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Fractions

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Chapter Notes

Overview
This chapter introduces the concept of fractions as parts of a whole. It explains how to identify, represent, compare, and generate equivalent fractions using real-life examples like chocolates, parathas, and pizzas. The big idea is that fractions represent equal parts of a whole, and the same fraction can have different names (equivalent fractions) depending on how the whole is divided.
Key Concepts (13)
Fraction
A fraction represents a part of a whole. It is written as numerator/denominator, where the numerator tells how many parts are taken and the denominator tells the total number of equal parts the whole is divided into.
e.g. If a chocolate is divided into 3 equal parts and you take 1 part, you have 1/3 of the chocolate.
Numerator
The top number in a fraction that tells how many equal parts are being considered or taken.
e.g. In 2/5, the numerator is 2, meaning 2 parts are taken out of 5.
Denominator
The bottom number in a fraction that tells the total number of equal parts the whole is divided into.
e.g. In 2/5, the denominator is 5, meaning the whole is divided into 5 equal parts.
Equivalent fractions
Fractions that represent the same part of a whole but have different numerators and denominators.
e.g. 1/2 = 2/4 = 3/6 = 4/8. All these fractions show the same shaded portion of a whole.
Whole
The complete object or quantity from which fractions are taken. The whole must be the same size when comparing fractions.
e.g. If you have two different-sized chocolates, 1/3 of the bigger chocolate can be larger than 1/2 of the smaller chocolate.
Comparing fractions with same denominator
When denominators are the same, the fraction with the larger numerator is greater.
e.g. 2/3 > 1/3 because 2 pieces of 1/3 are more than 1 piece of 1/3.
Comparing fractions with same numerator
When numerators are the same, the fraction with the smaller denominator is greater because the parts are larger.
e.g. 4/5 > 4/6 because 1/5 piece is larger than 1/6 piece.
Fractions greater than 1
A fraction where the numerator is greater than the denominator. It represents more than one whole.
e.g. 5/2 means 5 halves, which is equal to 2 whole parathas and 1/2 paratha (2 1/2).
Mixed fraction
A combination of a whole number and a proper fraction, representing a quantity greater than one whole.
e.g. 2 1/2 means 2 whole parathas and 1/2 paratha.
Number line representation
A line where fractions are placed between 0 and 1 (or beyond) by dividing the distance into equal parts based on the denominator.
e.g. To show 5/2 on a number line, divide each unit into 2 equal parts and count 5 steps from 0.
Comparing fractions with reference to 1
If a fraction is less than 1, it is a proper fraction. If it is greater than 1, it is an improper fraction. This helps compare fractions easily.
e.g. 7/8 < 8/6 because 7/8 is less than 1 and 8/6 is more than 1.
Comparing fractions with reference to 1/2
If a fraction is equal to 1/2, its numerator is half of its denominator. Fractions greater than 1/2 have numerators more than half the denominator, and vice versa.
e.g. 5/8 > 3/6 because 5/8 is more than 1/2 and 3/6 is exactly 1/2.
Fraction kit
A set of fraction pieces (like 1/2, 1/4, 1/8, etc.) used to physically explore equivalent fractions and making a whole.
e.g. One 1/2 piece and two 1/4 pieces together make a whole.
Formulas & Equations (6)
1/2 = 2/4 = 3/6 = 4/8 = ...
Reads as: One-half equals two-fourths equals three-sixths equals four-eighths and so on.
Variables: Numerator and denominator are multiplied or divided by the same number to get equivalent fractions.
Use when: When generating equivalent fractions or simplifying fractions.
⚠ Students often add or subtract the same number instead of multiplying or dividing both numerator and denominator.
1/3 = 2/6 = 3/9 = 4/12 = ...
Reads as: One-third equals two-sixths equals three-ninths equals four-twelfths and so on.
Variables: Multiply numerator and denominator by the same number to get an equivalent fraction.
Use when: When finding equivalent fractions for 1/3.
⚠ Forgetting to multiply both numerator and denominator by the same number.
2/5 = 4/10 = 6/15 = 8/20 = ...
Reads as: Two-fifths equals four-tenths equals six-fifteenths equals eight-twentieths and so on.
Variables: Multiply numerator and denominator by the same number to get an equivalent fraction.
Use when: When finding equivalent fractions for 2/5.
⚠ Only multiplying the numerator and leaving the denominator unchanged.
If denominator is same: larger numerator means larger fraction.
Reads as: When denominators are the same, the fraction with the larger numerator is greater.
Variables: Numerator: number of parts taken; Denominator: total equal parts.
Use when: Comparing fractions with the same denominator.
⚠ Comparing denominators instead of numerators.
If numerator is same: smaller denominator means larger fraction.
Reads as: When numerators are the same, the fraction with the smaller denominator is greater.
Variables: Numerator: number of parts taken; Denominator: size of each part (smaller denominator means larger parts).
Use when: Comparing fractions with the same numerator.
⚠ Thinking larger denominator means larger fraction.
Fraction > 1 if numerator > denominator.
Reads as: A fraction is greater than one if the numerator is greater than the denominator.
Variables: Numerator: number of parts; Denominator: number of parts in one whole.
Use when: Identifying improper fractions or comparing with 1.
⚠ Thinking any fraction with a large numerator is greater than 1, even if denominator is also large.
How It Works
Generating equivalent fractions
  1. Step 1: Take a given fraction, e.g., 1/3.
  2. Step 2: Multiply both the numerator and denominator by the same number (e.g., 2, 3, 4, etc.).
  3. Step 3: The new fraction (e.g., 2/6, 3/9, 4/12) represents the same part of the whole.
  4. Step 4: Continue to generate as many equivalent fractions as needed.
💡 You must multiply both numerator and denominator by the same number. Adding or subtracting will not give an equivalent fraction.
Comparing fractions with same denominator
  1. Step 1: Look at the denominators. If they are the same, compare the numerators.
  2. Step 2: The fraction with the larger numerator is greater.
  3. Step 3: Example: 2/3 > 1/3 because 2 pieces of 1/3 are more than 1 piece.
💡 Only compare numerators when denominators are equal. Do not compare denominators.
Comparing fractions with same numerator
  1. Step 1: Look at the numerators. If they are the same, compare the denominators.
  2. Step 2: The fraction with the smaller denominator is greater because the parts are larger.
  3. Step 3: Example: 4/5 > 4/6 because 1/5 piece is larger than 1/6 piece.
💡 Smaller denominator means larger parts, so the fraction is larger.
Comparing fractions using reference to 1
  1. Step 1: Check if the fraction is less than 1 (proper fraction) or greater than 1 (improper fraction).
  2. Step 2: Any fraction less than 1 is smaller than any fraction greater than 1.
  3. Step 3: Example: 7/8 < 8/6 because 7/8 < 1 and 8/6 > 1.
💡 A proper fraction (numerator < denominator) is always less than an improper fraction (numerator > denominator).
Comparing fractions using reference to 1/2
  1. Step 1: Check if the fraction is equal to 1/2 (numerator is half of denominator).
  2. Step 2: If numerator is more than half of denominator, the fraction is greater than 1/2.
  3. Step 3: If numerator is less than half of denominator, the fraction is less than 1/2.
  4. Step 4: Example: 5/8 > 3/6 because 5/8 > 1/2 and 3/6 = 1/2.
💡 To check if a fraction is half, see if the numerator is exactly half of the denominator.
Representing fractions greater than 1 on a number line
  1. Step 1: Divide the distance between 0 and 1 into equal parts based on the denominator.
  2. Step 2: Count the number of parts given by the numerator from 0.
  3. Step 3: If the numerator is greater than the denominator, you will go beyond 1.
  4. Step 4: Example: For 5/2, divide each unit into 2 parts. Count 5 steps from 0 to reach 2 1/2.
💡 Each whole unit is divided into the same number of parts as the denominator.
Using fraction kit to make a whole
  1. Step 1: Take fraction pieces of different sizes (e.g., 1/2, 1/4, 1/8).
  2. Step 2: Combine pieces so that their total equals 1 whole.
  3. Step 3: Example: One 1/2 piece and two 1/4 pieces make a whole.
  4. Step 4: Write the equivalent fractions observed, e.g., 1/2 = 2/4.
💡 The sum of the fractions must equal 1 whole. The pieces must be from the same whole.
Exam Traps — Don't Make These Mistakes
⚠ Comparing fractions from different wholes
✗ 1/3 of a big chocolate is always smaller than 1/2 of a small chocolate.
→ The wholes are different sizes. 1/3 of a large whole can be bigger than 1/2 of a small whole. Fractions can only be compared if they come from the same whole.
✓ We cannot compare 1/3 of one chocolate with 1/2 of another chocolate unless the chocolates are the same size. The wholes must be the same.
⚠ Confusing numerator and denominator in comparison
✗ 4/6 is greater than 4/5 because 6 is bigger than 5.
→ When numerators are the same, the fraction with the smaller denominator is larger because the parts are bigger. 4/5 means 4 parts of a whole divided into 5 parts, so each part is larger than in a whole divided into 6 parts.
✓ 4/5 > 4/6 because 1/5 is larger than 1/6, so 4 pieces of 1/5 are more than 4 pieces of 1/6.
⚠ Generating equivalent fractions by adding
✗ 1/3 = 2/4 because I added 1 to both numerator and denominator.
→ Equivalent fractions are generated by multiplying or dividing both numerator and denominator by the same number, not by adding or subtracting. Adding changes the value of the fraction.
✓ 1/3 = 2/6 (multiply numerator and denominator by 2). To get an equivalent fraction, multiply both by the same number.
⚠ Thinking all fractions with large numerators are greater than 1
✗ 7/9 is greater than 1 because 7 is a big number.
→ A fraction is greater than 1 only if the numerator is greater than the denominator. In 7/9, 7 < 9, so it is less than 1.
✓ 7/9 is less than 1 because the numerator (7) is smaller than the denominator (9). Only fractions like 9/7 or 8/6 are greater than 1.
⚠ Misidentifying half when numerator is not exactly half of denominator
✗ 3/6 is more than half because 3 is more than 2.
→ 3/6 is exactly half because 3 is half of 6. The fraction equals 1/2. To be more than half, the numerator must be more than half of the denominator.
✓ 3/6 = 1/2 because 3 is exactly half of 6. To check, see if numerator × 2 equals denominator.
Compare & Contrast
Rule for comparison
Comparing fractions with same denominator
Comparing fractions with same numerator
When denominators are the same, the fraction with the larger numerator is greater. When numerators are the same, the fraction with the smaller denominator is greater.
Comparison with 1
Proper fraction
Improper fraction
A proper fraction has numerator less than denominator (e.g., 3/4) and is less than 1. An improper fraction has numerator greater than denominator (e.g., 5/4) and is greater than 1.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA (2 marks), 1 LA (4 marks) — total 8 marks typically
Most asked: Identifying and generating equivalent fractions · Comparing fractions with same or different numerators/denominators · Representing fractions greater than 1 on a number line · Word problems involving fractions of a whole (e.g., paratha, pizza, chocolate)
Don't skip: Long answer questions involving comparison of fractions with different numerators and denominators using reference to 1 or 1/2; Questions requiring generation of equivalent fractions and filling in blanks
One-Line Revision (read aloud before the exam)
  1. A fraction represents a part of a whole, written as numerator/denominator.
  2. Equivalent fractions represent the same part of a whole with different numerators and denominators.
  3. To generate equivalent fractions, multiply or divide both numerator and denominator by the same number.
  4. When denominators are the same, the fraction with the larger numerator is greater.
  5. When numerators are the same, the fraction with the smaller denominator is greater.
  6. A fraction is greater than 1 if the numerator is greater than the denominator.
  7. A fraction is less than 1 if the numerator is smaller than the denominator.
  8. To compare fractions, the wholes must be the same size.
  9. 1/2 = 2/4 = 3/6 = 4/8 — all represent half of a whole.
  10. 1/3 = 2/6 = 3/9 = 4/12 — all represent one-third of a whole.
  11. 2/5 = 4/10 = 6/15 = 8/20 — all represent two-fifths of a whole.
  12. A mixed fraction combines a whole number and a proper fraction, e.g., 2 1/2.
  13. On a number line, fractions greater than 1 are placed beyond 1 by dividing each unit into equal parts.
  14. To check if a fraction is half, see if the numerator is exactly half of the denominator.
  15. A proper fraction is always less than an improper fraction.

MCQs — 10 Questions with Answers

Q1. Which of these is equivalent to 1/2?

  • A. 2/4 ✓
  • B. 1/3
  • C. 2/3
  • D. 1/4

Answer: A — When you multiply both numerator and denominator of 1/2 by 2, you get 2/4, which is equivalent to 1/2.

Q2. Tamanna ate 3/5 of her lunch and her brother ate 2/5. Who ate more?

  • A. Brother
  • B. Tamanna ✓
  • C. Both ate equal
  • D. Cannot be determined

Answer: B — When denominators are the same, compare numerators; 3/5 has larger numerator than 2/5, so Tamanna ate more.

Q3. How many 1/4 pieces make 1/2?

  • A. 1
  • B. 2 ✓
  • C. 3
  • D. 4

Answer: B — 1/2 = 2/4, so 2 pieces of 1/4 make 1/2.

Q4. Which fraction is greater: 4/6 or 4/8?

  • A. 4/8
  • B. 4/6 ✓
  • C. Both are equal
  • D. Cannot compare

Answer: B — When numerators are same, the fraction with smaller denominator is larger because 1/6 piece is bigger than 1/8 piece.

Q5. Complete the equivalent fraction: 1/3 = ___/9

  • A. 1
  • B. 2
  • C. 3 ✓
  • D. 6

Answer: C — Multiply numerator and denominator of 1/3 by 3: (1×3)/(3×3) = 3/9.

Q6. If Maa ate 5 pieces of 1/2 paratha, how many parathas did she eat?

  • A. 5 parathas
  • B. 2 1/2 parathas ✓
  • C. 1/2 paratha
  • D. 2 parathas

Answer: B — 5 pieces of 1/2 = 5/2 = 2 1/2 parathas (2 whole parathas and 1/2 paratha).

Q7. Which pair shows equivalent fractions?

  • A. 1/2 and 1/3
  • B. 2/3 and 4/6 ✓
  • C. 3/4 and 3/5
  • D. 1/4 and 1/6

Answer: B — 2/3 × 2/2 = 4/6, so these fractions are equivalent and represent the same shaded part.

Q8. Compare these fractions using < or >: 5/7 _____ 2/7

  • A. <
  • B. > ✓
  • C. =
  • D. Cannot compare

Answer: B — Same denominator, so compare numerators: 5 > 2, therefore 5/7 > 2/7.

Q9. What is 9/4 written as a mixed number?

  • A. 1 1/4
  • B. 2 1/4 ✓
  • C. 3 1/4
  • D. 4 1/4

Answer: B — 9/4 = 8/4 + 1/4 = 2 wholes + 1/4 = 2 1/4.

Q10. Raman ate 6 pieces of 1/4 paratha. How many parathas did he eat?

  • A. 1 1/2 parathas ✓
  • B. 1 1/4 parathas
  • C. 2 parathas
  • D. 6 parathas

Answer: A — 6 pieces of 1/4 = 6/4 = 4/4 + 2/4 = 1 + 1/2 = 1 1/2 parathas.

Flashcards

What are equivalent fractions?

Equivalent fractions represent the same part of a whole but have different numerators and denominators, like 1/2 = 2/4 = 3/6.

How do you make equivalent fractions?

Multiply or divide both the numerator and denominator by the same number, like 1/2 × 3/3 = 3/6.

When comparing fractions with same denominator, which is bigger: 3/5 or 2/5?

3/5 is bigger because 3 pieces of 1/5 are more than 2 pieces of 1/5.

When comparing fractions with same numerator, which is bigger: 4/6 or 4/8?

4/6 is bigger because 1/6 piece is larger than 1/8 piece.

What does 5/2 parathas mean?

5/2 means 5 pieces of 1/2 paratha, which equals 2 whole parathas plus 1/2 paratha, written as 2 1/2.

Why can't we compare 1/3 of one chocolate with 1/2 of another chocolate directly?

Because the wholes are different sizes, so the fractions represent different amounts.

How many 1/6 pieces make 1/3?

2 pieces of 1/6 make 1/3 because 1/3 = 2/6.

What fraction is equivalent to 2/5?

4/10 is equivalent to 2/5 because when you multiply both numerator and denominator by 2, you get 4/10.

If you have 9/4 paratha pieces, how many whole parathas is that?

9/4 equals 2 1/4 parathas, which means 2 whole parathas and 1/4 of another paratha.

When the numerators are the same, how do you know which fraction is smaller?

The fraction with the larger denominator is smaller because each piece is smaller, like 3/8 < 3/7.

Important Board Questions

Are 1/3 and 2/6 equivalent fractions? Write Yes or No. [1 mark]

Check if they show the same shaded part of the whole. Look at visual pictures or multiply 1/3 by 2/2.

Fill in the blank: 1/2 = ___/8. Also, explain how you found the answer. [2 marks]

Multiply both numerator and denominator by the same number. Which number do you need to multiply 2 by to get 8?

Explain with an example why we cannot compare 1/3 of a big chocolate with 1/3 of a small chocolate directly. Use a drawing if needed. [3 marks]

To compare fractions, the wholes must be the same size. Show how the same fraction can be different amounts if the wholes are different.

Dadaji ate 7 pieces of 1/2 paratha. Draw a number line and show 7/2 on it. How many whole parathas did he eat? Write as a mixed number. [5 marks]

Divide the space between 0 and 1 into 2 equal parts. Mark each 1/2. Count up to 7/2. Convert 7/2 to mixed number by dividing: 7÷2 = 3 remainder 1.

True or False: 4/6 is greater than 4/8. Give a reason for your answer. [2 marks]

When numerators are the same, compare the denominators. Which is larger: 1/6 piece or 1/8 piece?

Assertion-Reason Questions for Fractions

Q1. Assertion (A):

1/2 and 2/4 are equivalent fractions because they represent the same part of a whole.

Reason (R):

When you divide 1/2 into 2 equal parts, each part becomes 1/4, so 1/2 = 2/4.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The reason correctly explains why the assertion is true — breaking a half into two equal pieces gives two fourths, making them equivalent.

Q2. Assertion (A):

When comparing fractions with the same denominator, the fraction with the larger numerator is always larger.

Reason (R):

The numerator tells us how many pieces we have, so more pieces means more fraction.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are correct. The reason explains the assertion perfectly — when the pieces are the same size (same denominator), having more pieces (larger numerator) means a bigger fraction. For example, 4/5 > 3/5.

Q3. Assertion (A):

The fraction 9/4 is equal to 2 1/4 parathas.

Reason (R):

Improper fractions can be written as mixed numbers by dividing the numerator by the denominator.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — The assertion is true (9/4 = 2 1/4), and the reason is also true, but the reason does not directly explain why this assertion is true. The reason is a general rule about converting improper fractions, not a specific explanation of the 9/4 example.

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Case-Based Questions (CBQ) for Fractions

Case Study:

Asha's mother is making kheer (rice pudding) for Diwali. She needs to divide milk equally among 3 pots. The first pot gets 1/3 of the milk, the second pot also gets 1/3 of the milk, and the third pot gets the remaining milk. Her brother says that 1/3 of the milk in the first pot is less than 1/3 of the milk in a bigger container that she was using. Asha says this is wrong because both are 1/3 of their own wholes.

Q1. What fraction of the total milk does the third pot get?

  • A. 1/3 ✓
  • B. 2/3
  • C. 1/2
  • D. 3/3

Answer: A

Q2. Is Asha correct in her explanation? Why or why not?

  • A. No, because 1/3 of a small pot is always less than 1/3 of a big container
  • B. Yes, because to compare fractions of different wholes, the wholes must be the same size ✓
  • C. No, because fractions cannot be compared at all
  • D. Yes, because 1/3 is always equal to 1/3

Answer: B

Q3. How much milk did the first two pots get together?

  • A. 1/3 of the total milk
  • B. 2/3 of the total milk ✓
  • C. 1/2 of the total milk
  • D. All the milk

Answer: B

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