Pre-Algebra Class 6: Add, Subtract, Multiply & Divide Fractions and Mixed Numbers

Class 6 is the payoff for everything students learned in Class 5. Once a student can build equivalent fractions and simplify to lowest terms, they can attack the four operations — addition, subtraction, multiplication, and division — on any fractions and mixed numbers. Every SHSAT and SAT fraction question, every Regents rational-expression problem, and every Algebra 1 equation with fractions on both sides depends on the four moves in this class.

The two golden rules of fraction operations

The four moves you must know cold

Each operation has one clean rule. Master these four and every fraction problem — from Pre-Algebra to Algebra 2 — becomes an exercise in careful arithmetic.

Move 1 — Adding and subtracting fractions

The rule is one sentence: rewrite both fractions with a common denominator, then add or subtract the numerators and keep the denominator. The cleanest common denominator is the Least Common Denominator (LCD) — the LCM of the two denominators.

Why a common denominator? Fractions with different denominators have different-sized pieces. You cannot add 3 fifths to 2 thirds any more than you can add 3 apples to 2 oranges — until you rename them into the same unit. Fifteenths work for both: 3/5 = 9/15 (each fifth is 3 fifteenths), and 2/3 = 10/15 (each third is 5 fifteenths). Now the units match, and 9/15 + 10/15 = 19/15 = 1 4/15.

Finding the LCD in two steps:

1. Small denominators. Just list multiples until you find a match. Denominators 4 and 6: multiples of 4 are 4, 8, 12, 16… multiples of 6 are 6, 12, 18… LCD = 12.
2. Bigger denominators. Use prime factorization. Take the highest power of each prime that appears in either denominator. Denominators 12 and 18: 12 = 2² · 3, 18 = 2 · 3². Take 2² and 3² → LCD = 4 · 9 = 36.

Full example — 5/6 + 3/8:

1. LCD: 6 = 2 · 3, 8 = 2³ → LCD = 2³ · 3 = 24.
2. Rewrite: 5/6 = 20/24 (×4/4) and 3/8 = 9/24 (×3/3).
3. Add numerators: 20/24 + 9/24 = 29/24.
4. Simplify or convert if needed: 29/24 = 1 5/24 (already in lowest terms).

Move 2 — Multiplying fractions (the friendliest of the four)

Multiplying is the easiest of the four operations: multiply the numerators together, multiply the denominators together, simplify. No common denominator, no conversion, no reciprocal. Straight across.

a/b · c/d = (a · c) / (b · d)

Why. "Two-thirds of four-fifths" means: take the whole, cut it into fifths, take 4 of them, then cut each of those into thirds and take 2 of every 3. The whole is now cut into 3 · 5 = 15 equal little rectangles, and you have taken 2 · 4 = 8 of them. That is 8/15.

Cross-canceling — the pro shortcut. Before you multiply, cancel any common factor between a numerator and a denominator (even across fractions). This keeps the numbers small and often eliminates the need to simplify at the end.

Example. 4/9 × 3/8. The 4 in the top and 8 in the bottom share a 4 → 4/8 = 1/2. The 3 in the top and 9 in the bottom share a 3 → 3/9 = 1/3. So 4/9 × 3/8 = 1/3 × 1/2 = 1/6. Compare that to multiplying first (12/72) and simplifying at the end — same answer, more work.

"Of" means multiply. In word problems, "1/2 of 3/4" and "1/2 × 3/4" mean the same thing. The word "of" applied to fractions always translates to multiplication.

Move 3 — Dividing fractions with keep-change-flip

Dividing by a fraction is the same as multiplying by its reciprocal (its "flip" — swap numerator and denominator). The three-word chant is:

KEEP the first fraction  ·  CHANGE ÷ to ×  ·  FLIP the second fraction

a/b ÷ c/d = a/b · d/c

Why keep-change-flip works. "How many groups of 4/5 fit into 2/3?" is exactly what 2/3 ÷ 4/5 asks. To answer, multiply by how much of a whole one group is (that's 5/4 of a whole, since 4/5 is four of five equal pieces). Multiplying by 5/4 counts the number of 4/5-sized groups in the original 2/3. That's why "divide by a fraction = multiply by its reciprocal" is not just a trick — it is what division literally means when the divisor is smaller than 1.

Full example — 3/8 ÷ 9/16:

1. Keep 3/8, change ÷ to ×, flip 9/16 to 16/9. So 3/8 ÷ 9/16 = 3/8 × 16/9.
2. Cross-cancel. 3 and 9 share a 3 → 3/9 = 1/3. 16 and 8 share an 8 → 16/8 = 2/1.
3. Multiply the leftovers: 1/1 × 2/3 = 2/3.

Dividing by a whole number. A whole number n is really n/1. So 3/5 ÷ 2 = 3/5 ÷ 2/1 = 3/5 × 1/2 = 3/10. Dividing a fraction by a whole number always shrinks the fraction. Dividing a fraction by a fraction smaller than 1 always grows it — a fact that catches students off guard until they see the "how many groups fit" picture.

Move 4 — Working with mixed numbers

A mixed number like 2 3/4 is really two pieces glued together: a whole part (2) and a fraction part (3/4). Mixed numbers are lovely for reading (2 3/4 cups of flour) and terrible for arithmetic. The one rule that saves you every time:

Mixed → Improper. W a/b = (W · b + a) / b. Multiply the whole part by the denominator, add the numerator, put the result over the original denominator. Example: 2 3/4 = (2 · 4 + 3) / 4 = 11/4.

Improper → Mixed. Divide the numerator by the denominator. The quotient is the whole part; the remainder over the original denominator is the fraction part. Example: 23/6. 23 ÷ 6 = 3 with remainder 5, so 23/6 = 3 5/6. If the remainder is 0, the improper fraction is actually a whole number (24/6 = 4).

Do not multiply mixed numbers directly. Students sometimes try "1 1/2 × 2 1/3 = 2 1/6" by multiplying the whole parts and the fraction parts separately. This is wrong. Convert first: 1 1/2 = 3/2 and 2 1/3 = 7/3. Now 3/2 × 7/3 = 21/6 = 7/2 = 3 1/2. Very different from 2 1/6.

Order of operations still rules

When a problem mixes several fraction operations, PEMDAS still applies: parentheses first, then multiplication/division left to right, then addition/subtraction left to right. Never rearrange the operations because "fractions are hard." Example: 1/2 + 1/3 × 3/4 evaluates the multiplication first (1/3 × 3/4 = 1/4), then the addition (1/2 + 1/4 = 3/4). Not (1/2 + 1/3) × 3/4.

Fast benchmarks and sanity checks

These are not just tricks — they are how careful students catch arithmetic mistakes. If you get 4/15 for 2/3 + 1/4, the benchmark ("sum of two fractions less than 1 is close to 1") tells you the answer is way too small. That flag alone catches most common denominator errors.

Two mistakes students always make

Where this class shows up later

• Class 7–10 (ratios, proportions, percent). Every percent-change problem uses fraction multiplication, and every proportion (a/b = c/d) is solved by the multiplication and division rules from this class.
• Algebra 1. Solving equations with fractions on both sides depends on multiplying by a common denominator to clear them — and that only makes sense if you know how fraction operations work.
• Algebra 2 (rational expressions). Adding, subtracting, multiplying, and dividing rational expressions like (x+2)/(x−1) uses the exact four moves from Class 6 — only the numerators and denominators are polynomials instead of integers.
• SHSAT and SAT. Both tests love word problems where fractions and mixed numbers mix with percentages, ratios, or unit conversions. Students who cannot compute 2 1/2 ÷ 3/4 in under 30 seconds lose the time budget elsewhere.

The 25 questions below march from pure addition and subtraction through multiplication and division, then work through mixed numbers, and finish with real-life word problems that put all four operations together.

25 practice questions with hidden answers

About the Young Fermats Pre-Algebra course

Class 6 is one of 48 rolling classes in our Young Fermats Pre-Algebra program for grade 5–6 students (ages 10–12). The full arc runs from integers on the number line through systems of linear equations and probability — the direct on-ramp to Algebra 1 in 7th or 8th grade. Classes are small-group (max 6 students), 120 minutes per week, and students can start any Monday because the syllabus is rolling.

Every new family starts with a free 30-minute evaluation in our 226 W 79th Street classroom plus a written diagnostic delivered within 48 hours. The diagnostic maps your child's current level against the readiness signals for Pre-Algebra and Algebra 1 — and gives you a clear next step whether you enroll or not.

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