 8 Questions Solved
Q1. Define ratio and proportion. Explain the difference with examples.
Ratio:
A ratio is a comparison of two quantities of the same kind using division.
Ratio of a to b is written as a:b or a/b (read as "a to b").
Both quantities must have the same units.
Example:
If there are 5 boys and 3 girls in a class:
Ratio of boys to girls = 5:3 (read as "5 to 3")
Ratio of girls to boys = 3:5
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Proportion:
A proportion is an equation stating that two ratios are equal.
If a:b = c:d, then a/b = c/d, written as a:b::c:d (read as "a is to b as c is to d").
In this proportion, a and d are called extremes, and b and c are called means.
Example:
2:3 = 4:6
Because 2/3 = 4/6 = 0.667
Verification: 2 × 6 = 3 × 4 = 12 ✓
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Key Differences:
| Ratio | Proportion |
|-------|------------|
| Comparison of two quantities | Equality of two ratios |
| Written as a:b or a/b | Written as a:b::c:d or a/b = c/d |
| Single expression | Equation with two ratios |
| Example: 5:3 | Example: 5:3::10:6 |
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Property of Proportion:
In a:b::c:d, the product of extremes equals the product of means.
a × d = b × c (Cross multiplication)
Q2. Simplify the ratio 24:36 and verify if 24:36 is equal to 2:3.
Simplifying the ratio 24:36:
Step 1: Find the GCD (Greatest Common Divisor) of 24 and 36
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors: 1, 2, 3, 4, 6, 12
GCD = 12
Step 2: Divide both terms by the GCD
24 ÷ 12 = 2
36 ÷ 12 = 3
Simplified ratio = 2:3
Step 3: Express the result
24:36 = 2:3 (in simplest form)
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Verifying if 24:36 is equal to 2:3:
Method 1: Using fractions
24/36 = 24÷12/36÷12 = 2/3
2/3 = 2/3
Yes, they are equal ✓
Method 2: Cross multiplication
For 24:36 = 2:3
24 × 3 = 72
36 × 2 = 72
Both products are equal ✓
Therefore, 24:36 = 2:3 is TRUE.
Q3. If the ratio of two numbers is 3:5 and their sum is 40, find the numbers.
Given:
Ratio of two numbers = 3:5
Sum of the numbers = 40
Step 1: Represent the numbers using the ratio
Let the first number = 3x
Let the second number = 5x
(where x is a constant)
Step 2: Set up the equation using the sum
3x + 5x = 40
8x = 40
Step 3: Solve for x
x = 40 ÷ 8
x = 5
Step 4: Find the numbers
First number = 3x = 3 × 5 = 15
Second number = 5x = 5 × 5 = 25
Step 5: Verify
Ratio: 15:25 = 15÷5:25÷5 = 3:5 ✓
Sum: 15 + 25 = 40 ✓
Answer: The two numbers are 15 and 25.
Q4. What is a percentage? Convert 3/8 to percentage and 45% to a fraction.
Percentage:
A percentage is a ratio expressed as a fraction of 100.
The symbol '%' means "per hundred" or "out of 100".
Percentage is used to compare quantities easily.
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Converting 3/8 to Percentage:
Step 1: Convert the fraction to decimal
3/8 = 3 ÷ 8 = 0.375
Step 2: Multiply by 100 to get percentage
0.375 × 100 = 37.5%
Alternative method:
3/8 = 3/8 × 100% = (3 × 100)/8 % = 300/8 % = 37.5%
Answer: 3/8 = 37.5%
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Converting 45% to a Fraction:
Step 1: Write percentage as a fraction with denominator 100
45% = 45/100
Step 2: Simplify by finding GCD
GCD of 45 and 100 = 5
45 ÷ 5 = 9
100 ÷ 5 = 20
Simplified fraction = 9/20
Step 3: Verify
9/20 = 9 ÷ 20 = 0.45 = 45% ✓
Answer: 45% = 9/20
Q5. If a shirt costs Rs. 800 and a discount of 25% is given, what is the selling price?
Given:
Cost Price (CP) = Rs. 800
Discount = 25%
Step 1: Calculate the discount amount
Discount = 25% of 800
Discount = (25/100) × 800
Discount = 0.25 × 800
Discount = Rs. 200
Step 2: Calculate the Selling Price (SP)
Selling Price = Cost Price - Discount
SP = 800 - 200
SP = Rs. 600
Alternative method (Direct formula):
SP = CP × (100 - Discount%)/100
SP = 800 × (100 - 25)/100
SP = 800 × 75/100
SP = 800 × 0.75
SP = Rs. 600
Step 3: Verify
Discount given = Rs. 200
Selling Price = Rs. 600
Total = 200 + 600 = 800 ✓
Answer: The selling price of the shirt is Rs. 600.
Q6. What is simple interest? Calculate simple interest on Rs. 5000 at 8% per annum for 3 years.
Simple Interest (SI):
Simple interest is the interest calculated only on the principal amount.
Formula: SI = (P × R × T)/100
Where:
P = Principal (initial amount)
R = Rate of interest per annum (%)
T = Time period (in years)
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Given:
Principal (P) = Rs. 5000
Rate (R) = 8% per annum
Time (T) = 3 years
Step 1: Apply the simple interest formula
SI = (P × R × T)/100
SI = (5000 × 8 × 3)/100
Step 2: Calculate the numerator
5000 × 8 × 3 = 120,000
Step 3: Divide by 100
SI = 120,000/100
SI = Rs. 1200
Step 4: Calculate the Amount (Total money after interest)
Amount = Principal + Simple Interest
Amount = 5000 + 1200
Amount = Rs. 6200
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Answer:
Simple Interest = Rs. 1200
Amount after 3 years = Rs. 6200
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Note:
Simple interest is the same every year (Rs. 400 per year).
Year 1: Interest = 400
Year 2: Interest = 400
Year 3: Interest = 400
Total = 400 × 3 = 1200
Q7. In a class, the ratio of boys to girls is 4:5. If there are 36 boys, how many girls are there? What is the total number of students?
Given:
Ratio of boys to girls = 4:5
Number of boys = 36
Step 1: Set up the ratio
If boys : girls = 4:5
Then, boys/girls = 4/5
Step 2: Find the value of one ratio unit
Let one ratio unit = k
Then, boys = 4k
Since boys = 36:
4k = 36
k = 36/4
k = 9
Step 3: Find the number of girls
Girls = 5k
Girls = 5 × 9
Girls = 45
Step 4: Find the total number of students
Total = Boys + Girls
Total = 36 + 45
Total = 81
Step 5: Verify the ratio
Boys : Girls = 36 : 45
Simplify by dividing by 9: 36÷9 : 45÷9 = 4:5 ✓
Answer:
Number of girls = 45
Total number of students = 81
Q8. Compare the quantities: 60 km/h and 90 km/h. Express as a ratio and find what percentage is 60 km/h of 90 km/h.
Comparing 60 km/h and 90 km/h:
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Part 1: Express as a Ratio
Step 1: Write the ratio
60 : 90
Step 2: Simplify by finding GCD
GCD of 60 and 90 = 30
60 ÷ 30 = 2
90 ÷ 30 = 3
Simplified ratio = 2:3
This means for every 2 units of 60 km/h, there are 3 units of 90 km/h.
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Part 2: Find what percentage is 60 km/h of 90 km/h
Step 1: Set up the percentage formula
Percentage = (60/90) × 100%
Step 2: Simplify the fraction
60/90 = 2/3 = 0.6667
Step 3: Multiply by 100
Percentage = 0.6667 × 100% = 66.67% (approximately)
Exact value = (2/3) × 100% = 200/3 % = 66⅔%
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Answer:
Ratio = 2:3
60 km/h is 66⅔% (or 66.67%) of 90 km/h.
Alternatively: 60 km/h is 2/3 of 90 km/h.
Finding Common Ground — Practice MCQs with Answers
Attempt these 20 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. What is the smallest number that is a multiple of both 4 and 6?
- A. 12
- B. 18
- C. 24
- D. 30
Answer: A. 12 — The smallest number that is a multiple of both 4 and 6 is their Least Common Multiple (LCM). Multiples of 4 are 4, 8, 12, 16... Multiples of 6 are 6, 12, 18... The first common multiple is 12.
MCQ 2. Find the greatest common factor (HCF) of 12 and 18.
- A. 3
- B. 6
- C. 9
- D. 12
Answer: B. 6 — Factors of 12 are 1, 2, 3, 4, 6, 12. Factors of 18 are 1, 2, 3, 6, 9, 18. The greatest number that is common to both lists is 6.
MCQ 3. Which of the following is a common denominator for the fractions 1/3 and 2/5?
- A. 8
- B. 10
- C. 15
- D. 30
Answer: C. 15 — A common denominator must be a multiple of both 3 and 5. The least common multiple (LCM) of 3 and 5 is 15, which is a common denominator. 30 is also a common denominator, but 15 is the least.
MCQ 4. To compare 3/4 and 5/6, what is the least common denominator you would use?
- A. 10
- B. 12
- C. 18
- D. 24
Answer: B. 12 — The least common denominator is the LCM of the denominators. The LCM of 4 and 6 is 12.
MCQ 5. Two bells ring at intervals of 8 minutes and 12 minutes, respectively. If they ring together at 9:00 AM, at what time will they next ring together?
- A. 9:16 AM
- B. 9:20 AM
- C. 9:24 AM
- D. 9:30 AM
Answer: C. 9:24 AM — We need to find the LCM of 8 and 12. The LCM of 8 and 12 is 24. So they will ring together again 24 minutes after 9:00 AM, which is 9:24 AM.
MCQ 6. A shopkeeper wants to arrange 36 apples and 48 oranges into groups such that each group has the same number of fruits of the same type. What is the maximum number of fruits that can be in each group?
- A. 6
- B. 8
- C. 12
- D. 18
Answer: C. 12 — We need to find the HCF of 36 and 48 to determine the maximum number of fruits in each group. The HCF of 36 and 48 is 12.
MCQ 7. Which fraction is equivalent to 2/3?
- A. 4/5
- B. 6/8
- C. 8/12
- D. 10/18
Answer: C. 8/12 — To find an equivalent fraction, multiply both the numerator and denominator by the same non-zero number. Multiplying 2/3 by 4/4 gives 8/12.
MCQ 8. What is the sum of 1/4 and 3/8?
- A. 4/12
- B. 5/8
- C. 4/8
- D. 1/2
Answer: B. 5/8 — To add fractions, find a common denominator, which is 8 here. Convert 1/4 to 2/8. Then, 2/8 + 3/8 = 5/8.
MCQ 9. The product of two numbers is 180, and their HCF is 6. What is their LCM?
- A. 30
- B. 90
- C. 1080
- D. 186
Answer: A. 30 — For any two numbers, the product of the numbers is equal to the product of their HCF and LCM. So, LCM = (Product of numbers) / HCF = 180 / 6 = 30.
MCQ 10. Which of these numbers is a common factor of 20 and 30?
- A. 4
- B. 5
- C. 8
- D. 10
Answer: D. 10 — Factors of 20 are 1, 2, 4, 5, 10, 20. Factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. From the options, 10 is a common factor.
MCQ 11. Arrange the following fractions in ascending order: 1/2, 3/4, 2/5.
- A. 1/2, 2/5, 3/4
- B. 2/5, 1/2, 3/4
- C. 3/4, 1/2, 2/5
- D. 1/2, 3/4, 2/5
Answer: B. 2/5, 1/2, 3/4 — Convert to a common denominator (LCM of 2, 4, 5 is 20). 1/2 = 10/20, 3/4 = 15/20, 2/5 = 8/20. Thus, 8/20 < 10/20 < 15/20, which corresponds to 2/5 < 1/2 < 3/4.
MCQ 12. How many common multiples of 3 and 5 are less than 50?
- A. 2
- B. 3
- C. 4
- D. 5
Answer: C. 4 — The LCM of 3 and 5 is 15. The common multiples are multiples of 15. These are 15, 30, and 45. There are 3 common multiples less than 50.
MCQ 13. Which of the following decimals is greater than 0.75?
- A. 0.7
- B. 0.70
- C. 0.755
- D. 0.075
Answer: C. 0.755 — To compare decimals, align the decimal points and compare digits from left to right. 0.755 is greater than 0.750.
MCQ 14. Two numbers have an HCF of 4 and an LCM of 48. If one number is 16, what is the other number?
- A. 8
- B. 12
- C. 24
- D. 32
Answer: B. 12 — The product of two numbers equals the product of their HCF and LCM. So, 16 * Number2 = 4 * 48. Number2 = (4 * 48) / 16 = 12.
MCQ 15. What is a common prime factor of 10 and 15?
- A. 2
- B. 3
- C. 5
- D. 10
Answer: C. 5 — Prime factors of 10 are 2 and 5. Prime factors of 15 are 3 and 5. The common prime factor is 5.
MCQ 16. What is the value of 5.25 + 3.7 - 2.05?
- A. 6.9
- B. 6.09
- C. 6.99
- D. 7.0
Answer: A. 6.9 — First, add 5.25 + 3.70 = 8.95. Then, subtract 2.05 from the sum: 8.95 - 2.05 = 6.90.
MCQ 17. The ratio of two numbers is 3:4 and their HCF is 5. What is their LCM?
- A. 15
- B. 20
- C. 60
- D. 120
Answer: C. 60 — If the HCF is 5 and the ratio is 3:4, the numbers are 3*5 = 15 and 4*5 = 20. The LCM of 15 and 20 is 60.
MCQ 18. To subtract 1/6 from 3/4, what is the smallest common denominator required?
- A. 6
- B. 12
- C. 18
- D. 24
Answer: B. 12 — The smallest common denominator is the LCM of the denominators. The LCM of 4 and 6 is 12.
MCQ 19. Which of these is a common multiple of 2, 3, and 5?
- A. 10
- B. 15
- C. 20
- D. 30
Answer: D. 30 — A common multiple of 2, 3, and 5 must be divisible by all three numbers. The LCM of 2, 3, and 5 is 30.
MCQ 20. Which pair of numbers has 7 as a common factor?
- A. (14, 20)
- B. (21, 35)
- C. (28, 40)
- D. (30, 42)
Answer: B. (21, 35) — 21 is 3 x 7, and 35 is 5 x 7. Both numbers are multiples of 7, making 7 a common factor.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.