Class 7 — Mathematics (NCERT)
Fractions and Decimals: Multiplication, Division and Operations on Decimals
Class 7
- ✓By the end of this lesson students will be able to multiply fractions by whole numbers and other fractions.
- ✓By the end of this lesson students will be able to divide fractions by whole numbers and other fractions.
- ✓By the end of this lesson students will be able to perform multiplication of decimal numbers.
- ✓By the end of this lesson students will be able to perform division of decimal numbers.
- ✓By the end of this lesson students will be able to solve word problems involving operations on fractions and decimals.
Key concepts
To multiply a fraction by a whole number, we multiply the numerator of the fraction by the whole number and keep the denominator same. For example, (2/3) × 5 = (2×5)/3 = 10/3. To multiply two fractions, we multiply their numerators and multiply their denominators. For example, (2/3) × (1/4) = (2×1)/(3×4) = 2/12 = 1/6. Remember that 'of' means multiplication, e.g., 'half of 10' means (1/2) × 10.
The reciprocal of a non-zero fraction is obtained by interchanging its numerator and denominator. For example, the reciprocal of 2/3 is 3/2. The reciprocal of a whole number 'a' is 1/a.
To divide a fraction by another fraction or a whole number, we multiply the first fraction by the reciprocal of the second fraction (or the divisor). For example, (2/3) ÷ (1/4) = (2/3) × (4/1) = (2×4)/(3×1) = 8/3. Similarly, (2/3) ÷ 5 = (2/3) × (1/5) = (2×1)/(3×5) = 2/15.
To multiply decimal numbers, first multiply them as whole numbers, ignoring the decimal points. Then, count the total number of digits after the decimal point in both the numbers being multiplied. Place the decimal point in the product from the right, counting that total number of places. For example, 0.5 × 0.3: Multiply 5 × 3 = 15. There is 1 decimal place in 0.5 and 1 in 0.3, so total 2 decimal places. Thus, 0.5 × 0.3 = 0.15. When multiplying a decimal number by 10, 100, or 1000, the decimal point shifts to the right by as many places as there are zeros over 1.
To divide a decimal number by a whole number, perform the division as you would for whole numbers, and place the decimal point in the quotient exactly above the decimal point in the dividend. To divide a decimal number by 10, 100, or 1000, the decimal point shifts to the left by as many places as there are zeros over 1. To divide a decimal number by another decimal number, first shift the decimal point in the divisor to the right until it becomes a whole number. Shift the decimal point in the dividend by the same number of places to the right. Then perform the division as you would for a decimal by a whole number.
Key facts to remember
- 1To multiply fractions, multiply the numerators and multiply the denominators.
- 2The word 'of' in mathematics often indicates multiplication.
- 3The reciprocal of a fraction a/b is b/a.
- 4To divide by a fraction, multiply by its reciprocal.
- 5When multiplying decimals, count the total number of decimal places in the factors to place the decimal in the product.
- 6When dividing a decimal by 10, 100, or 1000, shift the decimal point to the left by the number of zeros.
- 7When multiplying a decimal by 10, 100, or 1000, shift the decimal point to the right by the number of zeros.
- 8To divide a decimal by another decimal, convert the divisor into a whole number by shifting decimal points in both numbers.
Worked examples
Example 1
Simplify: (3/4) × (8/9) + (5/6) ÷ (10/12)
Answer
5/3
Always simplify fractions at each step if possible to make calculations easier.
Example 2
Find the product of 4.35 and 2.7.
Answer
11.745
Example 3
A car travels a distance of 150.5 km in 3.5 hours. What is the average speed of the car in km per hour?
Answer
The average speed of the car is 43 km/h.
Remember to shift the decimal point in both the dividend and the divisor by the same number of places.
Common mistakes
- ✗Forgetting to find the reciprocal when dividing fractions, and instead multiplying directly.
- ✗Incorrectly placing the decimal point in the product or quotient of decimal numbers.
- ✗Adding or subtracting fractions by adding/subtracting numerators and denominators directly, instead of finding a common denominator.
- ✗Not simplifying fractions to their lowest terms after performing operations.
- ✗Confusing the rules for shifting decimal points (left for division by powers of 10, right for multiplication by powers of 10).
Exam tips
- ★Read the question carefully to identify whether it requires multiplication or division of fractions/decimals.
- ★Show all steps clearly, especially when dealing with multiple operations, to avoid errors and gain partial marks.
- ★For decimal operations, double-check the placement of the decimal point in your final answer.
- ★Practice converting word problems into mathematical expressions involving fractions and decimals.
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