Fractions and Decimals: NCERT Exercise 2.2 for Class 7

Understanding What Fraction Multiplication Really Means

Multiplying fractions might seem tricky, but it's simpler than you think! Let's break it down. When you multiply a whole number by a fraction, like 3 × (1/4), you're just doing repeated addition. It means you are adding 1/4 three times: (1/4) + (1/4) + (1/4) = 3/4. It’s like taking three quarter-pieces of a cake.

Now, what about multiplying a fraction by a fraction, like (1/2) × (1/4)? The multiplication sign here often means 'of'. So, this expression is asking, 'What is half of a quarter?' Imagine you have a chocolate bar and you break it into four equal pieces (quarters). If you then take one of those quarter pieces and break it in half, you get a smaller piece. How big is this new, smaller piece compared to the whole chocolate bar? It's 1/8. So, (1/2) × (1/4) = 1/8. This 'part of a part' is the core idea behind multiplying fractions. You are finding a fraction of an already existing fraction.

The Step-by-Step Process of Multiplying Fractions

1. Step 1: Convert to Improper Fractions — Before you multiply, make sure all numbers are in fraction form. Whole numbers can be written as a fraction with a denominator of 1 (e.g., 5 becomes 5/1). Mixed fractions must be converted into improper fractions (e.g., 2 ½ becomes (2*2+1)/2 = 5/2).
2. Step 2: Multiply the Numerators — Multiply the top numbers (the numerators) of the fractions together. This will be the numerator of your answer.
3. Step 3: Multiply the Denominators — Multiply the bottom numbers (the denominators) of the fractions together. This will be the denominator of your answer.
4. Step 4: Simplify the Result — The resulting fraction should be simplified to its lowest terms. You can do this by finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it. If the answer is an improper fraction, you can convert it back to a mixed fraction.

Worked Examples for Clarity

• Example 1: Multiply a whole number and a fraction: 7 × (2/5) Step 1: Write the whole number as a fraction: 7 = 7/1. Step 2: Set up the multiplication: (7/1) × (2/5). Step 3: Multiply numerators: 7 × 2 = 14. Step 4: Multiply denominators: 1 × 5 = 5. Step 5: Combine them: The result is 14/5. Step 6: Convert to a mixed fraction (optional but good practice): 14 ÷ 5 gives a quotient of 2 and a remainder of 4. So, 14/5 = 2 ⅘.
• Example 2: Using 'of' - Find ½ of 46 Step 1: Remember that 'of' means multiplication. So, the problem is ½ × 46. Step 2: Write the whole number as a fraction: 46 = 46/1. Step 3: The problem becomes (1/2) × (46/1). Step 4: Multiply numerators (1 × 46 = 46) and denominators (2 × 1 = 2). Step 5: The result is 46/2. Step 6: Simplify the fraction: 46 ÷ 2 = 23. So, ½ of 46 is 23.
• Example 3: Multiply two fractions and simplify: (3/8) × (4/5) Step 1: Multiply the numerators: 3 × 4 = 12. Step 2: Multiply the denominators: 8 × 5 = 40. Step 3: The result is 12/40. Step 4: Simplify the fraction. The greatest common divisor of 12 and 40 is 4. Divide both by 4. 12 ÷ 4 = 3 40 ÷ 4 = 10 Step 5: The simplified answer is 3/10.

Important Points & Common Mistakes to Avoid

• Don't Find a Common Denominator: Unlike addition and subtraction, you DO NOT need to find a common denominator when multiplying fractions. Simply multiply the tops and multiply the bottoms.
• Multiplying a Whole Number: When multiplying a whole number by a fraction, like 5 × (2/3), only multiply the whole number with the numerator. It becomes (5 × 2)/3 = 10/3. A common error is to multiply the 5 with both the numerator and the denominator.
• Always Simplify: Getting the right answer is great, but presenting it in its simplest form is even better and often required. Always check if your final fraction can be reduced.
• Convert Mixed Fractions First: Never try to multiply mixed fractions directly. Always convert them into improper fractions first, then multiply, and finally convert the answer back to a mixed fraction if needed.

Practice Questions with Solutions

• Q: Multiply and express as a mixed fraction: 5 × 3 ⅕ A: Step 1: Convert the mixed fraction to an improper fraction. 3 ⅕ = (3 × 5 + 1)/5 = 16/5. Step 2: Rewrite the problem as 5 × (16/5). Write the whole number 5 as 5/1. Step 3: The multiplication is (5/1) × (16/5). Step 4: Multiply the numerators: 5 × 16 = 80. Multiply the denominators: 1 × 5 = 5. The result is 80/5. Step 5: Simplify the fraction. 80 ÷ 5 = 16. Final answer: 16
• Q: Find the value of: (a) ½ of 2 ¾ (b) ½ of 4 ⅔ A: (a) ½ of 2 ¾ Step 1: 'Of' means multiplication. Convert the mixed fraction to an improper fraction: 2 ¾ = (2×4+3)/4 = 11/4. Step 2: The problem is (1/2) × (11/4). Step 3: Multiply numerators (1×11=11) and denominators (2×4=8). The result is 11/8. Step 4: Convert to a mixed fraction: 11 ÷ 8 is 1 with a remainder of 3. So, 1 ⅜. Final answer (a): 1 ⅜ (b) ½ of 4 ⅔ Step 1: Convert the mixed fraction: 4 ⅔ = (4×3+2)/3 = 14/3. Step 2: The problem is (1/2) × (14/3). Step 3: Multiply numerators (1×14=14) and denominators (2×3=6). The result is 14/6. Step 4: Simplify the fraction by dividing both parts by 2: 14/6 = 7/3. Step 5: Convert to a mixed fraction: 7 ÷ 3 is 2 with a remainder of 1. So, 2 ⅓. Final answer (b): 2 ⅓
• Q: Vidya and Pratap went for a picnic. Their mother gave them a water bottle that contained 5 litres of water. Vidya consumed ⅖ of the water. How much water did Vidya drink? A: Step 1: Identify the total amount of water and the fraction consumed. Total water = 5 litres. Fraction consumed by Vidya = ⅖. Step 2: The amount Vidya drank is ⅖ 'of' the total water. This means we need to calculate ⅖ of 5 litres. Step 3: Write the calculation: (2/5) × 5. Step 4: Set up the multiplication: (2/5) × (5/1). Step 5: Multiply numerators (2 × 5 = 10) and denominators (5 × 1 = 5). The result is 10/5. Step 6: Simplify the fraction: 10 ÷ 5 = 2. Final answer: Vidya drank 2 litres of water.
• Q: Multiply and reduce to lowest form: (2/3) × (5/7) A: Step 1: The fractions are already in their proper form, so we can multiply directly. Step 2: Multiply the numerators: 2 × 5 = 10. Step 3: Multiply the denominators: 3 × 7 = 21. Step 4: The result is 10/21. Step 5: Check if the fraction can be simplified. The factors of 10 are 1, 2, 5, 10. The factors of 21 are 1, 3, 7, 21. The only common factor is 1, so the fraction is already in its lowest form. Final answer: 10/21

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