IMO Practice Test — Subtraction Strategies

Explanation: Start at 20, count back 8 steps: 19,18,17,16,15,14,13,12 → 8 numbers said. If including starting number, 9 total. Question asks "how many numbers do you say" meaning count-back numbers: 8 numbers. Answer A.

Explanation: Fact family: three numbers. Sum = 27. Smallest = 4. Other two sum to 23. In a fact family, largest = sum of other two. So largest = 4 + middle? Wait, fact family: a, b, c where a+b=c. Sum of all three = a+b+c = c+c = 2c = 27 → c=13.5 not integer. Re-think: Sum of all three numbers in family = 27. If numbers are x, y, z with x+y=z, then sum = x+y+z = z+z = 2z = 27 → z=13.5 impossible. So maybe interpretation: three numbers in fact family (like 4,9,13) sum = 4+9+13=26. Not 27. For 4,11,15 sum=30. For 4,12,16 sum=32. For 4,8,12 sum=24. None 27. Likely intended: numbers 4,11,12? 4+11=15 not 12. 4+12=16. No. 4+13=17. Check option D: 4,11,12 doesn't work (4+11=15≠12). Option A: 4,9,14 → 4+9=13≠14. Option B: 4,8,15 → 4+8=12≠15. Option C: 4,10,13 → 4+10=14≠13. Option D: 4,11,12 → 4+11=15≠12. None work. Question may have typo. Assuming fact family where largest = sum, and sum of all three = 2×largest = 27 → largest = 13.5 impossible. Pass.

Explanation: First: 25 - ? = 13 → ? = 12. Then 8 + 12 = 20

Explanation: 17-9=8. Check each: 15-7=8, 14-6=8, 16-8=8, 13-5=8. All equal 8. Question asks "same answer" but all same? Actually 17-9=8, all options also =8. Question may test which related fact uses same numbers? 16-8 uses 8 which appears in both. But all correct. Possibly intended: which uses fact family 8,9,17? 17-9=8, 17-8=9. Only option with 8 and 9 is 17-9. None have 17. So answer is all. But IMO must pick one. Likely C (16-8) as alternate.

Explanation: Let missing number = x. Equation: x - 12 = x/2 (since answer is x/2). Multiply 2: 2x - 24 = x → x = 24. Check: 24-12=12, answer is 12, which is half of 24 ✓

Explanation: Start 30. Subtract 3: 27(1),24(2),21(3),18(4),15(5),12(6),9(7). After 7 steps he reaches 9 (<10). Steps = 7. Answer A.