Fractions are not broken, but almost every student’s first answer to a fraction problem is. Spot the mistake, fix it, and understand why the fix works.
Every fraction mistake follows a pattern. Learn the six patterns behind the mistakes, and fixing fractions stops being guesswork. It becomes a checklist you can run on any answer before you write it down.
Most students get fractions wrong in the same handful of ways: adding denominators together, forgetting to simplify, or comparing fractions by the wrong number. This guide flips the usual approach. Instead of only teaching the right method, it shows you the broken answer first, the one almost every student writes, and then fixes it step by step. When you have seen exactly where a method goes wrong, the correct method is much easier to remember.
The horizontal fraction bar that we use today is usually credited to Arab mathematicians of the Middle Ages, such as al-Ḥaṣṣār in the 12th century. Before that, fractions were often written out in words. Imagine writing “three-sevenths” every time instead of 3/7!
A fraction is a part of a whole, split into equal pieces. The bottom number (the denominator) says how many equal pieces the whole is cut into. The top number (the numerator) says how many of those pieces you have. So 3/4 means: cut the whole into 4 equal pieces, and take 3 of them.
A pizza is cut into 8 equal slices and 5 slices are eaten. Then 5/8 of the pizza is gone and 3/8 is left. The two parts add up to the whole: 5/8 + 3/8 = 8/8 = 1.
The word “equal” matters. If a chocolate bar is broken into one big piece and two small ones, you cannot call each piece 1/3. Fractions only work when the pieces are the same size.
This is the single most common fraction mistake: adding the numerators and the denominators.
The fix: when the denominators already match, only add the numerators. The denominator stays the same, because the size of each piece has not changed. 1/4 + 1/4 = 2/4, which simplifies to 1/2.
Think of pizza slices: two quarter-slices put together are still quarter-sized slices. There are just two of them now, not eight tiny pieces. The pieces do not shrink because you are adding.
Multiplying (or dividing) both the numerator and the denominator by the same number never changes the value of a fraction. It only changes how many pieces the whole is cut into.
1/2 = 2/4 = 4/8 = 50/100. Each fraction is the exact same amount, just sliced differently. Multiply top and bottom by 2: 1/2 → 2/4. Multiply by 4: 1/2 → 4/8. Multiply by 50: 1/2 → 50/100.
You can test whether two fractions are equivalent by simplifying both. For example, are 6/10 and 9/15 equivalent? 6/10 simplifies to 3/5 (divide by 2) and 9/15 simplifies to 3/5 (divide by 3). Both give 3/5, so yes.
You can only add, subtract or directly compare fractions once they describe pieces of the same size. Finding a common denominator means changing each fraction into an equivalent one that shares a denominator. The neatest choice is the lowest common multiple (LCM) of the denominators, because it keeps the numbers small.
The fix: convert 1/3 into sixths first: 1/3 = 2/6. Now both fractions share a denominator: 2/6 + 1/6 = 3/6, which simplifies to 1/2.
Work out 2/3 + 3/4.
Step 1: The LCM of 3 and 4 is 12.
Step 2: Convert: 2/3 = 8/12 (multiply top and bottom by 4) and 3/4 = 9/12 (multiply top and bottom by 3).
Step 3: Add the numerators: 8/12 + 9/12 = 17/12.
Step 4: Write as a mixed number: 17/12 = 1 5/12.
Answer: 1 5/12
Work out 5/6 − 1/4.
The LCM of 6 and 4 is 12. 5/6 = 10/12 and 1/4 = 3/12. Subtract the numerators: 10/12 − 3/12 = 7/12.
Answer: 7/12 (already in lowest terms, because 7 is prime and does not divide 12).
A fraction is in lowest terms when the numerator and denominator share no common factor other than 1. Simplifying means dividing both by their greatest common factor (GCF).
The fix: 6 and 8 share a common factor of 2. Divide both: 6 ÷ 2 = 3 and 8 ÷ 2 = 4. The simplified answer is 3/4. Always simplify unless the question says otherwise.
Simplify 36/60.
List the factors. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. The greatest common factor is 12.
Divide both by 12: 36 ÷ 12 = 3 and 60 ÷ 12 = 5.
Answer: 3/5
If you do not spot the GCF, divide by any common factor you can see and repeat. For example, 36/60 → 18/30 → 9/15 → 3/5. You reach the same answer in more steps.
Comparing fractions with different denominators is a trap if you do it by eye. Always convert to a common denominator, or use a benchmark such as 1/2.
The fix: convert to a common denominator (21): 1/3 = 7/21 and 2/7 = 6/21. Now compare the numerators directly: 7/21 > 6/21, so 1/3 is bigger. The conclusion is right, but the first reason was wrong. With a different pair, such as 1/3 and 2/5, the same bad reasoning would give the wrong answer.
(a) Which is bigger, 5/8 or 7/12? The LCM of 8 and 12 is 24. 5/8 = 15/24 and 7/12 = 14/24. So 5/8 is bigger.
Quick check by cross-multiplying: 5 × 12 = 60 and 7 × 8 = 56. Since 60 > 56, the first fraction is bigger. ✓
(b) Put 1/2, 3/5, 2/3 and 7/10 in order from smallest to largest. The LCM of 2, 5, 3 and 10 is 30. The fractions become 15/30, 18/30, 20/30 and 21/30.
Answer: 1/2 < 3/5 < 2/3 < 7/10
An improper fraction (like 9/4) has a numerator bigger than its denominator, so it is worth more than one whole. A mixed number (like 2¼) shows the same amount as whole numbers plus a leftover fraction. Both describe the identical quantity.
9/4 means 9 quarters. Four quarters make one whole, so 9 quarters make 2 wholes with 1 quarter left over: 9/4 = 2¼.
The fix: change the whole number into quarters first. 2 wholes = 2 × 4 = 8 quarters. Add the 1 extra quarter: 8 + 1 = 9 quarters. So 2¼ = 9/4. The shortcut is (whole × denominator + numerator) ÷ denominator.
Mixed to improper: 3 2/5 = (3 × 5 + 2)/5 = 17/5.
Improper to mixed: 23/6. Divide 23 by 6: 6 goes into 23 three times (18), with remainder 5. So 23/6 = 3 5/6.
Work out 2 1/3 + 1 3/4.
Method 1: add the wholes (2 + 1 = 3), then the fractions: 1/3 + 3/4 = 4/12 + 9/12 = 13/12 = 1 1/12. Total: 3 + 1 1/12 = 4 1/12.
Method 2: convert to improper fractions: 7/3 + 7/4 = 28/12 + 21/12 = 49/12 = 4 1/12. ✓ Both methods agree.
Multiplying and dividing do not need a common denominator. This is where many students waste time, or make mistakes by copying the rule for addition.
The fix: to divide, multiply by the reciprocal of the second fraction. 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2. It makes sense: how many quarters fit into one half? Two of them. The answer 1/8 is smaller than 1/2, which cannot be right when you are dividing by a number smaller than 1.
Work out 2/3 × 3/8. Multiply across: (2 × 3)/(3 × 8) = 6/24 = 1/4. Or cancel first: the 3 on top cancels with the 3 on the bottom, and 2/8 simplifies to 1/4, so the answer is 1/4.
“Of” means multiply: 3/4 of 20 = 3/4 × 20 = 60/4 = 15.
Work out 3/4 ÷ 3/8. Flip the second fraction and multiply: 3/4 × 8/3 = 24/12 = 2.
Check: 3/8 fits into 3/4 exactly twice, because 3/4 = 6/8 and 6/8 is two lots of 3/8. ✓
| Situation | Best Strategy | Why It Works |
|---|---|---|
| Adding/subtracting, same denominator | Keep denominator, combine numerators | Piece size has not changed |
| Adding/subtracting, different denominators | Find a common denominator first | Makes pieces the same size before combining |
| Answer looks “unfinished” | Simplify to lowest terms | Divide top and bottom by their greatest common factor |
| Comparing two fractions | Convert to a common denominator | Numerators become directly comparable |
| Fraction bigger than a whole | Convert to a mixed number | Separates whole amounts from the leftover piece |
| Multiplying | Multiply across, simplify | Taking a part of a part |
| Dividing | Flip the second fraction, then multiply | Asks “how many of these fit in that?” |
A recipe needs 3/4 cup of flour, but you are doubling it for a bigger batch. 3/4 + 3/4 = 6/4, which simplifies to 1½ cups. It is the same fixing process used above, now measuring real flour into a real bowl.
A family earns Rs 12,000 in a month (a small number, to keep the arithmetic easy). They spend 1/4 on rent and 1/3 on food. What fraction is left, and how much money is that?
Spent: 1/4 + 1/3 = 3/12 + 4/12 = 7/12.
Left: 1 − 7/12 = 12/12 − 7/12 = 5/12.
In rupees: 5/12 × 12,000 = Rs 5,000.
Check: rent = 1/4 × 12,000 = Rs 3,000 and food = 1/3 × 12,000 = Rs 4,000. Left = 12,000 − 3,000 − 4,000 = Rs 5,000. ✓
A tailor has a 5-metre roll of ribbon and cuts pieces that are each 5/8 of a metre long. How many pieces can be cut?
Number of pieces = 5 ÷ 5/8 = 5 × 8/5 = 8.
Answer: 8 pieces. Check: 8 × 5/8 = 5 metres. ✓
Ancient Egyptians mostly used “unit fractions”, which are fractions with a numerator of 1, like 1/3 or 1/7. To write 2/5, they had to add two different unit fractions (1/3 + 1/15) instead.
Run through this list each time you finish a fraction question. One quick check, “does my answer make sense?”, catches most mistakes. For example, if you add two positive fractions, the answer must be bigger than each of them. If you divide by a fraction less than 1, the answer must get bigger.
Each fraction below can be simplified. Type the fraction in lowest terms, then tap Check. Tap New Set for a fresh round — your best score is saved on this device.
1. 3/5 = 6/10, so 6/10 + 1/10 = 7/10. It is already in lowest terms.
2. The GCF of 24 and 36 is 12. 24 ÷ 12 = 2 and 36 ÷ 12 = 3. The answer is 2/3.
3. The common denominator is 63. 4/9 = 28/63 and 3/7 = 27/63. So 4/9 is bigger.
1. 2/5 = 4/10, so 4/10 + 3/10 = 7/10.
2. The LCM of 6 and 8 is 24. 5/6 = 20/24 and 3/8 = 9/24. So 20/24 − 9/24 = 11/24.
3. 2½ = 5/2. Then 5/2 ÷ 5/6 = 5/2 × 6/5 = 30/10 = 3.
4. Used = 2/3 × 3/4 = 6/12 = 1/2 of the whole tank. Still full = 3/4 − 1/2 = 3/4 − 2/4 = 1/4 of the tank.
Students often treat fractions as two separate whole numbers, which is why they add the tops and the bottoms. Concrete models help: paper strips, a chocolate bar, or a drawn bar split into equal parts. Ask the learner to draw the problem before using the rule. Use the “broken answer” approach in this guide: show a wrong answer and ask, “what went wrong, and how do we know?” Praise estimation (“will the answer be more or less than one half?”) as much as correct calculation. At home, cooking, sharing a pizza and measuring lengths are easy chances to use fractions.
10 questions. Select an answer for each, then submit to see your score instantly.