Multi-Concept Achievers
27 Class 6 IMO Mathematics Olympiad practice questions on multi-concept achievers, part of the HOTS / Achievers topic area. Every question below includes the correct answer and a full explanation.
Practice this topic interactively →1. A 3×3 grid contains the numbers 1 to 9, each used once. If every row and every column must have the same sum, what must that sum be?
- A.12
- B.15
- C.18
- D.20
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Answer: B. 15
1 to 9 sum to 45. Split across 3 equal rows, each row sums to 45÷3=15.
2. A student writes all numbers from 1 to 30. How many times does the digit 2 appear?
- A.12
- B.13
- C.14
- D.15
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Answer: B. 13
The digit 2 appears in the units place in 2, 12, 22 (3 times) and in the tens place in 20-29 (10 times). Total: 3+10=13.
3. A number when divided by 7 gives quotient 12 and remainder 5. What is the number?
- A.84
- B.87
- C.89
- D.91
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Answer: C. 89
Number = divisor×quotient + remainder = 7×12+5 = 89.
4. A table shows the pocket money of 4 friends forming a pattern: ₹50, ₹75, ₹100, ₹125. Continuing the same pattern, what is the 5th friend's pocket money?
- A.140
- B.145
- C.150
- D.155
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Answer: C. 150
Each friend gets ₹25 more than the last: 125 + 25 = 150.
5. A two-digit number has digits whose sum is 10. Reversing its digits makes the number 36 smaller. What is the original number?
- A.37
- B.46
- C.64
- D.73
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Answer: D. 73
Let the digits be t and u with t+u=10. Original−reversed = 9(t−u) = 36, so t−u=4. Solving with t+u=10: t=7, u=3 — the number is 73.
6. There are chickens and rabbits in a pen. There are 12 heads and 32 legs in total. How many rabbits are there?
- A.4
- B.5
- C.6
- D.8
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Answer: A. 4
If all 12 were rabbits, there'd be 48 legs — 16 too many. Each chicken instead of a rabbit removes 2 legs, so there are 16÷2=8 chickens and 12−8=4 rabbits. Check: 4×4+8×2=32.
7. A rectangle has area 96 cm² and its length is 4 cm more than its width. What is its perimeter?
- A.36 cm
- B.40 cm
- C.44 cm
- D.48 cm
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Answer: B. 40 cm
Width w satisfies w(w+4)=96, so w=8 and length=12. Perimeter = 2×(12+8) = 40 cm.
8. Using the digits 1, 2, 3 and 4 without repetition, how many three-digit even numbers can be formed?
- A.6
- B.8
- C.10
- D.12
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Answer: D. 12
The last digit must be 2 or 4 (2 choices). The remaining two positions are filled from the other 3 digits in 3×2=6 ways. Total = 2×6 = 12.
9. A number is chosen from 1 to 100. It is divisible by 3 but not by 6. How many such numbers are there?
- A.16
- B.17
- C.18
- D.19
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Answer: B. 17
Multiples of 3 from 1-100: 33. Multiples of 6 from 1-100: 16. Divisible by 3 but not 6: 33−16=17.
10. Three friends share ₹540. The second gets twice the first, and the third gets ₹60 more than the first. How much does the second friend get?
- A.₹180
- B.₹200
- C.₹240
- D.₹300
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Answer: C. ₹240
Let the first get x. Then x+2x+(x+60)=540, so 4x=480, x=120. The second gets 2×120=240.
11. A bag contains red, blue and green balls. Without looking, what is the minimum number of balls that must be picked to guarantee getting two balls of the same colour?
- A.2
- B.3
- C.4
- D.5
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Answer: C. 4
In the worst case, the first 3 picks could be one of each colour. The 4th pick must repeat a colour.
12. The sum of three consecutive even numbers is 90. What is the largest number?
- A.28
- B.30
- C.32
- D.34
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Answer: C. 32
Let the numbers be x, x+2, x+4. Their sum 3x+6=90 gives x=28, so the largest is x+4=32.
13. A shopkeeper marks up the price of an item by 25% and then gives a 20% discount on the marked price. What is the overall effect on the original price?
- A.No change
- B.10% increase
- C.10% decrease
- D.5% increase
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Answer: A. No change
1.25 × 0.80 = 1.00 — the markup and discount exactly cancel, so the final price equals the original price.
14. The digits of a two-digit number differ by 4. If the number is 4 times the sum of its digits, find the number.
- A.26
- B.48
- C.62
- D.84
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Answer: B. 48
Let tens=t, units=u. u=2t from the given ratio condition, and t−u=−4 gives t=4, u=8 → the number is 48. Check: 4×(4+8)=48.
15. A tank can be filled by pipe A in 20 minutes and by pipe B in 30 minutes. If both pipes are opened together, how long will it take to fill the tank?
- A.10
- B.12
- C.15
- D.18
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Answer: B. 12
Combined rate = 1/20+1/30 = 5/60 = 1/12 of the tank per minute, so it takes 12 minutes.
16. Statement I: The 6th triangular number is 21, which is also a Virahānka number. Statement II: Every hexagonal number is also a perfect square.
- A.Statement I is true, Statement II is false
- B.Statement I is false, Statement II is true
- C.Both are true
- D.Both are false
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Answer: A. Statement I is true, Statement II is false
The 6th triangular number is 21, and 21 also appears in the Virahānka sequence (1,2,3,5,8,13,21,...) — so I is true. But hexagonal numbers like 7, 19, and 37 aren't perfect squares — so II is false.
17. A shop's pictograph shows 1 symbol = 30 kg. Monday's row has 3 symbols, and this is twice what was sold on Tuesday. How many kg were sold on Tuesday?
- A.30
- B.40
- C.45
- D.50
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Answer: C. 45
Monday = 3×30 = 90 kg. Tuesday = 90 ÷ 2 = 45 kg.
18. The HCF of two co-prime numbers is always ___, and their LCM is always ___.
- A.1; the product of the two numbers
- B.1; the sum of the two numbers
- C.0; the product of the two numbers
- D.the smaller number; the larger number
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Answer: A. 1; the product of the two numbers
Co-prime numbers share no common factor besides 1, so their HCF is always 1 — and since they share nothing, their LCM must be their full product.
19. In a complete graph, the number of line segments equals the number of points. How many points does the graph have?
- A.2
- B.3
- C.4
- D.5
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Answer: B. 3
Number of segments = n(n−1)/2. Setting this equal to n gives (n−1)/2 = 1, so n = 3. Check: 3 points give 3×2/2 = 3 segments — equal to the number of points.
20. The temperature in a city was −6°C at midnight and rose steadily by 3°C every hour. What was the temperature after 5 hours?
- A.6°C
- B.9°C
- C.12°C
- D.15°C
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Answer: B. 9°C
−6 + (3×5) = −6 + 15 = 9°C.
21. A number leaves remainder 2 when divided by 5 and remainder 3 when divided by 4. Which is the smallest possible number greater than 10?
- A.13
- B.17
- C.22
- D.27
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Answer: D. 27
Numbers leaving remainder 2 mod 5: 7, 12, 17, 22, 27... Checking which also leaves remainder 3 mod 4: 27÷4=6 remainder 3. 27 is the smallest such number greater than 10.
22. In a class, 60% of students play cricket and 50% play football. If every student plays at least one of the two sports, what percentage play both?
- A.5%
- B.10%
- C.15%
- D.20%
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Answer: B. 10%
By the rule for overlapping groups: both = cricket% + football% − 100% = 60+50−100 = 10%.
23. A pen has 12 animals made of chickens and rabbits. There are 40 legs. How many rabbits are there?
- A.8
- B.9
- C.7
- D.10
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Answer: A. 8
If all 12 were rabbits there'd be 48 legs — 8 too many. Each chicken instead of a rabbit removes 2 legs, giving 4 chickens and 8 rabbits.
24. A collection has 640 stickers. 3/8 are round. Of the remaining stickers, 1/4 are triangular and the rest are rectangular. How many are rectangular?
- A.340
- B.200
- C.300
- D.240
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Answer: C. 300
640 × 3/8 = 240 round. Remaining 400: 1/4 are triangular (100), leaving 300 rectangular.
25. A number is divisible by 3 but not by 6. Which statement must be true?
- A.The number is even
- B.The number is divisible by 6
- C.The number is divisible by 2
- D.The number is odd
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Answer: D. The number is odd
Divisible by 6 requires divisibility by both 2 and 3. If it's divisible by 3 but not 6, it can't be divisible by 2 — so it must be odd.
26. A pen has 16 animals made of chickens and rabbits. There are 46 legs. How many rabbits are there?
- A.9
- B.7
- C.8
- D.6
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Answer: B. 7
If all 16 were rabbits there'd be 64 legs — 18 too many. Each chicken instead of a rabbit removes 2 legs, giving 9 chickens and 7 rabbits.
27. A pen has 8 animals made of chickens and rabbits. There are 26 legs. How many rabbits are there?
- A.5
- B.6
- C.7
- D.4
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Answer: A. 5
If all 8 were rabbits there'd be 32 legs — 6 too many. Each chicken instead of a rabbit removes 2 legs, giving 3 chickens and 5 rabbits.