Add, subtract, multiply, and divide fractions. - ISEE Upper Level Mathematics Achievement

SECTION 1

Historical Context & Motivation

Long before decimals or calculators existed, ancient civilizations needed a way to express quantities smaller than one whole unit. The concept of a fraction — literally meaning "something broken" from the Latin fractio — arose from everyday problems like dividing land, measuring grain, and distributing resources fairly. The ability to add, subtract, multiply, and divide these partial quantities became one of the most important skills in the history of mathematics.

~1800 BCE • Egyptian Unit Fractions

The Rhind Papyrus shows Egyptians using unit fractions (fractions with numerator 1) and special symbols to solve division problems for bread and beer rations.

~500 BCE • Babylonian Sexagesimal Fractions

Babylonians expressed fractions in a base-60 system, which is why we still divide hours into 60 minutes and circles into 360 degrees.

~300 CE • Chinese and Indian Notation

Mathematicians in China and India developed the stacked numerator-over-denominator form we use today, along with rules for all four operations on fractions.

~1200 CE • Fibonacci Brings Fractions to Europe

Leonardo of Pisa (Fibonacci) published his book Liber Abaci, introducing the Hindu-Arabic numeral system and modern fraction notation to European merchants and scholars.

Today, fraction operations appear throughout algebra, geometry, probability, and virtually every ISEE Mathematics Achievement question that involves ratios or parts of a whole. Mastering these four operations is not optional — it is foundational to every other topic on the exam. The question this lesson answers is straightforward: how do you combine, remove, scale, and split fractional quantities with speed and confidence?

SECTION 2

Core Principles & Definitions

Before diving into procedures, make sure you have the vocabulary locked down. A fraction consists of a numerator (the top number, which counts how many parts you have) and a denominator (the bottom number, which tells you the size of each part). Every operation on fractions follows rules designed to keep these two roles consistent and meaningful.

Common Denominator

To add or subtract fractions, both denominators must match. A common denominator is a shared multiple of both original denominators, preferably the least common multiple (LCM).

Simplify (Reduce)

After any operation, divide the numerator and denominator by their greatest common factor (GCF) to express the answer in lowest terms.

Reciprocal

The reciprocal of a fraction flips the numerator and denominator. Dividing by a fraction is the same as multiplying by its reciprocal.

Mixed Numbers vs. Improper Fractions

A mixed number like 2⅓ combines a whole number and a fraction. Convert it to the improper fraction 7/3 before performing operations.

Cross-Cancellation

When multiplying, you can divide any numerator and any denominator by a common factor before multiplying. This keeps numbers small and saves time on the ISEE.

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KEY TAKEAWAY

Think of fractions like slices of pizza. You can only combine slices directly (add or subtract) if all the pizzas are cut into the same number of equal slices — that is the common denominator. Multiplication and division do not need matching slices; they follow their own rules of "multiply straight across" and "flip and multiply."

SECTION 3

Visual Explanation

The Four Fraction Operations at a Glance

Each quadrant shows one operation. Notice that addition and subtraction require matching denominators, while multiplication and division work straight across (after flipping for division).

The diagram above is your roadmap for every fraction problem on the ISEE. When you see an addition or subtraction problem, your first question should always be: "Do the denominators match?" If yes, combine the numerators directly. If no, find the least common denominator (LCD) first. For multiplication, simply multiply straight across — numerator times numerator and denominator times denominator. For division, apply the "Keep-Change-Flip" mantra: keep the first fraction, change division to multiplication, and flip the second fraction.

SECTION 4

Mathematical Framework

Addition & Subtraction

ADDITION WITH LIKE DENOMINATORS

a/c + b/c = (a + b)/c

When denominators are identical, simply add the numerators and keep the denominator unchanged. For example, 3/7 + 2/7 = 5/7.

ADDITION WITH UNLIKE DENOMINATORS

a/b + c/d = (a × d + c × b) / (b × d)

Rewrite each fraction with the least common denominator (LCD). Multiply each numerator by the factor needed to reach the LCD, then add. Using the LCD rather than just b × d often produces smaller numbers.

Subtraction follows the exact same logic — find a common denominator, then subtract numerators instead of adding them. A common ISEE trap is forgetting to convert mixed numbers to improper fractions before subtracting. For instance, 3¼ − 1¾ is much easier once you rewrite it as 13/4 − 7/4 = 6/4 = 3/2.

Multiplication

MULTIPLICATION OF FRACTIONS

a/b × c/d = (a × c) / (b × d)

Multiply numerators together and denominators together. Before you do this, look for cross-cancellation opportunities: if any numerator shares a common factor with any denominator, divide both by that factor first.

Division

DIVISION OF FRACTIONS (KEEP-CHANGE-FLIP)

a/b ÷ c/d = a/b × d/c = (a × d) / (b × c)

Keep the first fraction. Change the ÷ sign to ×. Flip the second fraction (take its reciprocal). Then multiply as usual. For example, 5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4.

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ISEE Strategy Tip

On the ISEE, there is no penalty for wrong answers — always answer every question. If you are stuck on a fraction problem, try plugging each answer choice back into the equation to see which one works. Since numeric answer choices are ordered, you can also use estimation to eliminate choices quickly. For instance, ½ + ⅓ must be less than 1, which immediately eliminates any choice ≥ 1.

SECTION 5

Finding the LCD & Simplifying Results

The Least Common Denominator (LCD) is the smallest number that both denominators divide into evenly. Finding it quickly is the single biggest time-saver for fraction addition and subtraction on the ISEE. There are two reliable methods: listing multiples, or using prime factorization.

This diagram walks through both LCD methods and then applies the result to a complete addition problem. On the ISEE, the listing-multiples method is usually faster for small denominators, while prime factorization is more reliable for larger or less obvious numbers.

After performing any fraction operation, always check whether the result can be simplified. Divide the numerator and denominator by their greatest common factor (GCF). If the result is an improper fraction (numerator larger than denominator), check the answer choices — some ISEE problems expect improper fractions, while others expect mixed numbers. Read the choices before you convert.

SECTION 6

Worked Example

Let's work through a multi-step problem that combines several operations, the kind of challenge that earns you points on the ISEE.

Evaluate: 2⅓ × ¾ − ½

1
Step 1 — Convert the Mixed Number

Rewrite 2⅓ as an improper fraction. Multiply the whole number by the denominator and add the numerator: 2 × 3 + 1 = 7. So 2⅓ = 7/3.

7/3
2
Step 2 — Multiply 7/3 × 3/4

Before multiplying, look for cross-cancellation. The 3 in the denominator of the first fraction and the 3 in the numerator of the second fraction share a factor of 3. Cancel them: (7/1) × (1/4) = 7/4.

7/4
3
Step 3 — Find a Common Denominator for 7/4 − 1/2

The LCD of 4 and 2 is 4. Rewrite 1/2 as 2/4. Now the subtraction becomes 7/4 − 2/4.

7/4 − 2/4
4
Step 4 — Subtract and Simplify

Subtract numerators: 7 − 2 = 5. The denominator stays 4, giving 5/4. As a mixed number, this is 1¼. Check: GCF(5, 4) = 1, so the fraction is already in lowest terms.

5/4 = 1¼

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Order of Operations Reminder

Remember PEMDAS: multiplication comes before subtraction. In the problem above, you must compute 2⅓ × ¾ first, and then subtract ½. Skipping this step and subtracting first is a classic error the ISEE test-makers love to exploit.

SECTION 7

Common Errors & How to Avoid Them

Fraction problems are not inherently difficult, but they are full of small traps. The ISEE test-makers design wrong answer choices (called distractors) that match the exact results of common mistakes. Knowing these mistakes ahead of time turns traps into easy eliminations.

Common Error	What Goes Wrong	Correct Approach
Adding denominators	Writing 1/3 + 1/4 = 2/7 by adding tops and bottoms	Find the LCD (12), rewrite as 4/12 + 3/12 = 7/12
Forgetting to flip	Dividing 2/3 ÷ 4/5 by multiplying 2/3 × 4/5 instead of 2/3 × 5/4	Always flip the second fraction when dividing: Keep-Change-Flip
Not converting mixed numbers	Multiplying only the fraction part: 2½ × 4 = 2 instead of 10	Convert to improper first: 2½ = 5/2, then 5/2 × 4 = 20/2 = 10
Skipping simplification	Selecting 6/8 when the correct answer is written as 3/4	Always reduce to lowest terms before matching answer choices
Wrong order of operations	Adding before multiplying in a multi-step expression	Follow PEMDAS: multiplication and division before addition and subtraction

Five common fraction errors and their fixes

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KEY TAKEAWAY

Think of each wrong-answer distractor as a signpost saying "someone made this exact mistake." When you finish a fraction problem, take five seconds to ask: did I find a common denominator for addition/subtraction? Did I flip for division? Did I simplify? These three quick checks will catch the vast majority of errors.

SECTION 8

Fractions in Algebra & Beyond

The fraction skills you are building here extend directly into algebraic contexts that appear on the ISEE. Instead of numbers in the numerator and denominator, you may see variables, expressions, or even entire polynomials. The operations, however, follow exactly the same rules.

Numeric Fraction Skill	Algebraic Extension
Finding LCD of 4 and 6 → 12	Finding LCD of x and x² → x²
Simplifying 6/8 → 3/4	Simplifying 2x/(4x²) → 1/(2x)
Cross-cancelling in 3/4 × 8/9	Cancelling common factors in (x+1)/x × x²/(x+1)
Solving 1/2 + x = 3/4 for x	Solving equations with fractional coefficients

How numeric fraction skills map to algebra

On the ISEE, you may also encounter complex fractions — fractions that have a fraction in either the numerator, the denominator, or both. To simplify a complex fraction, multiply the top and bottom by the LCD of all the "inner" fractions. For example, to simplify (1/2 + 1/3) / (1/4), multiply the entire numerator and denominator by 12 to clear all fractions at once. This technique transforms the problem into simple integer arithmetic.

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Looking Ahead

Once you reach Algebra 2 and Pre-Calculus, you will add and subtract rational expressions — fractions whose numerators and denominators are polynomials. Every step you practice now with numbers (finding LCDs, cross-cancelling, simplifying) transfers directly to those more advanced problems.

SECTION 9

Practice Problems

These five problems mirror the style and difficulty you will encounter on the ISEE Mathematics Achievement section. Work each one without a calculator, just as you would on test day. Remember: there is no penalty for wrong answers, so always select an answer even if you need to guess.

Question 1: CONCEPTUAL

Which statement correctly explains why you need a common denominator to add fractions?

A. Common denominators make the numerators equal.
B. You can only combine parts that represent the same-sized piece.
C. Fractions must always be converted to decimals before adding.
D. Common denominators allow you to add the denominators together.

Show answer (B)

The correct answer is B. Fractions represent parts of a whole, and those parts must be the same size before you can count them together — that is the purpose of a common denominator. Choice A is wrong because matching denominators does not make numerators equal; it makes the unit sizes equal. Choice C is incorrect because converting to decimals is not required and is often less efficient. Choice D describes the classic error of adding denominators, which produces incorrect results.

Question 2: BASIC CALCULATION

What is 3/8 + 5/12?

A. 8/20
B. 2/3
C. 19/24
D. 5/6

Show answer (C)

The correct answer is C. The LCD of 8 and 12 is 24. Rewrite: 3/8 = 9/24 and 5/12 = 10/24. Adding gives 9/24 + 10/24 = 19/24. Choice A (8/20) results from the common error of adding numerators and denominators separately (3+5)/(8+12). Choice B (2/3 = 16/24) is too small. Choice D (5/6 = 20/24) would require a numerator of 20, not 19.

Question 3: INTERMEDIATE

What is 4/9 ÷ 2/3?

A. 8/27
B. 2/3
C. 4/3
D. 2

Show answer (B)

The correct answer is B. Apply Keep-Change-Flip: 4/9 ÷ 2/3 = 4/9 × 3/2. Cross-cancel the 4 and 2 (both divisible by 2) and the 9 and 3 (both divisible by 3): (2/3) × (1/1) = 2/3. Choice A (8/27) results from multiplying instead of dividing: 4/9 × 2/3 = 8/27. Choice C (4/3) comes from flipping the wrong fraction. Choice D (2) would require the numerator and denominator to be equal after the operation.

Question 4: APPLIED

A recipe calls for 2⅔ cups of flour. If you want to make ¾ of the recipe, how many cups of flour do you need?

A. 1½
B. 2
C. 2¼
D. 3 5/12

Show answer (B)

The correct answer is B. Convert 2⅔ to an improper fraction: (2 × 3 + 2)/3 = 8/3. Multiply by ¾: 8/3 × 3/4. Cross-cancel the 3s and reduce 8 and 4 by a factor of 4: (2/1) × (1/1) = 2. Choice A (1½) might come from incorrectly computing ¾ of just the fraction part. Choice C (2¼) comes from an arithmetic error in the numerator. Choice D (3 5/12) appears if you add instead of multiply: 8/3 + 3/4 = 32/12 + 9/12 = 41/12 = 3 5/12.

Question 5: CRITICAL THINKING

If 1/a − 1/b = 1/6 and a = 3, what is the value of b?

A. 3
B. 6
C. 9
D. 18

Show answer (B)

The correct answer is B. Substitute a = 3: 1/3 − 1/b = 1/6. Rearrange: 1/b = 1/3 − 1/6. Find the LCD of 3 and 6, which is 6: 1/3 = 2/6, so 1/b = 2/6 − 1/6 = 1/6. Therefore b = 6. Choice A (b = 3) would make 1/3 − 1/3 = 0, not 1/6. Choice C (b = 9) would give 1/3 − 1/9 = 3/9 − 1/9 = 2/9, not 1/6. Choice D (b = 18) would give 1/3 − 1/18 = 6/18 − 1/18 = 5/18, not 1/6.

SUMMARY

Lesson Summary

Summary

To add or subtract fractions, find the least common denominator (LCD), rewrite each fraction with that denominator, and then combine the numerators. To multiply fractions, multiply numerators together and denominators together — use cross-cancellation first to keep numbers manageable. To divide fractions, apply Keep-Change-Flip: keep the first fraction, change division to multiplication, and flip the second fraction.

Always convert mixed numbers to improper fractions before computing, and always simplify your final answer by dividing by the GCF. On the ISEE, glance at the answer choices first to know whether you need a fraction, a mixed number, or a decimal. Remember: there is no penalty for guessing, so use estimation to eliminate unlikely answers and always select a response.