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Fractions Made Understandable: The Complete Student Guide

Crystal-clear explanations with visual intuition, step-by-steps, and dozens of worked examples. Master simplifying, LCM, operations (with negatives), mixed numbers, and conversions to decimals and percents.

Published August 10, 20251 min read
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Fractions show parts of a whole. If they’ve ever felt messy or random, this guide fixes that. We’ll build from the ground up—using simple language, visual intuition, and a lot of worked examples you can practice right away in MathAI GPT.

1) What Is a Fraction? (The Core Idea)

A fraction a/b means: split something into b equal parts; take a of those parts. Read 3/4 as “three fourths”—the denominator names the kind of part.

  • Proper: numerator < denominator (e.g., 3/7) → less than 1.
  • Improper: numerator ≥ denominator (e.g., 17/5) → at least 1.
  • Mixed: whole + proper fraction (e.g., 3 2/5).
  • Reciprocal: flip (4/9 → 9/4). Used in division.

Visual: imagine a pizza. If it’s cut into 8 equal slices and you take 3, that’s 3/8.

2) Simplifying (Reducing) Fractions

A fraction is simplest when top and bottom share no common factor (other than 1). Use the greatest common divisor (GCD).

Worked Example: Simplifying 36/48

Step 1: Find GCD(36,48) = 12

Step 2: Divide both by 12: 36÷12 = 3, 48÷12 = 4

Step 3: Result: 36/48 = 3/4

Two fast ways to find GCD
  • Prime factors: 36=2·2·3·3; 48=2·2·2·2·3 → common = 2·2·3 = 12.
  • Euclidean Algorithm: gcd(48,36)=gcd(36,12)=gcd(12,0)=12.

3) Mixed ↔ Improper (and Why It Matters)

Mixed → Improper: (a b/c) → (a·c+b)/c. Example: 4 3/8 → 35/8.

Improper → Mixed: divide: 17/5 = 3 remainder 2 → 3 2/5.

Algebra tip: keep fractions improper while calculating; convert to mixed at the very end for readability.

4) Why Common Denominators Matter (Add & Subtract)

You can only add/subtract like parts. A common denominator makes the parts the same size. Use the least common multiple (LCM) to keep numbers small.

Find LCM by factors: 10 = 2·5, 15 = 3·5 → take highest powers → LCM = 2·3·5 = 30.

Worked Examples: Add & Subtract

Add: 3/10 + 7/15

LCM(10,15) = 30 → 9/30 + 14/30 = 23/30

Subtract: 5/12 − 1/18

LCM(12,18) = 36 → 15/36 − 2/36 = 13/36

Shortcut: Cross‑addition check (for two fractions only)

To compare or sanity‑check quickly: compare a·d vs b·c fora/b vs c/d. For sums, it can help detect arithmetic slips.

5) Multiplying Fractions (Cancel First)

Multiply tops together and bottoms together. Cancel any common factor before multiplying to avoid big numbers.

Worked Examples: Multiplication

Cancel first: (18/35)·(35/24)

Cancel 35 → 18/24 → Cancel 6 → 3/4

Mixed numbers: 2 1/2 · 3 3/4

Convert → (5/2)·(15/4) = 75/8 = 9 3/8

6) Dividing Fractions (Keep–Change–Flip)

a/b ÷ c/d = a/b · d/c, provided denominators are not 0. Invert the second fraction, then multiply; cancel where possible.

Worked Example: Keep-Change-Flip

Problem: 7/9 ÷ 14/27

Step 1: Keep first, change ÷ to ×, flip second → 7/9 × 27/14

Step 2: Cancel 7 and 9 → 1/1 × 3/2 = 3/2

7) Working With Negatives

  • −a/b = a/(−b) = −(a/b). (Same value.)
  • Add/subtract with LCM as usual; signs live in the numerators.
  • (−)·(−) = +, (−)·(+) = −, and same for division.

Write the minus on the numerator: prefer −3/5 to 3/−5.

8) Which Is Bigger? (Ordering Fractions)

For positive fractions a/b and c/d, compare cross‑products ad and bc. Bigger cross‑product means the bigger fraction.

Example: 7/12 vs 5/8 → 56 vs 60 → so 7/12 < 5/8.

9) Decimals ↔ Fractions ↔ Percents

  • Terminating decimals: count place value. 0.125 = 125/1000 = 1/8.
  • Repeating digit: x = 0.333… → 10x = 3.333… → 9x = 3 → x = 1/3.
  • Repeating block: x = 0.142142… → 1000x = 142.142… → 999x = 142 → x = 142/999.
  • Percent: multiply by 100%. 3/20 = 0.15 = 15%.

10) Word‑Problem Playbook (Step‑By‑Step)

  1. Underline what is asked (and units).
  2. Translate: “of” → multiply; “per” → divide; “left/remaining” → subtract.
  3. Keep improper fractions during work; convert to mixed only at the end if desired.
  4. Estimate to sanity‑check.

Example: You jog 3/4 of a 12‑mile route → (3/4)·12 = 9 miles.

11) Worked Examples (Fully Shown)

  1. Add: 5/6 + 7/8
    LCM(6,8)=24 → 20/24 + 21/24 = 41/24 = 1 17/24.
  2. Subtract (mixed): 2 3/5 − 1 7/10
    Convert: 13/5 − 17/10 = 26/10 − 17/10 = 9/10.
  3. Multiply (cancel): 21/28 × 8/9
    21/28 = 3/4 → (3/4)·(8/9) = 24/36 = 2/3.
  4. Divide (KCF): 7/12 ÷ 14/27
    7/12 · 27/14 → cancel → 9/8 = 1 1/8.
  5. Decimal to fraction (repeating block): 0.218757575…
    Let x = 0.2187575… (repeat 75). 100x = 21.87575…, 10000x = 2187.575… →9900x = 2165.7 → x = 21657/99000 → simplify → 7219/33000.

12) Common Mistakes (and How to Fix Them)

  • Adding denominators: 1/4 + 1/4 ≠ 2/8. Use an LCM; add numerators only.
  • Wrong mixed conversion: 3 2/5 ≠ (3·2)/5. Use (whole·den + num)/den.
  • Skipping cancellation: creates huge numbers; cancel early.
  • 0.3 vs 0.33…: 0.3 = 3/10; 0.33… = 1/3.
  • Zero denominator: never allowed.

13) Quick Reference Tables

OperationHowFast Check
Add/SubtractLCM → combine → simplifyReduced? Size reasonable?
MultiplyCancel → multiply acrossAny common factors left?
DivideInvert second → multiplyNo zero denominators?
Mixed → Improper(whole·den + num)/denAdded correctly?
Decimal → FractionPlace value / repeating algebraSimplified?
Percent ↔ Fractionx% = x/100; a/b → (100a/b)%Reasonable size?

14) Practice Sets (Answers Hidden)

A) Add & Subtract

  1. 1/3 + 5/12
  2. 7/10 − 3/25
  3. 3 1/4 + 2 5/6

B) Multiply & Divide

  1. 14/15 × 9/28
  2. 5/6 ÷ 10/9
  3. 1 2/3 × 3 3/5

C) Conversions & Ordering

  1. Convert 0.0625 to a fraction.
  2. Convert 2/9 to a percent (nearest tenth).
  3. Order from least to greatest: 3/5, 5/9, 7/12.
Show answers

A1: LCM 12 → 4/12 + 5/12 = 9/12 = 3/4.

A2: LCM 50 → 35/50 − 6/50 = 29/50.

A3: 13/4 + 17/6 = 39/12 + 34/12 = 73/12 = 6 1/12.

B1: cancel → 14/15 × 9/28 = 1/3.

B2: KCF → 5/6 · 9/10 = 45/60 = 3/4.

B3: (5/3)·(18/5) = 6.

C1: 0.0625 = 625/10000 = 1/16.

C2: 2/9 ≈ 0.222… → 22.2%.

C3: Compare cross‑products: order is 5/9 < 7/12 < 3/5.

15) Glossary (Quick Meanings)

  • Numerator: chosen parts (top).
  • Denominator: equal parts in the whole (bottom).
  • GCD: greatest common divisor.
  • LCM: least common multiple.
  • Unit fraction: numerator 1 (e.g., 1/7).

16) FAQ

Do I always need the least common denominator?

No. Any common denominator works; the least just keeps numbers small and tidy.

Should I convert mixed numbers before multiplying or dividing?

Yes—always convert to improper first.

How do I check if my answer is reasonable?

Estimate with friendly numbers (e.g., 2/3 ≈ 0.67). If way off, recheck steps.

Next Step

Open the MathAI GPT solver and paste two fraction problems you missed. Ask for: a step‑by‑step explanation, one harder variation, and a 30‑second “what to look for” checklist.

Bonus prompt: “Create a 10‑question mixed drill; hide answers until I click.”