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Practice Number Puzzles questions for the Cognitive Abilities Test — Grade 4 (Level 10). Every question includes a full explanation of why the correct answer is right and why the tempting distractors are wrong.
20 questions available · medium difficulty · CogAT Grade 4 · Free, no registration required
The child is given an equation with a missing value, or a pair of equations sharing a symbol, and works out what the unknown must be. It is the most school-like of the CogAT subtests and the closest thing on the test to early algebra, though it is pitched well below the arithmetic a child of the age meets in class.
Two equations that share a symbol, so the first has to be solved before the second can be. Multiplication and division appear, and a symbol stands for the same value in both lines, which makes deciding where to start part of the problem.
At level 10 — grade 4
Level 10 introduces paired equations sharing a symbol, along with multiplication and division. The new skill is deciding which equation to solve first, and it is worth teaching directly - fourth graders who work through the lines in the order given will stall on items they could otherwise complete.
When two equations share a symbol, one of them can be solved alone and the other cannot. Starting with the wrong line leads to a dead end, and the fix is to find the equation with only one unknown first.
A symbol means the same number in every line it appears in. Solving each equation independently produces two different values for the same shape, and that inconsistency is the signal the item has been misread.
The reasoning at this level is not difficult, so most lost marks are calculation errors rather than conceptual ones. Substituting back is what catches them, and it remains the most reliable few seconds a child can spend.
Write missing-number equations on paper and let your child fill them in, using a shape rather than a letter for the unknown. The aim is for the format to feel ordinary, since a good deal of the difficulty at this age is unfamiliarity rather than arithmetic. Insist on the check afterwards - read the equation back with the number in place and confirm both sides match.
It is the reasoning that algebra formalises - a symbol standing for an unknown - but with arithmetic well below the level a child of this age meets in class. No algebraic technique is needed or expected.
Because a shape is more approachable for children who have not met letters as variables. It means the same thing, and getting used to the convention is worth a few minutes of exposure.
Substituting the answer back into the equation and checking it balances. It takes seconds and catches nearly every error children make on these items.
10 example questions with full explanations. Use the interactive practice above to work through the complete set.
Correct answer: C
The missing number is how many sixes make 54, which is 54 ÷ 6 = 9. That makes C correct. Choice A comes from subtracting 6 from 54 instead of dividing. Choice B comes from adding 6, and choice D from multiplying 54 by 6.
Correct answer: D
Work out the side with no missing number first: 8 + 9 = 17. The other side must also come to 17, so you need the number that leaves 17 when taken from 25, which is 25 − 17 = 8. That makes D correct. Choice A is the total of the right-hand side rather than the missing number, which is the usual slip. Choice B comes from adding 25 and 17, and choice C from adding 25 and 9.
Correct answer: A
The missing number is the one that gives 12 after being split into 4 equal parts, so multiply back: 12 × 4 = 48. That makes A correct. Choice B comes from dividing 12 by 4 instead of multiplying. Choice C comes from adding 4 to 12, and choice D from subtracting 4.
Correct answer: B
Work out the left side first: 7 × 3 = 21. The right side must also come to 21, so the missing number plus 9 is 21, which makes it 12. That makes B correct. Choice A is the total of the left side rather than the missing number, which is the most common mistake here. Choice C comes from adding 9 to 21 instead of taking it away, and choice D repeats the 9 already shown.
Correct answer: D
The missing number is how many equal parts 40 must be split into to give 5 in each, which is 40 ÷ 5 = 8. That makes D correct. Choice A comes from multiplying 40 by 5 instead of dividing. Choice B comes from subtracting 5 from 40, and choice C from adding 5.
Correct answer: A
Undo the two steps in reverse order. Take the 3 away from 27, which leaves 24 for the multiplication part, then split 24 into 4 equal groups: 24 ÷ 4 = 6. That makes A correct. Choice B comes from stopping after taking away the 3 and never dividing. Choice C comes from multiplying the two small numbers shown, 4 × 3, instead of undoing anything. Choice D comes from multiplying 27 by 4 rather than dividing by it.
Correct answer: A
Work out the side with no missing number first: 71 − 5 = 66. The other side must also come to 66, so the missing number plus 26 is 66, which makes it 40. That makes A correct. Choice B is the total of the right-hand side rather than the missing number. Choice C comes from adding 26 to 66 instead of taking it away, and choice D comes from ignoring the 5 and working out 71 − 26.
Correct answer: C
Undo the steps in reverse. Take the 4 away from 12, which leaves 8 for the division part, then multiply back: 8 × 3 = 24. That makes C correct. Choice A comes from stopping after taking away the 4 and never multiplying. Choice B comes from adding the 4 to 12 first and then multiplying by 3, and choice D comes from multiplying 12 by 3 without undoing the 4 at all.
Correct answer: D
The missing number is how many equal parts 72 must be split into to give 9 in each, which is 72 ÷ 9 = 8. That makes D correct. Choice A comes from subtracting 9 from 72 instead of dividing. Choice B comes from multiplying 72 by 9, and choice C repeats the 9 already shown.
Correct answer: B
Undo the steps in reverse. Add the 6 back to 34, which gives 40 for the multiplication part, then divide: 40 ÷ 5 = 8. That makes B correct. Choice A comes from stopping after adding the 6 and never dividing. Choice C comes from subtracting the 6 instead of adding it back, and choice D repeats the 5 already shown in the puzzle.