Grade 7 Fractions & Rational Numbers: Master Adding, Subtracting, Multiplying (US Common Core)

What is a fraction?

Start with one whole thing — a pizza, a chocolate bar, a cake. Cut it into equal parts.

A fraction is just a way to say how many of those equal parts you have.

Every fraction is written as two numbers, one above the other, with a line between them.

In 3/4, the whole is cut into 4 equal parts and you take 3 of them.

Fraction

A number that names one or more equal parts of a whole, written as a/b where b is not 0.

Numerator (top)

The number of equal parts you take.

Denominator (bottom)

The number of equal parts the whole is divided into.

Memory tip: the Denominator is Down. It tells you how many pieces the whole is cut into.

A rational number is any number you can write as a fraction of two integers.
This is a big family:

► it includes whole numbers (3 = 3/1)
► negative numbers (−2 = −2/1)
► decimals that stop or repeat (0.5 = 1/2).

Rational number

Any number that can be written as a/b, where a and b are integers and b is not 0.

Use the tool below to build any fraction and see it as a shaded bar.

Watch how the simplest form and the decimal change as you move the numbers.

🍰 Fraction Visualizer

Numerator Denominator

Simplest form

As a decimal

0.75

Mixed number

Words you must know

These five ideas come back in every fractions question. Learn the short definitions and you are halfway there.

Equivalent fractions

The same amount can be written in many ways. If you cut a cake into 2 halves and take 1, that is the same as cutting it into 4 quarters and taking 2.

So 1/2 and 2/4 are equal in value — they just use different numbers.

Equivalent fractions

Two or more fractions that have the same value but different numerators and denominators.

Example: 1/2 = 2/4 = 3/6 = 4/8. Multiply (or divide) the top and bottom by the same number and the value does not change.

GCD and simplest form

To simplify a fraction means to write it with the smallest possible numbers.

You do this by dividing the top and bottom by the biggest number that fits into both — the GCD.

GCD

The Greatest Common Divisor: the largest number that divides both the numerator and the denominator exactly.

Simplest form

A fraction whose numerator and denominator share no common factor except 1.

Example: in 6/8, the GCD of 6 and 8 is 2. Divide both by 2: 6/8 = 3/4. Now 3 and 4 share nothing, so 3/4 is simplest.

Proper, improper & mixed

Proper fraction

The numerator is smaller than the denominator. Its value is less than 1. Example: 3/4.

Improper fraction

The numerator is equal to or bigger than the denominator. Its value is 1 or more. Example: 7/4.

Mixed number

A whole number plus a proper fraction, written side by side. Example: 7/4 = 1 and 3/4.

Reciprocal

The reciprocal is just the fraction turned upside down. You will need it for division.

Reciprocal

The fraction you get by swapping the numerator and denominator.
A number times its reciprocal always equals 1.

Example: the reciprocal of 2/3 is 3/2. Check: 2/3 × 3/2 = 6/6 = 1.

Working with fractions

This is your main study reference.

Follow the numbered steps for each operation, copy the worked examples, and use the teacher hacks to go faster.

1) Adding & Subtracting

The golden rule: you can only add or subtract fractions when the pieces are the same size — that means the denominators (bottoms) must match.

So there are two cases.

Case A The bottoms are already the same

Check the denominators — they match.
Add (or subtract) the numerators only.
Keep the same denominator.
Simplify if you can.

Example: 2/7 + 3/7

2/7 + 3/7

= 2 + 37 add the tops

= 5/7 keep the bottom; already simplest

Example (subtraction): 5/6 − 1/6

5/6 − 1/6

= 5 − 16

= 4/6 now simplify, GCD = 2

= 2/3

Hack — never touch the bottom: when bottoms match, work only the tops. 2/7 + 3/7 = 5/7, never 5/14. Adding the bottoms is the most common mistake.

Case B The bottoms are different

Find a common denominator (the LCD = lowest number both bottoms divide into).
Rename each fraction: multiply its top and bottom by the same number to reach the LCD.
Add (or subtract) the numerators.
Keep the common denominator.
Simplify.

Example: 1/2 + 1/3

LCD of 2 and 3 = 6

1/2 = 3/6 (×3 top and bottom)

1/3 = 2/6 (×2 top and bottom)

= 3/6 + 2/6

= 5/6

Hack — the butterfly (cross) method: for a/b ± c/d, the answer is (a×d) ± (c×b)b×d, then simplify. Example: 1/2 + 1/3 = (1×3) + (1×2)2×3 = 5/6. Fast when the LCD is hard to spot.

Hack — instant LCD: if one bottom divides the other, the bigger bottom is the LCD. Example: 1/4 + 1/8 → LCD is 8, just rename 1/4 = 2/8.

2) Multiplying

The easiest one — no common denominator needed.

Multiply the numerators (tops) together.
Multiply the denominators (bottoms) together.
Simplify.

Example: 2/3 × 4/5

tops: 2 × 4 = 8

bottoms: 3 × 5 = 15

= 8/15

Hack — cancel before you multiply: cross out common factors first to keep numbers small. 3/4 × 8/9 → cancel 3 with 9 (→1 and 3) and 4 with 8 (→1 and 2) → 1/1 × 2/3 = 2/3.

Hack — "of" means ×: "1/2 of 10" = 1/2 × 10 = 5. And any whole number is itself over 1: 10 = 10/1.

3) Dividing

Turn every division into a multiplication using three moves.

Keep the first fraction exactly as it is.
Flip the second fraction (its reciprocal).
Change the ÷ sign into ×.
Multiply across.
Simplify.

Example: 1/2 ÷ 1/4

keep 1/2, flip 1/4 → 4/1, change ÷ to ×

= 1/2 × 4/1

= 4/2

= 2

Hack — "KFC": Keep, Flip, Change. Only ever flip the second fraction — never the first.

4) Always simplify (the final step)

Find the GCD of the top and bottom.
Divide both by that GCD.

Hack — simplify in stages: if the GCD is hard to see, divide by 2 again and again while both are even, then try 3, then 5. Example: 12/18 → ÷2 = 6/9 → ÷3 = 2/3.

Bonus — compare two fractions fast: cross-multiply. For a/b vs c/d, compare a×d with c×b; the bigger product sits over the bigger fraction. Example: 3/5 vs 4/7 → 3×7 = 21 vs 4×5 = 20, so 3/5 is larger.

🧮 Fraction Operations Lab

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Test Yourself — 15 Exams

Each exam has 10 questions, and every answer comes with a step-by-step solution — even when you get it right.

Key takeaways

1. A fraction is parts of a whole: top = parts taken, bottom = equal parts in all.

2. To add or subtract, make the bottoms equal first. Example: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

3. To multiply, go straight across. Example: 2/3 × 4/5 = 8/15.

4. To divide, flip the second fraction and multiply. Example: 1/2 ÷ 1/4 = 1/2 × 4/1 = 2.

5. Always simplify using the GCD. Example: 6/8 = 3/4.