Fractions and Decimals Class 7 Maths Chapter Notes

Key Terms: Fractions and Decimals

Fraction

A number representing a part of a whole. It is written in the form a/b, where 'a' is the numerator and 'b' is the denominator (b ≠ 0).

Proper Fraction

A fraction where the numerator is smaller than the denominator (e.g., 2/5). Its value is always less than 1.

Improper Fraction

A fraction where the numerator is greater than or equal to the denominator (e.g., 7/4). Its value is always 1 or greater than 1.

Mixed Fraction

A combination of a whole number and a proper fraction (e.g., 1 ¾). It's another way to write an improper fraction.

Equivalent Fractions

Fractions that represent the same value, even though they have different numerators and denominators (e.g., 1/2 and 2/4).

Like Fractions

Fractions that have the same denominator (e.g., 1/7 and 5/7).

Unlike Fractions

Fractions that have different denominators (e.g., 1/3 and 2/5).

Reciprocal of a Fraction

The fraction obtained by interchanging the numerator and the denominator. The product of a fraction and its reciprocal is always 1. The reciprocal of a/b is b/a.

Decimal Fraction

A fraction whose denominator is a power of 10 (like 10, 100, 1000, etc.). It is commonly expressed using a decimal point.

Must-Remember Facts

• {"point":"To add or subtract unlike fractions, first find the LCM of the denominators and convert them into like fractions."}
• {"point":"The product of two proper fractions is always smaller than each of the individual fractions. (e.g., 1/2 × 1/3 = 1/6, which is smaller than both 1/2 and 1/3)."}
• {"point":"The product of two improper fractions is always greater than each of the individual fractions. (e.g., 3/2 × 4/3 = 12/6 = 2, which is greater than both 3/2 and 4/3)."}
• {"point":"To divide one fraction by another, multiply the first fraction by the reciprocal of the second fraction."}
• {"point":"The operator 'of' in mathematics represents multiplication. For example, 1/2 of 10 means (1/2) × 10."}
• {"point":"To multiply decimals, first multiply them as whole numbers, then place the decimal point. The number of decimal places in the product is the sum of decimal places in the given numbers."}
• {"point":"When dividing a decimal by 10, 100, 1000, etc., the decimal point shifts to the left by as many places as there are zeros in the divisor."}
• {"point":"When dividing a decimal by a whole number, place the decimal point in the quotient directly above the decimal point in the dividend."}
• {"point":"To divide a decimal by another decimal, shift the decimal point in both the divisor and the dividend to the right to make the divisor a whole number."}

Multiplication and Division of Fractions

Understanding how to multiply and divide fractions is essential. It's simpler than addition or subtraction because you don't need a common denominator.

Multiplication of Fractions:
To multiply two fractions, you simply multiply the numerators together and the denominators together.
Formula: (a/b) × (c/d) = (a × c) / (b × d)
For example, to multiply 2/3 by 4/5, you calculate (2 × 4) / (3 × 5) = 8/15. When multiplying a fraction by a whole number, you can write the whole number as a fraction with a denominator of 1. For example, 3 × 4/7 is the same as (3/1) × (4/7) = 12/7. Remember the word 'of' means multiplication, so '1/2 of 24' is simply (1/2) × 24 = 12.

Division of Fractions:
Division is the inverse of multiplication. To divide a fraction by another fraction, you use the 'invert and multiply' rule. This means you find the reciprocal of the second fraction (the divisor) and then multiply it by the first fraction.
The reciprocal is found by flipping the numerator and denominator. The reciprocal of c/d is d/c.
Formula: (a/b) ÷ (c/d) = (a/b) × (d/c) = (a × d) / (b × c)
For example, to divide 2/3 by 4/5, you calculate (2/3) × (5/4) = 10/12. This can be simplified to 5/6. This rule applies for dividing a whole number by a fraction, or a fraction by a whole number.

How to Add or Subtract Unlike Fractions

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Worked Examples

• {"heading":"Example 1: Multiplication of Fractions","body":"Question: Salil plants 4 saplings in a row in her garden. The distance between two adjacent saplings is 3/4 m. Find the distance between the first and the last sapling.\nSolution:\nThere are 4 saplings, so there are 3 gaps between them. \nDistance between two saplings = 3/4 m.\nTotal distance = 3 × (distance between two saplings)\nTotal distance = 3 × 3/4 = (3/1) × (3/4) = 9/4 m.\nAs a mixed fraction, this is 2 1/4 m."}
• {"heading":"Example 2: Division of Decimals","body":"Question: A car covers a distance of 89.1 km in 2.2 hours. What is the average distance covered by it in 1 hour?\nSolution:\nDistance covered = 89.1 km\nTime taken = 2.2 hours\nAverage distance per hour = Total Distance ÷ Total Time\n= 89.1 ÷ 2.2\nTo make the divisor a whole number, shift the decimal one place to the right in both numbers:\n= 891 ÷ 22\n= 40.5 km.\nSo, the car covers an average distance of 40.5 km in 1 hour."}
• {"heading":"Example 3: Comparing Fractions","body":"Question: Which is greater: 2/7 of 3/4 or 3/5 of 5/8?\nSolution:\nFirst part: 2/7 of 3/4 = (2/7) × (3/4) = 6/28 = 3/14.\nSecond part: 3/5 of 5/8 = (3/5) × (5/8) = 15/40 = 3/8.\nNow, compare 3/14 and 3/8. Since the numerators are the same, the fraction with the smaller denominator is greater.\nSince 8 < 14, it means 3/8 > 3/14.\nTherefore, 3/5 of 5/8 is greater."}

Fractions vs. Decimals

Exam Traps and Scoring Tips

Examiners often test your precision with fractions and decimals. A common mistake is forgetting to find the LCM when adding/subtracting unlike fractions and instead just adding the numerators and denominators separately - this is incorrect! Always find the LCM. For decimal multiplication, students often misplace the decimal point. A good tip is to estimate the answer first. For example, 3.9 × 2.1 is roughly 4 × 2 = 8. If your calculated answer is 0.819 or 81.9, you know you've made a mistake. The correct answer is 8.19, which is close to your estimate. Also, when a question involves both fractions and decimals, it's often easiest to convert everything to one format (usually decimals) before solving.

Quick Revision Check

• Q: Convert the improper fraction 11/4 into a mixed fraction. A: Divide 11 by 4. Quotient = 2, Remainder = 3. So, 11/4 = 2 3/4.
• Q: What is the value of 0.7 × 0.08? A: First, multiply 7 × 8 = 56. There is one decimal place in 0.7 and two in 0.08, for a total of three. So, the answer is 0.056.
• Q: Calculate 5/9 ÷ 10. A: 10 can be written as 10/1. To divide, multiply by the reciprocal. So, 5/9 × 1/10 = 5/90. Simplifying by dividing numerator and denominator by 5 gives 1/18.
• Q: A ribbon of length 5 1/4 m is cut into small pieces each of length 3/4 m. How many pieces will be cut? A: Convert 5 1/4 to an improper fraction: 21/4. Number of pieces = (Total length) ÷ (Length of one piece) = (21/4) ÷ (3/4) = (21/4) × (4/3) = 21/3 = 7 pieces.

Frequently Asked Questions

Frequently Asked Questions