SECTION 1
Where Did Fractions Come From?
Imagine you need to split a loaf of bread fairly among friends. That everyday problem is exactly why fractions were invented. Ancient civilizations needed a way to describe parts of things — land, food, and time — long before decimals existed.
~1800 BCE • Egyptian Fractions
~500 CE • Indian Fraction Notation
~1200 CE • The Fraction Bar Appears
1500s • Mixed Numbers Become Standard
Today, adding and subtracting fractions is one of the most tested skills on the ISEE. Understanding this topic helps you solve problems about recipes, distances, time, and much more. Let's build your skills step by step.
SECTION 2
Core Principles You Need to Know
Before you add or subtract any fractions, there are a few big ideas to lock in. Think of these as the rules of the game. Once you know them, every problem follows the same pattern.
Common Denominator
Least Common Denominator (LCD)
Equivalent Fractions
Simplify Your Answer
Convert Mixed Numbers
KEY TAKEAWAY
SECTION 3
Seeing Fractions Side by Side
A picture makes the idea of a common denominator click instantly. The diagram below shows what happens when you add 1/3 + 1/4. Notice how both bars are rewritten using twelfths so you can combine the shaded parts.
In the diagram, each small section of the bottom bar represents one twelfth. You can count seven shaded sections to verify the answer. This visual approach works for any pair of fractions — just find the common denominator, redraw, and combine.
SECTION 4
The Step-by-Step Method
Here are the formulas and procedures you will use on every fraction addition or subtraction problem. Memorize these steps and you will be ready for any ISEE question involving fractions.
Adding or Subtracting Simple Fractions
SAME DENOMINATOR
When the denominators match, just add (or subtract) the numerators. Keep the denominator the same.
DIFFERENT DENOMINATORS
Cross-multiply to create equivalent fractions with the same denominator, then add the numerators. Always simplify at the end.
Converting Mixed Numbers
MIXED → IMPROPER
Multiply the whole number (W) by the denominator (d), then add the numerator (n). Put the result over the original denominator. Example: 3⅖ → (3 × 5 + 2)/5 = 17/5.
IMPROPER → MIXED
Divide the numerator by the denominator. The quotient is the whole number, and the remainder goes over the denominator.
ISEE Test Tip
SECTION 5
Subtracting with Borrowing (Regrouping)
Sometimes when you subtract mixed numbers, the fraction part you are taking away is bigger than the fraction part you have. In that case, you need to borrow (or regroup) from the whole number. This is similar to borrowing in whole-number subtraction.
The trick is that when you borrow 1, you rewrite it as a fraction whose numerator and denominator are the same as the denominator you're already working with. If the denominator is 4, you write the borrowed 1 as 4/4. If the denominator is 6, you write it as 6/6. Then add that to the fraction you already have.
Quick Check
SECTION 6
Worked Example: Adding Mixed Numbers
Let's walk through a full problem from start to finish. This is the kind of question you will see on the ISEE. Watch each step carefully.
Solve: 2 2/3 + 4 3/5
- 1
Step 1 — Convert to Improper Fractions
For 2⅔: multiply 2 × 3 = 6, then add 2 to get 8. Write 8/3. For 4⅗: multiply 4 × 5 = 20, then add 3 to get 23. Write 23/5.8/3 + 23/5 - 2
Step 2 — Find the LCD
The denominators are 3 and 5. List multiples of each: 3, 6, 9, 12, 15… and 5, 10, 15… The smallest number in both lists is 15.LCD = 15 - 3
Step 3 — Create Equivalent Fractions
For 8/3, multiply top and bottom by 5: 8 × 5 = 40 and 3 × 5 = 15, giving 40/15. For 23/5, multiply top and bottom by 3: 23 × 3 = 69 and 5 × 3 = 15, giving 69/15.40/15 + 69/15 - 4
Step 4 — Add the Numerators
Add 40 + 69 = 109. Keep the denominator 15.109/15 - 5
Step 5 — Convert Back to a Mixed Number and Simplify
Divide 109 ÷ 15 = 7 remainder 4. So the answer is 7 4/15. Check: the GCF of 4 and 15 is 1, so this fraction is already in simplest form.7 4/15
Alternative Method
SECTION 7
Comparing Two Approaches
There are two main ways to add or subtract mixed numbers. Both give the same answer. The table below compares them so you can choose the one that works best for you on test day.
| Feature | Improper Fraction Method | Separate Parts Method |
|---|---|---|
| What you do | Convert all mixed numbers to improper fractions, find LCD, add/subtract, convert back. | Handle whole numbers and fractions separately, then combine. |
| Best for | Subtraction with borrowing; complex problems with many fractions. | Simple addition; when the fractions don't create an improper result. |
| Risk | Larger numbers can lead to multiplication errors. | You might forget to borrow when subtracting. |
| Speed | Slightly slower but very reliable. | Faster for easy problems. |
| ISEE Recommendation | Great default strategy. Works every time. | Use when both fractions have the same denominator. |
Comparison of two methods for mixed number operations
TEST DAY STRATEGY
SECTION 8
Connection to Harder Topics
Adding and subtracting fractions is a foundation skill that connects to many more advanced math topics. The ISEE tests some of these connections, and you'll use fractions even more in algebra and beyond.
| This Lesson | Where It Leads |
|---|---|
| Finding a common denominator | Solving equations with fractions (e.g., x/3 + x/5 = 8) |
| Converting mixed numbers | Multiplying and dividing mixed numbers |
| Simplifying fractions | Simplifying ratios and proportions |
| Borrowing in subtraction | Working with negative numbers and integers |
How fraction skills connect to future topics
Every time you master fraction operations, you are building a skill that makes the next math topic easier. Think of each lesson as a building block — and this one is one of the most important blocks in the whole structure.
SECTION 9
Practice Problems
Try these five problems. They go from easier to harder, just like the ISEE. Remember: there is no penalty for guessing, so always pick an answer! Use process of elimination to cross out choices that don't make sense.
Question 1: CONCEPTUAL
- A. 5/14
- B. 5/7
- C. 6/7
- D. 1
Show answer (B)
Question 2: BASIC CALCULATION
- A. 2/10
- B. 5/12
- C. 1/2
- D. 7/12
Show answer (B)
Question 3: INTERMEDIATE
- A. 5 1/5
- B. 5 2/5
- C. 5 5/6
- D. 6 1/6
Show answer (C)
Question 4: APPLIED
- A. 1 3/4
- B. 1 7/8
- C. 2 1/4
- D. 7
Show answer (A)
Question 5: CRITICAL THINKING
- A. 3 5/12
- B. 3 7/12
- C. 4 1/12
- D. 4 5/12
Show answer (A)
SUMMARY
Putting It All Together
Summary
To add or subtract fractions, first make sure the denominators are the same by finding the least common denominator (LCD). Rewrite each fraction as an equivalent fraction with that LCD. Then add or subtract the numerators and keep the denominator. Always simplify your final answer to lowest terms.
For mixed numbers, you can convert to improper fractions first, or handle the whole numbers and fractions separately. When subtracting, watch out for the need to borrow (regroup) from the whole number. On the ISEE, remember there is no penalty for guessing — always answer every question, and use process of elimination to improve your odds!