Fractions: Meaning, Comparing and Adding
The question in plain words: What does a fraction like 1/4 mean? How do we tell which of two fractions is bigger? How do we add fractions that have the same denominator?
After this lesson you can:
- read and write a fraction
- name its numerator and denominator
- compare two fractions with the same denominator, or with the same numerator
- make equivalent fractions
- add fractions with the same denominator
1. A story: Maa's roti at lunch
Radha lives in a village near Tonk. At lunch, Maa has one hot roti left, and four children are waiting: Radha, Gopal, Meena and little Kanha.
Maa folds the roti and tears it into 4 equal pieces. Each child gets one piece.
"Each of you got one fourth of the roti," says Maa.
The next day Gopal is at his nani's house. Only three children are waiting. Maa tears the same kind of roti into 3 equal pieces.
Kanha asks, "Today we are fewer. Is my piece bigger or smaller than yesterday?"
Keep Kanha's question in mind. By the end of this lesson you will answer it with full confidence.
2. Step 1: what is a fraction?
- A fraction is a part of a whole, when the whole is cut into equal parts.
- The roti is cut into 4 equal parts. One part is 1/4 of the roti.
- We read 1/4 as "one by four" or "one fourth" (एक चौथाई).
- Take 3 of those parts and you have 3/4 of the roti ("three fourths").
3. Step 2: numerator and denominator
- Every fraction has a number on top, a line, and a number below.
- The number below the line is the denominator (हर). It tells how many equal parts the whole is cut into.
- The number on top is the numerator (अंश). It tells how many of those parts we take.
- In 3/4: numerator = 3, denominator = 4.
Read a fraction: 3/4
- Numerator (अंश)
- 3 parts we take
- Denominator (हर)
- 4 equal parts in the whole
The denominator is never 0.
4. Step 3: the parts must be equal
- A fraction works only when all the parts are equal.
- If a roti breaks into 4 pieces of different sizes, one piece is not 1/4 of the roti.
- Field example: Bapu shares his field equally among his 3 sons. Each son gets 1/3 of the field.
- If one son gets a bigger piece, the pieces are no longer thirds.
5. Step 4: compare fractions with the same denominator
- 2/5 and 3/5 have the same denominator, 5.
- So the whole is cut into 5 equal parts in both. Every part has the same size.
- 3 parts are more than 2 parts. So 3/5 > 2/5.
- Rule: same denominator → the bigger numerator gives the bigger fraction.
6. Step 5: compare fractions with the same numerator
- Back to Kanha. Yesterday he got 1/4 of a roti. Today he gets 1/3.
- Both fractions have the same numerator, 1. Now look at the denominator.
- A roti shared among 3 gives bigger pieces than a roti shared among 4.
- So 1/3 > 1/4. Kanha's piece is bigger today.
- Rule: same numerator → the smaller denominator gives the bigger fraction.
- So 1/2 > 1/3 > 1/4 > 1/5.
7. Step 6: equivalent fractions
- Half a roti is the same amount as 2 quarters, and the same as 4 eighths.
- So 1/2 = 2/4 = 4/8. These are equivalent fractions (तुल्य भिन्न): different names, the same amount.
- How to make one: multiply the numerator and the denominator by the same number.
- 1/2 = (1 × 2)/(2 × 2) = 2/4
- 1/2 = (1 × 4)/(2 × 4) = 4/8
- You can also divide both by the same number: 6/8 = (6 ÷ 2)/(8 ÷ 2) = 3/4.
- Changing only one of the two numbers changes the amount.
8. Step 7: add fractions with the same denominator
- Maa gives Radha 1/4 of a roti, and then 2/4 more.
- The pieces are all quarters, so the piece size does not change. 1 quarter + 2 quarters = 3 quarters.
- So 1/4 + 2/4 = 3/4.
- Rule: add the numerators. Keep the denominator the same.
Question
Ramu ploughs 2/7 of his field in the morning and 3/7 in the evening. How much of the field did he plough in all?
Solution
- The denominators are the same (7), so the parts are of the same size.
- Add the numerators: 2 + 3 = 5.
- Keep the denominator: 7.
Answer: 5/7 of the field.
Watch out: 2/7 + 3/7 is not 5/14. Do not add the denominators.
- If the sum has the same numerator and denominator, it is one whole: 2/5 + 3/5 = 5/5 = 1.
- Subtraction with the same denominator works the same way: 5/7 − 2/7 = 3/7.
9. Shortcuts (after you understand)
Same denominator: look only at the numerators. The bigger numerator wins. Same numerator: look only at the denominators. The smaller denominator wins: fewer sharers, bigger share. Equal or not? a/b and c/d are equal when a × d = b × c. Example: 2/4 and 3/6 → 2 × 6 = 12 and 4 × 3 = 12, so they are equal. Adding with the same denominator: add the tops, keep the bottom. One whole: when the numerator equals the denominator, the fraction is 1 (4/4 = 1).
10. Quick check
Which of these fractions is the greatest?
Show answer
Correct answer: A
All four have the same numerator, 1. The smaller the denominator, the bigger each part. 2 is the smallest denominator, so 1/2 is the greatest.
Option B: 1/3 is less than 1/2: a roti cut into 3 gives smaller pieces than one cut into 2.
Option C: 1/5 is smaller still: the roti is cut into 5 parts.
Option D: 1/6 is the smallest of the four: 6 is the biggest denominator.
Hint: Fewer sharers, bigger share.
2/9 + 5/9 = ?
Show answer
Correct answer: B
The denominators are the same. Add the numerators (2 + 5 = 7) and keep the denominator 9. The answer is 7/9.
Option A: Here the denominators were added too (9 + 9 = 18). The size of the parts does not change.
Option C: Here the numerators were multiplied (2 × 5 = 10) instead of added.
Option D: Here the numerators were subtracted (5 − 2 = 3) instead of added.
Hint: Add the tops, keep the bottom.
Which fraction is equivalent to 3/4?
Show answer
Correct answer: A
Multiply the numerator and the denominator by 2: (3 × 2)/(4 × 2) = 6/8.
Option B: Adding 1 to both numbers does not keep the amount. 4/5 is more than 3/4.
Option C: Only the denominator was changed. 3/8 is half of 3/4.
Option D: Only the numerator was changed. 6/4 is more than one whole.
Hint: Multiply both numbers by the same number.
A field is divided into 6 equal parts. Wheat is sown in 1 part and mustard in 3 parts. What fraction of the field is sown?
Show answer
Correct answer: B
1/6 + 3/6 = 4/6. The parts are of the same size, so add the numerators and keep the denominator 6. (4/6 is the same as 2/3.)
Option A: Here the denominators were added too (6 + 6 = 12).
Option C: 3/6 is only the mustard part.
Option D: 1/6 is only the wheat part.
Hint: Count the sown parts out of 6.
A Class 3 child writes "1/4 is bigger than 1/3, because 4 is bigger than 3". What should the teacher do first?
Show answer
Correct answer: B
The child sees with their own eyes that a roti cut into 3 parts gives bigger pieces. Learning moves from concrete things to the symbol, and the mistake is used to teach.
Option A: A rule learnt by heart without meaning is soon forgotten or used wrongly.
Option C: More sums do not fix a wrong idea; the child may repeat the same mistake.
Option D: Only marking it wrong does not show the child why the idea is wrong.
Hint: From concrete to abstract.
"1/4 is bigger than 1/3, because 4 is bigger than 3." No. The denominator tells how many parts the whole is cut into. More parts means smaller parts. So 1/3 > 1/4. "1/4 + 2/4 = 3/8." No. Do not add the denominators. The parts are still quarters, so the answer is 3/4. "Any 4 pieces of a roti are quarters." No. Only equal pieces make quarters.
Start with real things: paper rotis, a sheet folded into equal parts, or a pile of 12 pebbles shared equally. Fold, then write: children fold a paper into 4 equal parts, colour one part and only then write 1/4. Compare by placing: put a 1/3 piece on top of a 1/4 piece cut from the same-size rotis; children see which is bigger. Use the error: when a child says "1/4 is bigger than 1/3", ask them to check with paper pieces instead of just marking it wrong. Include everyone: give a child with low vision a roti cut from thick card, with raised lines between the parts, to feel.
Sign in to keep your progress. Sign in