SECTION 1
Where Did These Operations Come From?
People have been adding, subtracting, multiplying, and dividing for thousands of years. Ancient civilizations needed these skills to trade goods, measure land, and build structures. The four basic operations are the building blocks of all mathematics, and every math problem you will ever solve uses at least one of them.
3000 BCE • Ancient Egypt & Mesopotamia
500 CE • Indian Mathematicians
1200 CE • Fibonacci Brings It to Europe
1557 CE • The Equals Sign Is Born
On the ISEE, you do not have a calculator. That means your ability to add, subtract, multiply, and divide quickly and correctly is your biggest advantage. Let's sharpen those skills so you feel confident on test day.
SECTION 2
Core Principles of the Four Operations
Each operation has its own purpose and set of rules. Understanding what each one does — and how they relate to each other — is the key to solving problems accurately.
Addition (+)
Subtraction (−)
Multiplication (×)
Division (÷)
KEY TAKEAWAY
ISEE Test Strategy
SECTION 3
Seeing the Four Operations
A picture can make abstract ideas click. The diagram below shows how the four operations connect to each other. Notice how each operation sits across from its inverse.
Notice the dashed arrows between opposite pairs. Whenever you add, you can undo it by subtracting. Whenever you multiply, you can undo it by dividing. This relationship is your secret weapon for checking answers on the ISEE.
SECTION 4
Key Properties and Rules
Certain rules — called properties — help you compute faster and avoid mistakes. Let's look at the most important ones.
COMMUTATIVE PROPERTY
You can swap the order of numbers when adding or multiplying. This does not work for subtraction or division! 10 − 3 ≠ 3 − 10.
IDENTITY PROPERTY
Adding 0 or multiplying by 1 leaves any number unchanged. Zero is the additive identity; one is the multiplicative identity.
DISTRIBUTIVE PROPERTY
This lets you break tough multiplications into easier parts. For example, 7 × 23 = 7 × 20 + 7 × 3 = 140 + 21 = 161.
ORDER OF OPERATIONS (PEMDAS)
When a problem has more than one operation, follow this order. Multiply and divide go left to right, then add and subtract go left to right.
Common ISEE Trap
SECTION 5
Techniques for Accurate Computation
Speed matters on the ISEE, but accuracy matters more. A quick answer that is wrong earns you zero points. Here are proven techniques to stay fast and accurate.
- ✓ Estimate first. Round the numbers, do a quick mental calculation, and use that to eliminate answer choices that are way too big or way too small.
- ✓ Line up your columns. In addition and subtraction, make sure the ones place, tens place, and hundreds place are perfectly aligned. One misaligned digit can ruin the entire answer.
- ✓ Watch for borrowing and carrying. The most common mistakes in subtraction come from forgetting to regroup (borrow). Double-check any column where the top digit is smaller than the bottom.
- ✓ Check with the inverse. After subtracting, add your answer to the smaller number. After dividing, multiply your answer by the divisor. If you get back to the original number, you are correct.
SECTION 6
Worked Example: Multi-Step Problem
ISEE problems often require more than one operation. Let's work through a realistic problem step by step.
A school orders 24 boxes of pencils. Each box holds 36 pencils. If the school gives away 158 pencils and divides the rest equally among 7 classrooms, how many pencils does each classroom get?
- 1
Step 1 — Find the Total Number of Pencils
Multiply the number of boxes by the number of pencils per box: 24 × 36. Break it up: 24 × 30 = 720 and 24 × 6 = 144.24 × 36 = 720 + 144 = 864 pencils - 2
Step 2 — Subtract the Pencils Given Away
Subtract the 158 pencils that were given away from the total: 864 − 158. Line up the digits carefully and regroup (borrow) where needed.864 − 158 = 706 pencils remaining - 3
Step 3 — Divide Among 7 Classrooms
Divide 706 by 7. Think: 7 × 100 = 700, so 706 ÷ 7 = 100 with remainder 6. But since we're dividing pencils, each classroom gets 100 pencils and there are 6 left over.706 ÷ 7 = 100 pencils per classroom (with 6 left over) - 4
Step 4 — Quick Check with Estimation
Estimate: 24 × 36 ≈ 25 × 36 = 900. Then 900 − 158 ≈ 740. Finally 740 ÷ 7 ≈ 106. Our answer of 100 is close and reasonable. ✓Answer: 100 pencils per classroom
SECTION 7
Common Mistakes and How to Avoid Them
On the ISEE, the wrong answer choices are designed to match common errors. If you know the typical mistakes, you can avoid falling into these traps.
| Mistake | Example | How to Fix It |
|---|---|---|
| Forgetting to regroup (borrow) in subtraction | 503 − 267: writing 344 instead of 236 | Always check: add your answer to the bottom number. 236 + 267 = 503 ✓ |
| Misaligning place values in addition | 1,245 + 83: treating 83 as 830 | Line up the ones digits first, then work left column by column. |
| Ignoring order of operations | 8 + 2 × 5 = 50 instead of 18 | Always multiply and divide before adding and subtracting (PEMDAS). |
| Dropping a zero in multiplication | 405 × 3 = 125 instead of 1,215 | Be careful with zeros as placeholders. Estimate first: 400 × 3 = 1,200. |
| Wrong remainder in division | 83 ÷ 9 = 9 R 1 instead of 9 R 2 | Multiply quotient × divisor, then subtract: 9 × 9 = 81, and 83 − 81 = 2. |
Five common arithmetic mistakes and their fixes
KEY TAKEAWAY
SECTION 8
Connecting to Bigger ISEE Topics
The four operations don't just appear in basic arithmetic problems. They are the engine behind almost every other topic on the ISEE. Once you master them, other topics become much easier.
| ISEE Topic | How the Four Operations Are Used |
|---|---|
| Fractions | You add, subtract, multiply, and divide fractions using the same core operations, just with extra steps like finding common denominators. |
| Percents | Finding 25% of 80 means multiplying: 0.25 × 80 = 20. Percent increase and decrease use subtraction and division too. |
| Equations | Solving x + 15 = 42 requires subtraction. Solving 3x = 72 requires division. Every equation is solved by doing inverse operations. |
| Area & Perimeter | Perimeter uses addition. Area uses multiplication. Finding a missing side from the area requires division. |
| Mean (Average) | To find the mean, you add all the values and then divide by how many there are. Both operations must be accurate. |
How the four operations connect to advanced ISEE topics
As you move into more advanced math, the four operations never go away — they just get dressed up in new situations. If your arithmetic is solid, you can focus your brainpower on the new concepts instead of worrying about computation errors.
SECTION 9
Practice Problems
Try these five problems. Remember: no calculator, just like on the real ISEE! Take your time, show your work on scratch paper, and use estimation to check your answers.
Question 1: CONCEPTUAL
- A. 6,802
- B. 6,812
- C. 6,912
- D. 7,012
Show answer (C)
Question 2: BASIC CALCULATION
- A. 2,245
- B. 2,255
- C. 2,345
- D. 3,245
Show answer (A)
Question 3: INTERMEDIATE
- A. 1,580
- B. 1,680
- C. 1,780
- D. 1,880
Show answer (B)
Question 4: APPLIED
- A. $96
- B. $108
- C. $120
- D. $144
Show answer (C)
Question 5: CRITICAL THINKING
- A. 6
- B. 18
- C. 21
- D. 36
Show answer (C)
SUMMARY
Wrapping It All Up
Summary
The four basic operations — addition, subtraction, multiplication, and division — are the foundation of every ISEE math problem. Addition and subtraction are inverse operations, and so are multiplication and division. Use this relationship to check your work.
Always follow PEMDAS (order of operations) when a problem has more than one operation. Use estimation to eliminate wrong answer choices quickly, and rely on the distributive property to break hard multiplications into easier pieces. Line up your columns, watch for regrouping (borrowing and carrying), and always double-check with the inverse operation. You've got this!