💡 To save as PDF, press Ctrl + P (or ⌘ + P on Mac) and choose “Save as PDF”
CHAPTER 1 — LARGE NUMBERS AROUND US
Class VII Mathematics — Challenge Paper Set 4
Preeti Kushwah Classes
General Instructions:
1. All questions are compulsory.
2. Section A: 6 questions of 1 mark each (6 marks).
3. Section B: 5 questions of 2 marks each (10 marks).
4. Section C: 4 questions of 3 marks each (12 marks).
5. Section D: 2 questions of 4 marks each (8 marks).
6. Bonus: 1 question of 4 marks (optional).
7. This is an advanced paper. Think carefully before writing!
1. All questions are compulsory.
2. Section A: 6 questions of 1 mark each (6 marks).
3. Section B: 5 questions of 2 marks each (10 marks).
4. Section C: 4 questions of 3 marks each (12 marks).
5. Section D: 2 questions of 4 marks each (8 marks).
6. Bonus: 1 question of 4 marks (optional).
7. This is an advanced paper. Think carefully before writing!
Section A — (1 Mark Each) [6 × 1 = 6]
Q1.1
How many 7-digit numbers are there in all?
Q2.1
What is the difference between the largest 8-digit number and the smallest 9-digit number?
Q3.1
Replace the * in 38,*4,*16 so that the number is divisible by 11. (Divisibility rule: difference of sums of alternate digits = 0 or multiple of 11)
Q4.1
A number rounded to the nearest lakh gives 27,00,000. What is the largest possible value of the number?
Q5.1
If 7 × 10a + 3 × 10b + 5 = 70,30,005, find the values of a and b.
Q6.1
How many zeros are there in the product: 25 × 58 × 3?
Section B — (2 Marks Each) [5 × 2 = 10]
Q7.2
A number N is such that when each digit is increased by 1, the resulting number is exactly double of N. Find N.
Hint: N is a 3-digit number.
Hint: N is a 3-digit number.
Q8.2
Without computing, determine which is greater and by how much: 111,111,111 × 9 or 999,999,999?
Q9.2
Express each of the following as a product of a number and a power of 10:
(a) 4,70,00,000 (b) 30,05,000
Represent each in both Indian and International comma placement.
(a) 4,70,00,000 (b) 30,05,000
Represent each in both Indian and International comma placement.
Q10.2
The sum of two numbers is 1,00,00,000. One number has all 9s and is the largest 6-digit number. Find the other number and write it in words (Indian system).
Q11.2
Ramu’s calculator has only two buttons: ×2 and +1. Starting from 0, what is the minimum number of button presses to reach 100? Show the sequence.
Section C — (3 Marks Each) [4 × 3 = 12]
Q12.3
Study the pattern and answer:
1 × 9 + 1 = 10
12 × 9 + 2 = 110
123 × 9 + 3 = 1110
1234 × 9 + 4 = ?
12345 × 9 + 5 = ?
(a) Complete the next two lines without calculating directly.
(b) Verify the 4th line by actual multiplication.
(c) Write the general term: 123...n × 9 + n = ?
1 × 9 + 1 = 10
12 × 9 + 2 = 110
123 × 9 + 3 = 1110
1234 × 9 + 4 = ?
12345 × 9 + 5 = ?
(a) Complete the next two lines without calculating directly.
(b) Verify the 4th line by actual multiplication.
(c) Write the general term: 123...n × 9 + n = ?
Q13.3
India’s GDP (Gross Domestic Product) is approximately ₹2,70,00,000 crore per year.
(a) Express this in International system using “trillion” (1 trillion = 1012).
(b) If India’s population is 140 crore, what is the approximate GDP per person (per capita GDP)?
(c) If GDP grows by 7% next year, what will the new GDP be? Express in crores.
(a) Express this in International system using “trillion” (1 trillion = 1012).
(b) If India’s population is 140 crore, what is the approximate GDP per person (per capita GDP)?
(c) If GDP grows by 7% next year, what will the new GDP be? Express in crores.
Q14.3
Find the number of digits in each of the following products WITHOUT multiplying. State the rule and verify any one:
(a) 9,999 × 9,999
(b) 1,00,001 × 99
(c) 254 × 44
(a) 9,999 × 9,999
(b) 1,00,001 × 99
(c) 254 × 44
Q15.3
A school has three branches with the following number of books in their libraries:
(a) Estimate the total books (round each to nearest ten thousand first).
(b) Calculate the actual total. What is the estimation error?
(c) If the school wants to have a total of 5,00,000 books across all branches, how many more books do they need? Express in International system words.
| Branch | Books |
|---|---|
| North | 1,24,850 |
| South | 2,07,436 |
| East | 98,715 |
(b) Calculate the actual total. What is the estimation error?
(c) If the school wants to have a total of 5,00,000 books across all branches, how many more books do they need? Express in International system words.
Section D — (4 Marks Each) [2 × 4 = 8]
Q16.4
ISRO’s Chandrayaan-3 spacecraft travelled approximately 3,84,000 km to reach the Moon. The Mangalyaan Mars orbiter travelled approximately 68,00,00,000 km to reach Mars.
(a) How many times is the Mars distance compared to the Moon distance? Round to the nearest hundred.
(b) If a spacecraft travels at 36,000 km/hour, how many days would each journey take?
(c) Express the Mars distance in the International system. How many millions is it?
(d) Light travels at 3,00,000 km per second. How many seconds does light take to reach from Earth to Moon? And to Mars?
(a) How many times is the Mars distance compared to the Moon distance? Round to the nearest hundred.
(b) If a spacecraft travels at 36,000 km/hour, how many days would each journey take?
(c) Express the Mars distance in the International system. How many millions is it?
(d) Light travels at 3,00,000 km per second. How many seconds does light take to reach from Earth to Moon? And to Mars?
Q17.4
A mobile phone company sells 3 models:
(a) Calculate the daily revenue from each model. Which model earns the most?
(b) Find the total daily revenue.
(c) What is the total annual revenue (365 days)? Express in crores (round to nearest crore).
(d) If the company targets ₹1,000 crore annual revenue, how many extra Premium phones must they sell daily (assuming other sales stay the same)?
| Model | Price (₹) | Daily Sales |
|---|---|---|
| Basic | 8,999 | 1,250 |
| Standard | 17,499 | 840 |
| Premium | 42,999 | 320 |
(b) Find the total daily revenue.
(c) What is the total annual revenue (365 days)? Express in crores (round to nearest crore).
(d) If the company targets ₹1,000 crore annual revenue, how many extra Premium phones must they sell daily (assuming other sales stay the same)?
Bonus Question (Optional) [4 Marks]
Q18.4HARD
★ Observe the pattern:
12 = 1
112 = 121
1112 = 12321
11112 = 1234321
(a) Write 111112 using the pattern (without calculating).
(b) How many digits does 1111111112 (nine 1s squared) have?
(c) What is the sum of digits of 111112?
(d) Does this pattern hold for 11111111112 (ten 1s)? Explain why or why not.
12 = 1
112 = 121
1112 = 12321
11112 = 1234321
(a) Write 111112 using the pattern (without calculating).
(b) How many digits does 1111111112 (nine 1s squared) have?
(c) What is the sum of digits of 111112?
(d) Does this pattern hold for 11111111112 (ten 1s)? Explain why or why not.
Preeti Kushwah Classes
Scan for more notes & papers
www.preetikushwahclasses.com
Scan for more notes & papers
www.preetikushwahclasses.com
Answer Key & Detailed Solutions
Q1. [1 Mark]
Smallest 7-digit number = 10,00,000
Largest 7-digit number = 99,99,999
Total 7-digit numbers = 99,99,999 − 10,00,000 + 1 = 90,00,000
= Ninety lakh
Largest 7-digit number = 99,99,999
Total 7-digit numbers = 99,99,999 − 10,00,000 + 1 = 90,00,000
= Ninety lakh
Q2. [1 Mark]
Largest 8-digit number = 9,99,99,999
Smallest 9-digit number = 10,00,00,000
Difference = 10,00,00,000 − 9,99,99,999 = 1
Smallest 9-digit number = 10,00,00,000
Difference = 10,00,00,000 − 9,99,99,999 = 1
Q3. [1 Mark]
Number: 38,*4,*16 = 3 8 * 4 * 1 6
Positions (from right): 6(1) 1(2) *(3) 4(4) *(5) 8(6) 3(7)
Odd positions sum: 6 + * + * + 3 = 9 + 2*
Wait — let me re-index. The number is 3,8,a,4,b,1,6
Odd positions (1st, 3rd, 5th, 7th from left): 3 + a + b + 6 = 9 + a + b
Even positions (2nd, 4th, 6th from left): 8 + 4 + 1 = 13
For divisibility by 11: |9 + a + b − 13| = 0 or 11
|a + b − 4| = 0 or 11
Case 1: a + b = 4 (possible: e.g., a=0,b=4 or a=1,b=3 or a=2,b=2 etc.)
Case 2: a + b = 15 (possible: a=6,b=9 or a=7,b=8 etc.)
One answer: a = 2, b = 2 → 38,24,216
(Multiple answers exist)
Positions (from right): 6(1) 1(2) *(3) 4(4) *(5) 8(6) 3(7)
Odd positions sum: 6 + * + * + 3 = 9 + 2*
Wait — let me re-index. The number is 3,8,a,4,b,1,6
Odd positions (1st, 3rd, 5th, 7th from left): 3 + a + b + 6 = 9 + a + b
Even positions (2nd, 4th, 6th from left): 8 + 4 + 1 = 13
For divisibility by 11: |9 + a + b − 13| = 0 or 11
|a + b − 4| = 0 or 11
Case 1: a + b = 4 (possible: e.g., a=0,b=4 or a=1,b=3 or a=2,b=2 etc.)
Case 2: a + b = 15 (possible: a=6,b=9 or a=7,b=8 etc.)
One answer: a = 2, b = 2 → 38,24,216
(Multiple answers exist)
Q4. [1 Mark]
Rounded to nearest lakh = 27,00,000
This means: 26,50,000 ≤ N ≤ 27,49,999
Largest possible value = 27,49,999
This means: 26,50,000 ≤ N ≤ 27,49,999
Largest possible value = 27,49,999
Q5. [1 Mark]
70,30,005 = 7,000,000 + 30,000 + 5
= 7 × 106 + 3 × 104 + 5
Comparing: 7 × 10a = 7 × 106 → a = 6
3 × 10b = 3 × 104 → b = 4
= 7 × 106 + 3 × 104 + 5
Comparing: 7 × 10a = 7 × 106 → a = 6
3 × 10b = 3 × 104 → b = 4
Q6. [1 Mark]
25 × 58 × 3
= 25 × 55 × 53 × 3
= (2 × 5)5 × 125 × 3
= 105 × 375
= 375 × 1,00,000
= 3,75,00,000
Number of trailing zeros = 5
(The number of trailing zeros = min(power of 2, power of 5) = min(5, 8) = 5)
= 25 × 55 × 53 × 3
= (2 × 5)5 × 125 × 3
= 105 × 375
= 375 × 1,00,000
= 3,75,00,000
Number of trailing zeros = 5
(The number of trailing zeros = min(power of 2, power of 5) = min(5, 8) = 5)
Q7. [2 Marks]
Let N = abc = 100a + 10b + c [1 mark for setup]
New number = 100(a+1) + 10(b+1) + (c+1) = 100a + 10b + c + 111 = N + 111
Given: N + 111 = 2N
111 = N
N = 111 [1 mark]
Verification: Increasing each digit by 1: 222. Is 222 = 2 × 111? Yes! ✅
New number = 100(a+1) + 10(b+1) + (c+1) = 100a + 10b + c + 111 = N + 111
Given: N + 111 = 2N
111 = N
N = 111 [1 mark]
Verification: Increasing each digit by 1: 222. Is 222 = 2 × 111? Yes! ✅
Q8. [2 Marks]
111,111,111 × 9 [1 mark]
= 111,111,111 × (10 − 1)
= 1,111,111,110 − 111,111,111
= 999,999,999
So 111,111,111 × 9 = 999,999,999 [1 mark]
They are equal! The difference is 0.
= 111,111,111 × (10 − 1)
= 1,111,111,110 − 111,111,111
= 999,999,999
So 111,111,111 × 9 = 999,999,999 [1 mark]
They are equal! The difference is 0.
Q9. [2 Marks]
(a) 4,70,00,000 [1 mark]
= 47 × 105 or 4.7 × 107
Indian: 4,70,00,000 | International: 47,000,000
(b) 30,05,000 [1 mark]
= 3005 × 103 or 30.05 × 105
Indian: 30,05,000 | International: 3,005,000
= 47 × 105 or 4.7 × 107
Indian: 4,70,00,000 | International: 47,000,000
(b) 30,05,000 [1 mark]
= 3005 × 103 or 30.05 × 105
Indian: 30,05,000 | International: 3,005,000
Q10. [2 Marks]
Largest 6-digit number = 9,99,999 [1 mark]
Other number = 1,00,00,000 − 9,99,999 = 90,00,001
In words: Ninety lakh one [1 mark]
Other number = 1,00,00,000 − 9,99,999 = 90,00,001
In words: Ninety lakh one [1 mark]
Q11. [2 Marks]
Work backwards from 100: [1 mark for approach]
100 ÷ 2 = 50, 50 ÷ 2 = 25, 25 − 1 = 24, 24 ÷ 2 = 12, 12 ÷ 2 = 6, 6 ÷ 2 = 3, 3 − 1 = 2, 2 ÷ 2 = 1, 1 − 1 = 0
Forward sequence (starting from 0): [1 mark]
0 →(+1)→ 1 →(×2)→ 2 →(+1)→ 3 →(×2)→ 6 →(×2)→ 12 →(×2)→ 24 →(+1)→ 25 →(×2)→ 50 →(×2)→ 100
Minimum presses = 9
100 ÷ 2 = 50, 50 ÷ 2 = 25, 25 − 1 = 24, 24 ÷ 2 = 12, 12 ÷ 2 = 6, 6 ÷ 2 = 3, 3 − 1 = 2, 2 ÷ 2 = 1, 1 − 1 = 0
Forward sequence (starting from 0): [1 mark]
0 →(+1)→ 1 →(×2)→ 2 →(+1)→ 3 →(×2)→ 6 →(×2)→ 12 →(×2)→ 24 →(+1)→ 25 →(×2)→ 50 →(×2)→ 100
Minimum presses = 9
Q12. [3 Marks]
(a) Pattern completion: [1 mark]
1234 × 9 + 4 = 11110
12345 × 9 + 5 = 111110
Pattern: result is (n ones followed by a zero) for n-digit 123...n
(b) Verification of 4th line: [1 mark]
1234 × 9 = 11,106
11,106 + 4 = 11,110 = 11110 ✅
(c) General term: [1 mark]
123...n × 9 + n = 111...10 (n ones followed by a zero)
= 1111...10 where there are n ones
1234 × 9 + 4 = 11110
12345 × 9 + 5 = 111110
Pattern: result is (n ones followed by a zero) for n-digit 123...n
(b) Verification of 4th line: [1 mark]
1234 × 9 = 11,106
11,106 + 4 = 11,110 = 11110 ✅
(c) General term: [1 mark]
123...n × 9 + n = 111...10 (n ones followed by a zero)
= 1111...10 where there are n ones
Q13. [3 Marks]
GDP = ₹2,70,00,000 crore = ₹2.7 × 107 crore = ₹2.7 × 107 × 107 = ₹2.7 × 1014
(a) In International system: [1 mark]
1 trillion = 1012
2.7 × 1014 = 270 × 1012 = 270 trillion rupees
(≈ $3.3 trillion approximately)
(b) Per capita GDP: [1 mark]
140 crore = 1,40,00,00,000 = 1.4 × 109
GDP per person = 2.7 × 1014 ÷ 1.4 × 109
= (2.7/1.4) × 105 = 1.93 × 105
≈ ₹1,93,000 per person per year
(c) 7% growth: [1 mark]
New GDP = 2,70,00,000 × 1.07 = 2,88,90,000 crore
≈ Two crore eighty-eight lakh ninety thousand crore rupees
(a) In International system: [1 mark]
1 trillion = 1012
2.7 × 1014 = 270 × 1012 = 270 trillion rupees
(≈ $3.3 trillion approximately)
(b) Per capita GDP: [1 mark]
140 crore = 1,40,00,00,000 = 1.4 × 109
GDP per person = 2.7 × 1014 ÷ 1.4 × 109
= (2.7/1.4) × 105 = 1.93 × 105
≈ ₹1,93,000 per person per year
(c) 7% growth: [1 mark]
New GDP = 2,70,00,000 × 1.07 = 2,88,90,000 crore
≈ Two crore eighty-eight lakh ninety thousand crore rupees
Q14. [3 Marks]
Rule: Product of m-digit and n-digit numbers has (m+n−1) to (m+n) digits.
(a) 9,999 × 9,999: m=4, n=4 → 7 to 8 digits [1 mark]
Actual: 9,9992 = 9,99,80,001 → 8 digits ✅
(b) 1,00,001 × 99: m=6, n=2 → 7 to 8 digits [1 mark]
Actual: 1,00,001 × 99 = 99,00,099 → 7 digits ✅
(c) 254 × 44: [1 mark]
= (25 × 4)4 = 1004 = 108 = 10,00,00,000
This has 9 digits.
254 = 3,90,625 (6 digits), 44 = 256 (3 digits)
Rule predicts: 8 to 9 digits. Actual is 9 digits ✅
(a) 9,999 × 9,999: m=4, n=4 → 7 to 8 digits [1 mark]
Actual: 9,9992 = 9,99,80,001 → 8 digits ✅
(b) 1,00,001 × 99: m=6, n=2 → 7 to 8 digits [1 mark]
Actual: 1,00,001 × 99 = 99,00,099 → 7 digits ✅
(c) 254 × 44: [1 mark]
= (25 × 4)4 = 1004 = 108 = 10,00,00,000
This has 9 digits.
254 = 3,90,625 (6 digits), 44 = 256 (3 digits)
Rule predicts: 8 to 9 digits. Actual is 9 digits ✅
Q15. [3 Marks]
(a) Estimation: [1 mark]
North: 1,24,850 ≈ 1,20,000
South: 2,07,436 ≈ 2,10,000
East: 98,715 ≈ 1,00,000
Estimated total = 1,20,000 + 2,10,000 + 1,00,000 = 4,30,000
(b) Actual total and error: [1 mark]
1,24,850 + 2,07,436 + 98,715 = 4,31,001
Error = |4,31,001 − 4,30,000| = 1,001 (about 0.23% error)
(c) More books needed: [1 mark]
5,00,000 − 4,31,001 = 68,999
International: Sixty-eight thousand nine hundred ninety-nine
North: 1,24,850 ≈ 1,20,000
South: 2,07,436 ≈ 2,10,000
East: 98,715 ≈ 1,00,000
Estimated total = 1,20,000 + 2,10,000 + 1,00,000 = 4,30,000
(b) Actual total and error: [1 mark]
1,24,850 + 2,07,436 + 98,715 = 4,31,001
Error = |4,31,001 − 4,30,000| = 1,001 (about 0.23% error)
(c) More books needed: [1 mark]
5,00,000 − 4,31,001 = 68,999
International: Sixty-eight thousand nine hundred ninety-nine
Q16. [4 Marks]
(a) Mars distance ÷ Moon distance: [1 mark]
68,00,00,000 ÷ 3,84,000 = 1,770.8...
Rounded to nearest hundred: 1,800 times
(b) Journey time at 36,000 km/hr: [1 mark]
Moon: 3,84,000 ÷ 36,000 = 10.67 hours ≈ 11 hours (less than 1 day)
Mars: 68,00,00,000 ÷ 36,000 = 18,889 hours
In days: 18,889 ÷ 24 = 787 days (≈ 2 years 2 months)
(c) International system: [1 mark]
68,00,00,000 = 680,000,000 = 680 million km
(d) Light travel time: [1 mark]
Moon: 3,84,000 ÷ 3,00,000 = 1.28 seconds
Mars: 68,00,00,000 ÷ 3,00,000 = 2,267 seconds ≈ 37.8 minutes
68,00,00,000 ÷ 3,84,000 = 1,770.8...
Rounded to nearest hundred: 1,800 times
(b) Journey time at 36,000 km/hr: [1 mark]
Moon: 3,84,000 ÷ 36,000 = 10.67 hours ≈ 11 hours (less than 1 day)
Mars: 68,00,00,000 ÷ 36,000 = 18,889 hours
In days: 18,889 ÷ 24 = 787 days (≈ 2 years 2 months)
(c) International system: [1 mark]
68,00,00,000 = 680,000,000 = 680 million km
(d) Light travel time: [1 mark]
Moon: 3,84,000 ÷ 3,00,000 = 1.28 seconds
Mars: 68,00,00,000 ÷ 3,00,000 = 2,267 seconds ≈ 37.8 minutes
Q17. [4 Marks]
(a) Daily revenue per model: [1 mark]
Basic: 8,999 × 1,250 = ₹1,12,48,750
Standard: 17,499 × 840 = ₹1,46,99,160
Premium: 42,999 × 320 = ₹1,37,59,680
Standard earns the most!
(b) Total daily revenue: [1 mark]
1,12,48,750 + 1,46,99,160 + 1,37,59,680 = ₹3,97,07,590
(c) Annual revenue: [1 mark]
3,97,07,590 × 365 = ₹14,49,32,70,350
≈ ₹1,449 crore (rounded to nearest crore)
(d) Extra Premium phones for ₹1,000 crore: [1 mark]
Target annual: 1,000 crore = 10,00,00,00,000
Current annual: 14,49,32,70,350 (already exceeds ₹1,000 crore!)
No extra phones needed — current revenue already exceeds the target.
(If the question means ₹1,000 crore EXTRA: need 10,00,00,00,000 ÷ 365 ÷ 42,999 ≈ 64 extra Premium phones/day)
Basic: 8,999 × 1,250 = ₹1,12,48,750
Standard: 17,499 × 840 = ₹1,46,99,160
Premium: 42,999 × 320 = ₹1,37,59,680
Standard earns the most!
(b) Total daily revenue: [1 mark]
1,12,48,750 + 1,46,99,160 + 1,37,59,680 = ₹3,97,07,590
(c) Annual revenue: [1 mark]
3,97,07,590 × 365 = ₹14,49,32,70,350
≈ ₹1,449 crore (rounded to nearest crore)
(d) Extra Premium phones for ₹1,000 crore: [1 mark]
Target annual: 1,000 crore = 10,00,00,00,000
Current annual: 14,49,32,70,350 (already exceeds ₹1,000 crore!)
No extra phones needed — current revenue already exceeds the target.
(If the question means ₹1,000 crore EXTRA: need 10,00,00,00,000 ÷ 365 ÷ 42,999 ≈ 64 extra Premium phones/day)
Q18. Bonus [4 Marks]
(a) 111112 using the pattern: [1 mark]
Following the pattern: 123454321
(Digits go up to the number of 1s, then come back down)
(b) 1111111112 (nine 1s): [1 mark]
Pattern gives: 12345678987654321
Count digits: 9 going up + 8 coming down = 17 digits
(c) Sum of digits of 111112 = 123454321: [1 mark]
1+2+3+4+5+4+3+2+1 = 25
(Alternatively: 52 = 25, since the sum of digits of 111...1n2 = n2)
(d) Does pattern hold for ten 1s? [1 mark]
No! The pattern breaks when there are more than 9 ones.
With 10 ones, the “middle” would need digit “10” which doesn’t fit in one digit place.
Carrying occurs: 11111111112 = 1,23456790,8765432,1 — the carries propagate and the simple palindrome pattern breaks.
Actual: 11111111112 = 1234567900987654321
Following the pattern: 123454321
(Digits go up to the number of 1s, then come back down)
(b) 1111111112 (nine 1s): [1 mark]
Pattern gives: 12345678987654321
Count digits: 9 going up + 8 coming down = 17 digits
(c) Sum of digits of 111112 = 123454321: [1 mark]
1+2+3+4+5+4+3+2+1 = 25
(Alternatively: 52 = 25, since the sum of digits of 111...1n2 = n2)
(d) Does pattern hold for ten 1s? [1 mark]
No! The pattern breaks when there are more than 9 ones.
With 10 ones, the “middle” would need digit “10” which doesn’t fit in one digit place.
Carrying occurs: 11111111112 = 1,23456790,8765432,1 — the carries propagate and the simple palindrome pattern breaks.
Actual: 11111111112 = 1234567900987654321