1 · Multiply before you add
11 + 6 × 3 is not 17 × 3. Multiplication and division go first, then addition and subtraction. 6 × 3 = 18, then 11 + 18 = 29.
20 − 8 ÷ 2: 8 ÷ 2 = 4 first, then 20 − 4 = 16. Brackets beat everything: (20 − 8) ÷ 2 = 6.
2 · Factors
A factor divides a number with nothing left over. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Find them in pairs: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6.
"Which is a factor of 48?" Test each choice: does 48 ÷ it come out even?
3 · Multiples
A multiple is what you get by multiplying: multiples of 7 are 7, 14, 21, 28, 35... Multiples of 5 end in 0 or 5. Multiples of 10 end in 0.
A number that is a multiple of both 4 and 6: 12, 24, 36. The smallest is 12.
4 · Odd and even rules
Even + even = even. Odd + odd = even. Even + odd = odd. Any number × even = even. Odd × odd = odd.
So the sum of two odd numbers is always even, and 3 × 7 × 9 must be odd. You can rule out choices without calculating.
5 · Consecutive numbers
Consecutive means in a row: 20, 21, 22. "Three consecutive numbers add up to 63." The middle number is 63 ÷ 3 = 21, so they are 20, 21, 22. The largest is 22.
For three in a row, the middle one is always the total ÷ 3.
6 · Number riddles
"A number is greater than 29, less than 40, is odd, and its digits add up to 10." Take the clues in order. Between 29 and 40: 30 to 39. Odd: 31, 33, 35, 37, 39. Digits add to 10: 3 + 7 = 10. The number is 37.
Each clue crosses out more numbers. Keep a list.
7 · Working backwards, two steps
"A number is multiplied by 6, then 4 is subtracted, giving 50." Undo in reverse order: add 4 back (54), then divide by 6 (9).
If the question then asks for double the number, do not stop at 9: 2 × 9 = 18. Read the last sentence twice.
8 · Two-rule machines
"Put in 5 → 15, put in 9 → 23." Try × 2 + 5: 5 × 2 + 5 = 15 ✓, 9 × 2 + 5 = 23 ✓. So 14 gives 14 × 2 + 5 = 33.
Trick to find the rule: when the In goes up by 1, how much does the Out go up? That number is the multiplier.
9 · Patterns that multiply
2, 6, 18, 54: each number is × 3 the one before. Next: 162.
If the gaps grow fast, suspect multiplying. If the gaps are the same, it is adding.
10 · Halfway and between
"Exactly halfway between 200 and 800": add and halve, (200 + 800) ÷ 2 = 500. Or half the gap: 600 ÷ 2 = 300, then 200 + 300.
It works for any two numbers: halfway between 720 and 1,320 is 1,020.
11 · Which must be true?
"Sam has more than Ali. Ali has more than Kim." What must be true? Sam has more than Kim. What might be true but is not forced? Kim has 5.
Pick only what the clues force. If you can imagine a case where it fails, it is not "must."
12 · Estimate, then check
Before choosing an answer, ask "about how big should it be?" 23 × 14 is about 20 × 15 = 300, so 322 is believable and 3,220 is not.
A quick estimate catches most slips. You are ready for the test! 🎉
Ready to check what stuck?
Take the free 24-question test. It starts easy and builds to harder, ISEE-style reasoning. It's auto-scored, and you can review every answer afterward.