1. Chapter at a Glance
When one roti is shared equally by 4 children, each child gets one part out of 4 equal parts — a fraction of the roti. Fractions let us talk about parts of a whole, measure lengths that are not whole numbers, and share things fairly. In this chapter you will meet fractional units, place fractions on the number line, convert between mixed and improper fractions, find equivalent fractions, compare fractions, and add and subtract them.
- Fractional units like ½, ⅓, ¼ and fractions as multiples of them.
- Reading and writing fractions; numerator and denominator.
- Fractions on the number line.
- Mixed fractions and improper fractions.
- Equivalent fractions and the simplest form.
- Comparing, adding and subtracting fractions.
- How Indian mathematicians wrote and worked with fractions.
2. Fractional Units and Fractions
When a whole is divided into n equal parts, each part is the fractional unit 1/n (read "one n-th"). For example, ¼ is one of 4 equal parts.
A fraction such as 3/4 means 3 fractional units of ¼, i.e. ¼ + ¼ + ¼.
- The bottom number is the denominator — into how many equal parts the whole is divided.
- The top number is the numerator — how many of those parts are taken.
- The parts must be equal; otherwise the pieces are not fractions of the same kind.
The more parts a whole is cut into, the smaller each part becomes: ½ > ⅓ > ¼ > ⅕ > ….
3 rotis are shared equally among 4 children. How much does each child get?
Solution: Cut each roti into 4 equal parts — 12 quarters in all. Each child gets 12 ÷ 4 = 3 quarters = ¾ of a roti. In general, a things shared among b people give a/b each.
3. Fractions on the Number Line
To mark ¾ on a number line, divide the gap between 0 and 1 into 4 equal parts. Each part is ¼. Count 3 parts from 0 — that point is ¾.
Fractions greater than 1, like 5/4, lie beyond 1: count 5 quarters from 0.
4. Mixed and Improper Fractions
- A proper fraction has numerator smaller than denominator (less than 1): 3/5.
- An improper fraction has numerator equal to or larger than the denominator (1 or more): 7/4.
- A mixed fraction has a whole-number part and a proper fraction: 1¾.
- Improper → mixed: divide numerator by denominator; the quotient is the whole part, the remainder is the new numerator. 23/5 = 4 remainder 3 → 4⅗.
- Mixed → improper: (whole × denominator + numerator) / denominator. 3⅖ = (15 + 2)/5 = 17/5.
5. Equivalent Fractions and Simplest Form
Fractions that name the same amount are equivalent. ½ = 2/4 = 3/6 = 4/8 = 50/100.
A fraction is in its simplest form (lowest terms) when the numerator and denominator have no common factor other than 1. To simplify, divide both by their HCF.
Write 36/48 in simplest form.
Solution: HCF(36, 48) = 12. 36 ÷ 12 = 3 and 48 ÷ 12 = 4. So 36/48 = 3/4.
Checking equivalence by cross-multiplication
a/b and c/d are equivalent if a × d = b × c. Example: 4/6 and 10/15 → 4 × 15 = 60 and 6 × 10 = 60, so they are equivalent.
6. Comparing Fractions
| Situation | Rule | Example |
|---|---|---|
| Same denominator | Larger numerator → larger fraction | 5/9 > 4/9 |
| Same numerator | Smaller denominator → larger fraction | 3/5 > 3/8 |
| Different both | Make denominators equal (use LCM) or cross-multiply | 2/3 vs 3/5 → 10/15 vs 9/15 |
| Compare with ½ | Is numerator more or less than half the denominator? | 5/12 < ½ < 4/7 |
Arrange 3/4, 5/6, 2/3 in ascending order.
Solution: LCM of 4, 6, 3 is 12. 3/4 = 9/12, 5/6 = 10/12, 2/3 = 8/12. Ascending: 2/3 < 3/4 < 5/6.
7. Adding and Subtracting Fractions
Same denominator
Add or subtract the numerators; keep the denominator. 2/7 + 3/7 = 5/7; 8/9 − 5/9 = 3/9 = 1/3.
Different denominators
- Find the LCM of the denominators.
- Change each fraction into an equivalent fraction with the LCM as denominator.
- Add or subtract the numerators.
- Simplify the answer and change to a mixed fraction if needed.
Find 3/4 + 5/6.
Solution: LCM(4, 6) = 12. 3/4 = 9/12 and 5/6 = 10/12. Sum = 19/12 = 1 7/12.
Find 2⅓ − 1¾.
Solution: Convert to improper: 7/3 − 7/4. LCM 12: 28/12 − 21/12 = 7/12.
A pinch of history
Indian mathematicians wrote fractions with one number above the other — without a line — as early as the Bakhshali manuscript. Brahmagupta (7th century) gave clear rules for adding and subtracting fractions by first making their denominators equal, the same method we use today. The horizontal fraction bar was added later by Arab mathematicians.
8. Fraction of a Quantity and the Fraction Wall
Very often we need a fraction of something — ¾ of a class, ⅖ of a sum of money, ⅔ of a kilogram. To find a fraction of a quantity, first find one fractional unit, then take as many units as the numerator says.
- Divide the quantity by the denominator to find the value of one part.
- Multiply that value by the numerator.
Find ⅗ of ₹450.
Solution: One-fifth of 450 = 450 ÷ 5 = 90. Three-fifths = 3 × 90 = ₹270.
A rope is 2 m long. Find ⅜ of it in centimetres.
Solution: 2 m = 200 cm. One-eighth = 200 ÷ 8 = 25 cm. Three-eighths = 3 × 25 = 75 cm.
The fraction wall
A fraction wall is a set of strips of equal length placed one below another. The first strip is one whole, the next is cut into halves, the next into thirds, then quarters, fifths, sixths and so on. Looking down the wall you can see at once that 2/4 lines up with 1/2 and 3/6, that 1/3 is smaller than 1/2, and that 2/3 is bigger than 3/5. A fraction wall is an excellent tool for comparing fractions and finding equivalent fractions without any calculation.
Fractions in measurement
Fractions are part of everyday measurement: half a kilogram (500 g), a quarter of an hour (15 minutes), three-quarters of a metre (75 cm) and half a litre (500 mL). Converting a fraction of a larger unit into a smaller unit, as in Example 7, is a very common exam question.
9. Estimating with Fractions
Before calculating, it helps to estimate the answer using benchmark fractions 0, ½ and 1. This tells you whether your final answer is sensible.
- 7/8 is close to 1, and 1/9 is close to 0. So 7/8 + 1/9 should be close to 1. (Exact value: 63/72 + 8/72 = 71/72, just under 1.)
- 5/12 is a little less than ½ and 4/7 is a little more than ½, so their sum is close to 1.
- 2⅞ − 1⅒ is about 3 − 1 = 2, so an answer like 4 or ½ must be wrong.
Without calculating exactly, is 3/5 + 4/9 more or less than 1?
Solution: 3/5 is more than ½ (since 3 is more than half of 5) and 4/9 is a little less than ½ (4 is less than 4.5). Together they are close to 1. Exact check: 27/45 + 20/45 = 47/45, which is a little more than 1.
10. Master Formula Table
| Rule | Formula | Example |
|---|---|---|
| Fair share | a items ÷ b people = a/b | 3 rotis ÷ 4 = ¾ |
| Equivalent fractions | a/b = (a × k)/(b × k) | 2/5 = 6/15 |
| Simplest form | divide by HCF | 18/24 = 3/4 |
| Equivalence test | a × d = b × c | 3/4 and 9/12 |
| Mixed → improper | (w × d + n)/d | 2¾ = 11/4 |
| Improper → mixed | quotient and remainder | 17/5 = 3⅖ |
| Same denominators | a/d ± b/d = (a ± b)/d | 4/9 + 2/9 = 6/9 |
| Different denominators | convert using LCM | 1/4 + 1/6 = 5/12 |
11. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Fractional unit | 1/n = one of n equal parts | Equal sharing |
| Number line | Divide each unit into equal parts | Mark fractions |
| Mixed fractions | Whole + proper fraction | Conversions |
| Equivalent fractions | Same value, different form | Simplest form, missing numbers |
| Comparing | Common denominator | Ordering lists |
| Add/subtract | Use LCM of denominators | Word problems |
Chart: size of fractional units
12. Mind Map
- Fractional units
- Numerator, denominator
- Fair sharing
- Proper, improper
- Mixed fractions
- Number line
- Multiply/divide both
- Simplest form (HCF)
- Cross-multiplication
- Compare with LCM
- Add, subtract
- Brahmagupta's method
13. Tips and Tricks
- To compare two fractions quickly, cross-multiply: for a/b and c/d, compare a × d with b × c.
- Compare with benchmark fractions 0, ½ and 1 before doing long calculations.
- Simplify fractions before adding — smaller numbers mean fewer mistakes.
- For mixed fractions, you may add whole parts and fraction parts separately: 2¼ + 1½ = 3 + ¾ = 3¾.
- Adding denominators: 1/2 + 1/3 is not 2/5 (it is 5/6).
- Thinking 1/8 > 1/6 because 8 > 6 — more parts means smaller parts.
- Using unequal parts in a picture of a fraction.
- Forgetting to simplify the final answer.
14. Important One-liners
- A fraction names equal parts of a whole.
- The denominator tells how many equal parts the whole is divided into.
- The numerator tells how many parts are taken.
- a items shared equally among b people gives a/b each.
- Proper fractions are less than 1; improper fractions are 1 or more.
- Equivalent fractions have the same value.
- A fraction is in simplest form when the HCF of its terms is 1.
- With the same numerator, the fraction with the smaller denominator is larger.
- To add unlike fractions, first make the denominators equal.
- Brahmagupta gave rules for adding fractions by equalising denominators.
- Every fraction has a unique simplest form.
15. Exam-pattern Questions with Detailed Answers
Which fraction is equivalent to 2/3?
- 4/9
- 6/9
- 3/4
- 5/6
Show answer
Multiply numerator and denominator of 2/3 by 3: 6/9.
Answer: (B) 6/9
Convert 29/6 into a mixed fraction.
Show answer
29 ÷ 6 = 4 remainder 5, so 29/6 = 4⅚.
Answer: 4⅚
Which is larger: 5/8 or 5/12?
Show answer
Same numerator; the smaller denominator (8) means bigger parts. So 5/8 > 5/12.
Answer: 5/8
Find 5/6 − 2/9.
Show answer
LCM(6, 9) = 18. 5/6 = 15/18, 2/9 = 4/18. Difference = 11/18.
Answer: 11/18
Arrange in descending order: 7/10, 3/5, 13/20, 1/2.
Show answer
LCM 20: 7/10 = 14/20, 3/5 = 12/20, 13/20, 1/2 = 10/20.
Descending: 14/20, 13/20, 12/20, 10/20.
Answer: 7/10 > 13/20 > 3/5 > 1/2
Fill in the box: 3/7 = □/35 = 24/□.
Show answer
35 = 7 × 5, so numerator = 3 × 5 = 15. 24 = 3 × 8, so denominator = 7 × 8 = 56.
Answer: 15 and 56
Ravi walked 1¾ km in the morning and 2⅓ km in the evening. How far did he walk in all?
- 3 1/12 km
- 4 1/12 km
- 3 7/12 km
- 4 7/12 km
Show answer
Whole parts: 1 + 2 = 3. Fraction parts: ¾ + ⅓ = 9/12 + 4/12 = 13/12 = 1 1/12.
Total = 3 + 1 1/12 = 4 1/12 km.
Answer: (B) 4 1/12 km
5 chocolates are shared equally by 3 children. How much does each get? Write as a mixed fraction.
Show answer
Each gets 5/3 of a chocolate = 1⅔ chocolates.
Answer: 1⅔ chocolates each
A tank is 3/8 full. After 120 litres are added, it becomes 5/8 full. What is the capacity of the tank?
Show answer
The added water fills 5/8 − 3/8 = 2/8 = 1/4 of the tank.
So 1/4 of the capacity = 120 L, and the capacity = 4 × 120 = 480 L.
Answer: 480 litres
Which is greater: 7/9 or 11/14? Use cross-multiplication.
Show answer
Compare 7 × 14 = 98 with 9 × 11 = 99. Since 98 < 99, 7/9 < 11/14.
Answer: 11/14 is greater
Find a fraction between 2/5 and 3/5 whose denominator is 10, and another whose denominator is 20.
Show answer
2/5 = 4/10 and 3/5 = 6/10, so 5/10 = 1/2 lies between them.
2/5 = 8/20 and 3/5 = 12/20, so 9/20, 10/20 or 11/20 lie between them.
Answer: 5/10 and (for example) 9/20
Of the students in a class, 1/3 like cricket, 1/4 like football and the rest like kabaddi. What fraction like kabaddi? If the class has 36 students, how many is that?
Show answer
Cricket + football = 1/3 + 1/4 = 4/12 + 3/12 = 7/12. Kabaddi = 1 − 7/12 = 5/12.
5/12 of 36 = 36 ÷ 12 × 5 = 15 students.
Answer: 5/12; 15 students
16. Quick Revision Summary
- A fraction a/b means a parts out of b equal parts; a things shared among b people gives a/b each.
- Mark fractions on the number line by dividing each unit into equal parts.
- Proper < 1; improper ≥ 1; mixed = whole + proper fraction; convert using division.
- Equivalent fractions: multiply/divide both terms by the same number; simplest form uses the HCF.
- Compare and add/subtract by making denominators equal (LCM).