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Working with Fractions

NCERT Class 7 · Mathematics Based on NCERT Class 7 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter builds on the basic understanding of fractions to teach multiplication and division of fractions. It explains how to multiply a fraction by a whole number, a whole number by a fraction, and a fraction by another fraction using both visual models (area of a rectangle) and the standard formula (multiply numerators and denominators). The chapter also introduces the concept of reciprocals and uses them to divide fractions, showing how division can be rewritten as multiplication by the reciprocal. The one big idea is that operations on fractions follow the same logical rules as whole numbers, but require careful handling of numerators and denominators, and that the product or quotient can be smaller or larger than the original numbers depending on whether the numbers are between 0 and 1 or greater than 1.
Key Concepts (12)
Multiplication of a Fraction by a Whole Number
To multiply a whole number by a fraction, first divide the whole number by the denominator of the fraction, then multiply the result by the numerator. Alternatively, rewrite the whole number as a fraction with denominator 1 and multiply numerators and denominators.
e.g. Aaron walks 3 km in 1 hour. To find how far he walks in 2/5 hours: First find distance in 1/5 hour (3 ÷ 5 = 3/5 km), then multiply by 2 (2 × 3/5 = 6/5 km). So 2/5 × 3 = 6/5.
Multiplication of Two Fractions
To multiply two fractions, multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. This is given by the formula a/b × c/d = (a × c) / (b × d).
e.g. The tortoise walks 1/4 km in 1 hour. To find how far it walks in 1/2 hour: 1/2 × 1/4 = (1×1)/(2×4) = 1/8 km.
Cancelling Common Factors (Apavartana)
Before multiplying fractions, if the numerator of one fraction and the denominator of another (or the same fraction) share a common factor, divide both by that factor to simplify the multiplication. This is called cancelling.
e.g. To multiply 12/7 × 5/24: 12 and 24 share a common factor of 12. Cancel: 12/7 × 5/24 = (12÷12)/(7) × 5/(24÷12) = 1/7 × 5/2 = 5/14.
Reciprocal of a Fraction
The reciprocal of a fraction a/b is b/a. When a fraction is multiplied by its reciprocal, the product is always 1.
e.g. The reciprocal of 3/5 is 5/3. Check: 3/5 × 5/3 = 15/15 = 1.
Division of Fractions
To divide one fraction by another, find the reciprocal of the divisor and multiply it by the dividend. The formula is a/b ÷ c/d = a/b × d/c = (a × d) / (b × c).
e.g. To find 2/3 ÷ 3/5: Reciprocal of 3/5 is 5/3. So 2/3 ÷ 3/5 = 2/3 × 5/3 = 10/9.
Product of Fractional Units
When two unit fractions (fractions with numerator 1) are multiplied, the product is 1 divided by the product of their denominators. 1/b × 1/d = 1/(b × d).
e.g. 1/12 × 1/18 = 1/(12×18) = 1/216.
Mixed Fraction
A number consisting of a whole number and a proper fraction, e.g., 1 1/4. It can be converted to an improper fraction for calculations.
e.g. 1 1/4 hours = 5/4 hours. Cost of 5/4 hour of internet at ₹8 per hour = 5/4 × 8 = 40/4 = ₹10.
Area of a Rectangle with Fractional Sides
The area of a rectangle whose sides are fractional lengths is found by multiplying the two fractional lengths together. This is a geometric interpretation of fraction multiplication.
e.g. A rectangle of length 1/2 unit and breadth 1/4 unit has area = 1/2 × 1/4 = 1/8 square units.
Product Comparison Rule
When multiplying: if one number is between 0 and 1, the product is less than the other number. If one number is greater than 1, the product is greater than the other number. If both are between 0 and 1, the product is less than both.
e.g. 3/4 × 2/5 = 6/20 = 3/10. Here 3/10 is less than both 3/4 and 2/5 because both are between 0 and 1.
Quotient Comparison Rule
When dividing: if the divisor is between 0 and 1, the quotient is greater than the dividend. If the divisor is greater than 1, the quotient is less than the dividend.
e.g. 6 ÷ 1/4 = 24. Here divisor (1/4) is between 0 and 1, so quotient (24) is greater than dividend (6).
Brahmagupta's Formula for Multiplication
The general rule for multiplying fractions, first stated by Brahmagupta in 628 CE: 'Multiplication of two or more fractions is obtained by taking the product of the numerators divided by the product of the denominators.'
e.g. 5/12 × 7/18 = (5×7)/(12×18) = 35/216.
Brahmagupta's Formula for Division
The general rule for dividing fractions: a/b ÷ c/d = a/b × d/c = (a × d)/(b × c). First stated by Brahmagupta and clarified by Bhāskara II.
e.g. 2/3 ÷ 3/5 = 2/3 × 5/3 = 10/9.
Formulas & Equations (4)
a/b × c/d = (a × c) / (b × d)
Reads as: a over b multiplied by c over d equals a times c over b times d
Variables: a, c are numerators; b, d are denominators (non-zero integers)
Use when: Multiplying any two fractions, or a fraction and a whole number (write whole number as n/1)
⚠ Students often add the numerators and denominators instead of multiplying, e.g., writing 1/2 × 1/3 = 2/5 instead of 1/6.
1/b × 1/d = 1/(b × d)
Reads as: one over b times one over d equals one over b times d
Variables: b, d are denominators (non-zero integers)
Use when: Multiplying two unit fractions (fractions with numerator 1)
⚠ Students forget to multiply the denominators, writing 1/b × 1/d = 1/(b+d) instead of 1/(b×d).
a/b ÷ c/d = a/b × d/c = (a × d) / (b × c)
Reads as: a over b divided by c over d equals a over b multiplied by d over c equals a times d over b times c
Variables: a, c are numerators; b, d are denominators (non-zero integers); c/d is the divisor
Use when: Dividing any fraction by another fraction, or a whole number by a fraction (write whole number as n/1)
⚠ Students invert the dividend instead of the divisor, e.g., writing 2/3 ÷ 3/5 = 3/2 × 3/5 instead of 2/3 × 5/3.
Reciprocal of a/b = b/a
Reads as: The reciprocal of a over b is b over a
Variables: a, b are non-zero integers
Use when: Finding the reciprocal to perform division of fractions, or to check if two fractions are reciprocals (product = 1)
⚠ Students think the reciprocal of a whole number n is n/1 instead of 1/n. For example, reciprocal of 5 is 1/5, not 5/1.
How It Works
Multiplying a Whole Number by a Fraction (Step-by-Step Method)
  1. Step 1: Divide the whole number (multiplicand) by the denominator of the fraction (multiplier).
  2. Step 2: Multiply the result from Step 1 by the numerator of the fraction.
  3. Step 3: The final result is the product. If it is an improper fraction, convert to a mixed fraction if required.
💡 This method works because multiplying by a fraction like 2/5 means 'take 2 parts out of 5 equal parts of the whole number'. First find one part (divide by 5), then take 2 parts (multiply by 2).
Multiplying Two Fractions Using Area Model
  1. Step 1: Draw a unit square to represent the whole (1).
  2. Step 2: Divide the square into rows equal to the denominator of the first fraction (multiplicand). Shade the number of rows equal to its numerator.
  3. Step 3: Divide the same square into columns equal to the denominator of the second fraction (multiplier). Shade the number of columns equal to its numerator using a different colour or pattern.
  4. Step 4: The overlapping (double-shaded) region represents the product. Count the number of small squares in the overlap (numerator) and total small squares in the whole (denominator).
💡 The total number of small squares is always the product of the two denominators. The number of overlapping squares is the product of the two numerators.
Multiplying Fractions by Cancelling Common Factors
  1. Step 1: Write the multiplication as a single fraction: (numerator1 × numerator2) / (denominator1 × denominator2).
  2. Step 2: Look for any common factor between any numerator and any denominator (they can be from different fractions).
  3. Step 3: Divide both the numerator and the denominator by that common factor (cancel them).
  4. Step 4: Repeat Step 2-3 until no common factors remain.
  5. Step 5: Multiply the remaining numerators together and the remaining denominators together to get the simplified product.
💡 Cancelling can be done across fractions, not just within the same fraction. For example, in 14/15 × 25/42, cancel 14 and 42 (common factor 14), and cancel 15 and 25 (common factor 5).
Dividing Fractions Using Reciprocals
  1. Step 1: Identify the dividend (first fraction) and the divisor (second fraction).
  2. Step 2: Find the reciprocal of the divisor by swapping its numerator and denominator.
  3. Step 3: Rewrite the division problem as multiplication: dividend × reciprocal of divisor.
  4. Step 4: Multiply the fractions using the standard multiplication method (multiply numerators, multiply denominators).
  5. Step 5: Simplify the result to its lowest terms if possible.
💡 You must invert the divisor, not the dividend. A common trick to remember: 'Keep, Change, Flip' — Keep the dividend, Change ÷ to ×, Flip the divisor.
Solving Word Problems Involving Fraction Multiplication and Division
  1. Step 1: Read the problem carefully and identify what is being asked (total, part, number of groups, etc.).
  2. Step 2: Determine the operation: 'of' usually means multiplication; 'how many groups' or 'sharing' usually means division.
  3. Step 3: Write the mathematical expression (e.g., 5 × 2/3, or 1/4 ÷ 5).
  4. Step 4: Perform the operation using the appropriate method (multiplication or division rule).
  5. Step 5: Write the answer with the correct unit and in simplest form (mixed fraction if needed).
💡 Always check if the answer makes sense in the context. For example, if you are dividing a smaller quantity into more parts, the answer should be smaller than the original.
Exam Traps — Don't Make These Mistakes
⚠ Inverting the Wrong Fraction in Division
✗ 2/3 ÷ 3/5 = 3/2 × 3/5 = 9/10
→ The student inverted the dividend (2/3 became 3/2) instead of the divisor (3/5). The rule is to invert only the divisor, not the dividend.
✓ 2/3 ÷ 3/5 = 2/3 × 5/3 = 10/9
⚠ Adding Instead of Multiplying Numerators and Denominators
✗ 1/2 × 1/3 = 2/5
→ The student added the numerators (1+1=2) and added the denominators (2+3=5), confusing multiplication with addition. Multiplication requires multiplying numerators and denominators.
✓ 1/2 × 1/3 = (1×1)/(2×3) = 1/6
⚠ Forgetting to Convert Mixed Fractions Before Multiplying
✗ 1 1/4 × 8 = 1 × 8 + 1/4 × 8 = 8 + 2 = 10
→ While the answer is accidentally correct here, this method is unreliable and often wrong for other problems. The standard method is to convert the mixed fraction to an improper fraction first.
✓ 1 1/4 = 5/4, so 5/4 × 8 = (5×8)/4 = 40/4 = 10
⚠ Misinterpreting 'of' in Word Problems
✗ Find 1/2 of 1/4 km: 1/2 ÷ 1/4 = 2
→ The word 'of' in fraction problems means multiplication, not division. 'Half of a quarter' means 1/2 × 1/4.
✓ 1/2 × 1/4 = 1/8 km
⚠ Cancelling Only Within the Same Fraction
✗ 12/7 × 5/24 = 12/7 × 5/24 (no cancellation done, then multiplied to get 60/168, then simplified)
→ While not technically wrong, it is inefficient and increases chance of arithmetic errors. Students often do not realise they can cancel across fractions (12 with 24).
✓ 12/7 × 5/24 = (12÷12)/(7) × 5/(24÷12) = 1/7 × 5/2 = 5/14
⚠ Confusing Product Comparison Rules
✗ 3/4 × 5 = 15/4, and since 15/4 > 5, the product is greater than both numbers.
→ The product (15/4 = 3.75) is greater than 3/4 (0.75) but less than 5. The rule states: when one number is between 0 and 1 (3/4) and the other is greater than 1 (5), the product lies between them.
✓ 3/4 × 5 = 15/4. The product is less than 5 (the number >1) and greater than 3/4 (the number between 0 and 1).
Compare & Contrast
Operation and Method
Multiplication of Fractions
Division of Fractions
In multiplication, you directly multiply numerators and denominators (a/b × c/d = ac/bd). In division, you first find the reciprocal of the divisor, then multiply (a/b ÷ c/d = a/b × d/c = ad/bc).
Size of Product Relative to Factors
Product When Both Numbers are Between 0 and 1
Product When Both Numbers are Greater Than 1
When both numbers are between 0 and 1 (e.g., 3/4 × 2/5 = 3/10), the product is less than both numbers. When both numbers are greater than 1 (e.g., 4/3 × 4 = 16/3), the product is greater than both numbers.
Size of Quotient Relative to Dividend
Quotient When Divisor is Between 0 and 1
Quotient When Divisor is Greater Than 1
When the divisor is between 0 and 1 (e.g., 6 ÷ 1/4 = 24), the quotient is greater than the dividend. When the divisor is greater than 1 (e.g., 6 ÷ 3 = 2), the quotient is less than the dividend.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 Very Short Answer (1 mark), 1 Short Answer (2 marks) on multiplication/division, 1 Long Answer (3-4 marks) on word problems involving fractions
Most asked: Multiplication of fractions (including cancelling common factors) · Division of fractions using reciprocals · Word problems involving 'of' (multiplication) and sharing (division) · Converting mixed fractions to improper fractions for operations · Product and quotient comparison rules
Don't skip: Word problems combining multiplication and division of fractions (e.g., Somu's land problem, cistern filling problem); Finding area of rectangle with fractional sides; Simplifying products by cancelling common factors before multiplying
One-Line Revision (read aloud before the exam)
  1. To multiply a fraction by a whole number, divide the whole number by the denominator, then multiply by the numerator.
  2. To multiply two fractions, multiply the numerators together and the denominators together: a/b × c/d = (a×c)/(b×d).
  3. Cancelling common factors before multiplying fractions makes calculations easier and reduces errors.
  4. The reciprocal of a fraction a/b is b/a, and their product is always 1.
  5. To divide fractions, multiply the dividend by the reciprocal of the divisor: a/b ÷ c/d = a/b × d/c.
  6. When multiplying, if one factor is between 0 and 1, the product is less than the other factor.
  7. When multiplying, if both factors are between 0 and 1, the product is less than both factors.
  8. When dividing, if the divisor is between 0 and 1, the quotient is greater than the dividend.
  9. When dividing, if the divisor is greater than 1, the quotient is less than the dividend.
  10. A mixed fraction must be converted to an improper fraction before performing multiplication or division.
  11. The area of a rectangle with fractional sides is found by multiplying the fractional lengths.
  12. In word problems, 'of' usually means multiplication, and 'how many groups' or 'sharing equally' usually means division.
  13. The product of two unit fractions is 1 divided by the product of their denominators: 1/b × 1/d = 1/(b×d).
  14. Always simplify your final answer to its lowest terms or convert to a mixed fraction if required.

MCQs — 10 Questions with Answers

Q1. Aaron walks 3 km in 1 hour. How far does he walk in 2 hours?

  • A. 2 × 3 = 6 km ✓
  • B. 3 ÷ 2 = 1.5 km
  • C. 3 + 2 = 5 km
  • D. 2/3 km

Answer: A — Walking is repeated equally, so multiply hours by distance per hour: 2 × 3 = 6 km.

Q2. A tortoise walks 1/4 km in 1 hour. How far in 3 hours?

  • A. 3/4 km ✓
  • B. 1/12 km
  • C. 4/3 km
  • D. 1/4 + 3 km

Answer: A — Multiply 3 × 1/4 = 3/4 km (repeat the fraction three times).

Q3. What is 1/2 × 1/4 using the unit square method?

  • A. 1/6
  • B. 1/8 ✓
  • C. 2/4
  • D. 1/2

Answer: B — Dividing into 2 rows and 4 columns creates 8 parts; one part shaded = 1/8.

Q4. To multiply fractions 5/12 × 7/18, what do you compute?

  • A. (5 + 7)/(12 + 18)
  • B. (5 × 7)/(12 × 18) ✓
  • C. 5/12 + 7/18
  • D. (5 - 7)/(12 - 18)

Answer: B — Multiply numerators together and denominators together: (5 × 7)/(12 × 18) = 35/216.

Q5. A farmer gives 2/3 acre to each of 5 grandchildren. Total land given?

  • A. 5 × 2/3 = 10/3 acres ✓
  • B. 2/3 ÷ 5 = 2/15 acres
  • C. 2/3 + 5 = 5 2/3 acres
  • D. 5/3 acres

Answer: A — 5 grandchildren each get 2/3 acre, so multiply: 5 × 2/3 = 10/3 or 3 1/3 acres.

Q6. Internet time costs ₹8 per hour. Cost for 1 1/4 hours?

  • A. ₹8
  • B. ₹9
  • C. ₹10 ✓
  • D. ₹12

Answer: C — 1 1/4 = 5/4 hours; 5/4 × 8 = (5 × 8)/4 = 40/4 = ₹10.

Q7. When multiplying 12/7 × 5/24, you can cancel 12 and 24 because?

  • A. 12 and 24 are both even
  • B. 12 and 24 share a common factor of 12 ✓
  • C. Numerators and denominators are always cancelled
  • D. 24 ÷ 12 = 2

Answer: B — Common factors can be divided out before multiplying to simplify: 12/24 = 1/2, giving 1 × 5/(7 × 2) = 5/14.

Q8. Manju's team can build 1 km of canal in 8 days. In 1 day, they build?

  • A. 8 km
  • B. 1/8 km ✓
  • C. 1 × 8 = 8 km
  • D. 8/1 km

Answer: B — Divide the work equally: 1 km ÷ 8 days = 1/8 km per day (or 1 × 1/8).

Q9. The Moon sets 5/6 hour later each day. After 4 days from Monday 10 pm, when does it set on Friday?

  • A. 10 + 4 × (5/6) = 10 + 20/6 hours after original ✓
  • B. 10 pm sharp
  • C. 5/6 + 4 hours
  • D. 6/5 hours later

Answer: A — Multiply days by delay: 4 × 5/6 = 20/6 = 3 1/3 hours delay from Monday 10 pm.

Q10. What must you do before multiplying 14/15 × 25/42 to make it easier?

  • A. Add all numbers first
  • B. Convert to decimal
  • C. Cancel 14 with 42 and 25 with 15 by their common factors ✓
  • D. Multiply all numerators and denominators immediately

Answer: C — Cancelling reduces large numbers: 14÷7=2, 42÷7=6, 25÷5=5, 15÷5=3, leaving (2×5)/(6×3)=5/9 instead of 350/630.

Flashcards

What is 3 × 1/4 in km if tortoise walks 1/4 km per hour?

3 × 1/4 = 3/4 km (add the fraction three times: 1/4 + 1/4 + 1/4).

How do you multiply two fractions like 3/4 × 2/5?

Multiply numerators together and denominators together: (3 × 2)/(4 × 5) = 6/20 = 3/10.

Why can you cancel 12 and 24 before multiplying 12/7 × 5/24?

Both 12 and 24 share a common factor of 12, so dividing both by 12 simplifies to 1 × 5/(7 × 2) = 5/14 without large numbers.

What does the unit square represent in fraction multiplication?

The unit square represents 1 whole, and dividing it into rows and columns shows how fractions of fractions create smaller pieces.

If 1 hour costs ₹8, how much does 1 1/4 hours cost using multiplication?

Convert 1 1/4 to 5/4 hours, then 5/4 × 8 = (5 × 8)/4 = 40/4 = ₹10.

What is 1/2 × 1/4 using the unit square method?

Divide the unit square into 2 rows and 4 columns (8 parts total), shade 1 part: 1/2 × 1/4 = 1/8.

How do you convert a whole number to a fraction for multiplication?

Write the whole number as a fraction with denominator 1, e.g., 3 = 3/1, then multiply using the fraction rule.

What is 14/15 × 25/42 after cancelling common factors?

Cancel 14 with 42 (÷7) and 25 with 15 (÷5): (1 × 5)/(3 × 3) = 5/9.

If Tenzin drinks 1/2 glass of milk daily, how much in 7 days?

7 × 1/2 = 7/2 = 3 1/2 glasses (add 1/2 seven times or multiply 7 by numerator, keep denominator).

What is the area of a rectangle with sides 1/2 unit and 1/4 unit?

Area = length × breadth = 1/2 × 1/4 = 1/8 square unit (shown by 1 shaded rectangle out of 8 in unit square).

Important Board Questions

Tenzin drinks 1/2 glass of milk every day. How many glasses does he drink in one week? [1 mark]

Multiply 7 days by 1/2 glass per day using the rule: whole number × fraction = (whole × numerator) ÷ denominator.

A water tap fills 7/10 of a tank in 1 hour. How much of the tank is filled in 3/4 hour? Show your working. [2 marks]

Use the formula fraction × fraction: (3/4) × (7/10) = (3×7)/(4×10). Simplify if needed before multiplying.

A farmer distributes 2/3 acre of land to each of her 5 grandchildren. How much total land does she give? Convert your answer to a mixed fraction and explain using the multiplication rule for fraction × whole number. [3 marks]

Multiply 5 × 2/3 = (5×2)/3 = 10/3. Convert improper fraction to mixed fraction: 10÷3 = 3 remainder 1, so 3 1/3 acres.

The government took 1/6 of Somu's land for a road. Of the remaining land, she gives half to her daughter Krishna. (a) What fraction of the original land remains after the road? (b) What fraction does Krishna receive? (c) Draw a unit square to show all divisions. Show all steps clearly. [5 marks]

Step 1: Find remaining land = 1 - 1/6 = 5/6. Step 2: Krishna gets half of remaining = 1/2 × 5/6 = 5/12 (multiply numerators and denominators). Step 3: Use unit square divided into 6 rows; shade 5 parts for remaining land, then divide those 5 parts in half (creating 12 total parts).

Assertion-Reason Questions for Working with Fractions

Q1. Assertion (A):

When a fraction is multiplied by a whole number, we multiply the whole number by the numerator and keep the denominator unchanged.

Reason (R):

Multiplying a fraction by a whole number means repeating the fraction that many times, so the denominator remains the number of equal parts in the original fraction.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. When we calculate 5 × 2/3, we are adding 2/3 five times, which gives (5 × 2)/3 = 10/3. The denominator stays as 3 because we're still dividing into thirds. The reason correctly explains why the denominator doesn't change.

Q2. Assertion (A):

The product 2/3 × 4/5 can be found by multiplying numerators and denominators separately: (2 × 4)/(3 × 5) = 8/15.

Reason (R):

This method works because it represents the area of a rectangle with sides 2/3 and 4/5 units.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — The assertion is correct: 2/3 × 4/5 = 8/15. However, the reason is not the complete or most fundamental explanation. While the area model does illustrate this multiplication, the reason is not why the rule works—the rule comes from the definition of fraction multiplication and the distributive property. Both statements are true, but the reason doesn't correctly explain the assertion.

Q3. Assertion (A):

To find 1/2 hour's distance when 1 hour distance is 3 km, we calculate 1/2 × 3 = 3/2 km.

Reason (R):

The distance is proportional to time, so multiplying 1/2 by the hourly distance gives the distance for half an hour.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are correct. If 3 km is covered in 1 hour, then in 1/2 hour, the distance is 1/2 × 3 = 3/2 km. The reason (proportionality) correctly explains why we perform this multiplication operation.

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Case-Based Questions (CBQ) for Working with Fractions

Case Study:

Priya is preparing snacks for a school event. She has a recipe that calls for 2/3 cup of sugar to make one batch of cookies. She needs to make 5 batches for the event. Additionally, her friend Rahul asks her to make 3/4 of a batch for his sister's birthday. After calculating the sugar needed, Priya realizes she should also prepare a special dessert that requires 1/2 of the sugar she originally calculated for the 5 batches.

Q1. How much sugar does Priya need for 5 batches of cookies?

  • A. 10/3 cups ✓
  • B. 15 cups
  • C. 5/3 cups
  • D. 6 cups

Answer: A

Q2. How much sugar is needed for Rahul's 3/4 batch?

  • A. 2/4 cup
  • B. 1/2 cup ✓
  • C. 3/2 cup
  • D. 9/12 cup

Answer: B

Q3. How much sugar is needed for the special dessert (which is 1/2 of the sugar for 5 batches)?

  • A. 5/3 cups
  • B. 10/6 cups
  • C. 2/3 cup
  • D. Both A and B ✓

Answer: D

Q4. What is the total sugar needed for all three items (5 batches + 3/4 batch + dessert)?

  • A. 20/3 cups
  • B. 16/3 cups
  • C. 7 cups
  • D. 19/3 cups ✓

Answer: D

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