Determine missing numbers that make an equation or relationship true - CogAT Grades 3 Plus

SECTION 1

Where Did This Idea Come From?

People have been solving for missing numbers for thousands of years. Ancient civilizations needed to figure out unknown amounts when trading goods, building structures, and dividing land. The idea of finding a hidden value in a math sentence is one of the oldest problems in all of mathematics.

On the CogAT Quantitative Battery, you will see equations or number relationships with a blank or question mark. Your job is to figure out which number makes the statement true. This skill tests how well you reason with numbers.

1800 BCE • Ancient Babylon

Babylonian scribes carved clay tablets with problems like "what number added to 5 gives 12?" They solved missing-number puzzles for trade and taxes.

300 BCE • Greek Mathematics

Greek thinkers like Euclid studied relationships between numbers. They explored how equal quantities stay balanced, much like a scale.

820 CE • Al-Khwarizmi's Algebra

The Persian mathematician Al-Khwarizmi wrote the first textbook on algebra. The word "algebra" comes from his book title. He taught systematic ways to find unknowns.

1960s • Cognitive Ability Testing

Researchers began using missing-number problems to measure reasoning ability. These questions became a key part of tests like the CogAT.

Today, finding missing numbers is not just a math skill. It is a way to show how well you can think logically. The CogAT uses these questions to see how you reason with number relationships and patterns.

SECTION 2

Core Principles & Definitions

Before you start solving, you need to understand a few key ideas. These principles are the foundation for every missing-number problem you will encounter.

Balance

An equation (a math sentence with an equals sign) must have the same value on both sides. Think of it like a seesaw that stays level.

Inverse Operations

Inverse operations (opposite operations) undo each other. Addition undoes subtraction. Multiplication undoes division. Use them to isolate the missing number.

Number Relationships

Sometimes you are not given an equation. Instead, you see a pattern or rule connecting numbers. Find the rule first, then apply it to find the missing piece.

Substitution Check

After you find your answer, always plug it back in. If both sides are equal (or the pattern holds), you know you are correct. This is called checking your work.

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KEY TAKEAWAY

Think of an equation like a balanced scale. If you know the total weight on one side and part of the weight on the other side, you can figure out the missing weight. The equals sign is the middle of the scale — both sides must match!

SECTION 3

Seeing the Balance

A great way to understand missing-number problems is to picture a balance scale. The diagram below shows how the equation 5 + ? = 12 works. The left side holds 5 and an unknown amount. The right side holds 12. For the scale to balance, the unknown must be 7.

The blue box holds 5, the yellow box holds the unknown (?), and the green box holds 12. Both sides of the scale must equal the same total. Since 12 − 5 = 7, the missing number is 7.

Notice the dashed arrow in the diagram. It connects the unknown (?) to the solution box below. The key step is using the inverse operation. Since the equation uses addition, you undo it with subtraction. This is the most important strategy for CogAT number equation questions.

SECTION 4

The Math Behind Finding Missing Numbers

Every missing-number problem follows a simple idea: use the opposite operation to undo what is done to the unknown. Here are the four main types you will see.

ADDITION TYPE

a + ? = c → ? = c − a

If a number is added to the unknown, subtract it from the total to find the missing value. Example: 8 + ? = 15, so ? = 15 − 8 = 7.

SUBTRACTION TYPE

a − ? = c → ? = a − c

If the unknown is subtracted, find the difference between the starting number and the result. Example: 20 − ? = 13, so ? = 20 − 13 = 7.

MULTIPLICATION TYPE

a × ? = c → ? = c ÷ a

If the unknown is multiplied, divide the product by the known factor. Example: 6 × ? = 42, so ? = 42 ÷ 6 = 7.

DIVISION TYPE

a ÷ ? = c → ? = a ÷ c

If the unknown is a divisor, divide the original number by the result. Example: 56 ÷ ? = 8, so ? = 56 ÷ 8 = 7.

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CogAT Tip

On the CogAT, you usually pick from multiple-choice answers. A quick way to check is to plug each answer choice back into the equation. The one that makes both sides equal is correct!

SECTION 5

Types of Missing-Number Problems on the CogAT

The CogAT Quantitative Battery does not always give you a simple equation with one blank. You might also see number analogies, number series, and number puzzles. The diagram below shows these three categories and how they work.

This diagram shows the three main categories of missing-number problems: Equations (balance both sides), Series (find and extend a pattern), and Analogies (apply the same rule to a new pair).

Problem Type	What You See	What to Do
Equation	? + 14 = 30	Use inverse operations: 30 − 14 = 16
Series	4, 8, 16, 32, ?	Find the rule (×2), then apply it: 32 × 2 = 64
Analogy	[10 → 5] [18 → ?]	Find the rule (÷2), then apply it: 18 ÷ 2 = 9
Two-step	3 × ? + 2 = 17	Undo addition first (17 − 2 = 15), then division (15 ÷ 3 = 5)

Common problem types on the CogAT Quantitative Battery

SECTION 6

Worked Example: Step by Step

Let's walk through a CogAT-style problem together. Read each step carefully and notice how we use inverse operations to find the answer.

Find the missing number: 4 × ? − 6 = 22

1
Step 1 — Read the Problem

We need to find the number that, when multiplied by 4 and then reduced by 6, gives 22. There are two operations here: multiplication and subtraction.
2
Step 2 — Undo the Last Operation First

The last thing done to the number was subtracting 6. To undo subtraction, we add 6 to both sides.

4 × ? = 22 + 6 = 28
3
Step 3 — Undo the Remaining Operation

Now we have 4 × ? = 28. To undo multiplication by 4, we divide by 4.

? = 28 ÷ 4 = 7
4
Step 4 — Check Your Answer

Plug 7 back in: 4 × 7 − 6 = 28 − 6 = 22. ✓ Both sides match, so our answer is correct!

The missing number is 7.

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Remember!

When a problem has two operations, always undo the last operation first. Work backwards, like rewinding a video.

SECTION 7

Strategies, Strengths & Common Mistakes

Knowing the math is only half the battle. On a timed test like the CogAT, you also need smart strategies. Here is a comparison of the best approaches and common pitfalls to avoid.

Strategy	When to Use It	Watch Out For
Inverse Operations	Standard equations with one unknown	Using the wrong inverse (e.g., adding instead of subtracting)
Plug and Check	Multiple-choice questions when you are unsure	Takes more time; start with the middle answer choice to save time
Find the Pattern	Series and analogy problems	Assuming addition when the pattern uses multiplication or another rule
Work Backwards	Multi-step equations	Forgetting to undo the last operation first
Estimation	Eliminating wrong answer choices quickly	Relying on estimation alone without checking

Strategy comparison for missing-number problems

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KEY TAKEAWAY

Think of solving a missing-number problem like unlocking a combination lock. Each step you "undo" is like turning the dial one click closer to the answer. If you undo the steps in the wrong order, the lock won't open. Always reverse the last step first!

SECTION 8

Connecting to Algebra and Beyond

Finding missing numbers is actually the beginning of algebra. In algebra, the question mark becomes a letter like x or n. The strategies you learn here will carry you through high school math and beyond.

What You See on CogAT	What It Looks Like in Algebra
? + 9 = 15	x + 9 = 15
3 × ? = 21	3x = 21
2 × ? + 5 = 19	2x + 5 = 19
[4 → 16] [7 → ?]	f(n) = n², find f(7)

CogAT problems are early algebra in disguise

As you move into higher grades, you will work with equations that have two unknowns, exponents, and even negative numbers. But the core idea stays the same: use inverse operations and logical reasoning to uncover what is hidden. Mastering this now gives you a huge head start.

SECTION 9

Practice Problems

Try these five problems on your own. They get harder as you go. After you try each one, read the answer to see if you are right and learn from the explanation.

Question 1: CONCEPTUAL

In the equation 9 + ? = 17, what does the question mark stand for? What operation would you use to find it?

Show answer

The question mark stands for the missing number that, when added to 9, gives 17. To find it, you use the inverse of addition, which is subtraction. You calculate 17 − 9 = 8. The missing number is 8. You can check: 9 + 8 = 17. ✓

Question 2: BASIC CALCULATION

Find the missing number: 7 × ? = 63

Show answer

Since the unknown is multiplied by 7, use the inverse operation: divide. Calculate 63 ÷ 7 = 9. The missing number is 9. Check: 7 × 9 = 63. ✓ This is a straightforward multiplication/division problem.

Question 3: INTERMEDIATE

Find the missing number: 5 × ? − 8 = 32

Show answer

This is a two-step problem. First, undo the subtraction by adding 8 to both sides: 5 × ? = 32 + 8 = 40. Next, undo the multiplication by dividing by 5: ? = 40 ÷ 5 = 8. The missing number is 8. Check: 5 × 8 − 8 = 40 − 8 = 32. ✓ Remember to always undo the last operation first.

Question 4: APPLIED

Look at this number analogy: [6 → 36] [8 → 64] [11 → ?]. What is the missing number?

Show answer

First, find the rule. Notice that 6 × 6 = 36 and 8 × 8 = 64. The rule is to square the number (multiply it by itself). So for 11, calculate 11 × 11 = 121. The missing number is 121. This is a number analogy problem. The key was noticing the squaring pattern, not just adding or multiplying by a fixed number.

Question 5: CRITICAL THINKING

In the number series 3, 5, 9, 17, ?, each number follows a rule based on the previous number. What is the missing number? Describe the rule.

Show answer

Look at the differences: 5 − 3 = 2, 9 − 5 = 4, 17 − 9 = 8. The differences are 2, 4, 8 — each difference doubles! The next difference would be 8 × 2 = 16. So the missing number is 17 + 16 = 33. Another way to see the rule: each number is doubled and then 1 is subtracted (3 × 2 − 1 = 5, 5 × 2 − 1 = 9, 9 × 2 − 1 = 17, 17 × 2 − 1 = 33). On the CogAT, try looking at both the differences between numbers and how each number relates to the one before it.

SUMMARY

Putting It All Together

Summary

Finding missing numbers is about keeping things balanced. Whether you face an equation, a number series, or a number analogy, the process follows the same path. First, figure out the operation or pattern. Then use inverse operations or extend the rule to solve for the unknown. Finally, check your answer by plugging it back in.

On the CogAT Quantitative Battery, use smart strategies to save time: try working backwards for multi-step problems and plug and check when you are stuck. Remember, addition undoes subtraction, and multiplication undoes division. These inverse pairs are your most powerful tools. Master them now, and you will be ready for algebra and beyond!