CBSE Class 7 Maths Chapter 4 Another Peek Beyond the Point Notes 2025-26

Decimals allow us to extend the Indian place value system to represent fractions like tenths, hundredths, and thousandths easily. For example, the number 27.53 represents 2 tens, 7 ones, 5 tenths, and 3 hundredths. Decimals help us show smaller parts of a whole number in a simple notation, which is important for many calculations in daily life.

Each digit in a decimal has a specific place value depending on its position from the decimal point. To the right of the decimal, the first place is tenths, then hundredths, then thousandths, and so on. For example, in the decimal 0.254, the digit 2 is in the tenths place, 5 in the hundredths place, and 4 in the thousandths place. Decimals are just another way to write fractions whose denominators are 10, 100, 1000, etc. For example: 254/1000 is the same as 0.254.

To convert a fraction to a decimal, you divide the numerator by the denominator. For example, 24 ÷ 100 = 0.24, or 678 ÷ 1000 = 0.678. An easy method for dividing by numbers like 10, 100, or 1000 is to move the decimal point to the left by as many zeros as in the denominator. So, dividing 123 by 10 (one zero) gives 12.3, dividing by 100 (two zeros) is 1.23, and by 1000 (three zeros) is 0.123.

When multiplying decimals, multiply the numbers as if they were whole numbers. Then, in the final answer, count the total number of decimal places in both numbers you multiplied and place the decimal point that far from the right in the answer. For instance, 9.5 × 5 is calculated as 95 × 5 = 475. Since there is one digit after the decimal in 9.5, the product should have one digit after the decimal: 47.5. Another example: 12.5 × 7.5 = 125 × 75 = 9375. Both numbers have one decimal place, so their product should have two decimal places: 93.75.

• Buying 5 pens at ₹9.5 each: 9.5 × 5 = ₹47.5.
• Car travels 12.5 km with 1 litre; with 7.5 litres, it covers 12.5 × 7.5 = 93.75 km.
• Ajay walks 0.827 km one way to school, so 0.827 × 2 × 6 = 9.924 km in a week.
• Finding area: Area = length × breadth. Example: 5.7 cm × 13.3 cm = 75.81 cm².

Product of two decimals can be smaller or bigger than the numbers being multiplied, depending on their size. If both numbers are less than 1, the product is even smaller. If both are more than 1, the product will be greater. If one is less than 1 and the other is bigger, the product lies between them.

If you know a product of two whole numbers, you can quickly find the product of the same numbers with decimal points by adjusting for the total decimal places. For example, if 596 × 248 = 147808, then 5.96 × 24.8 = 147.808 (since 5.96 has two decimal places and 24.8 has one, total is three decimal places in the answer).

• To multiply by 10, move the decimal one place to the right. Example: 2.5 × 10 = 25.
• To multiply by 100, move two places to the right. Example: 3.47 × 100 = 347.
• For 0.425 × 1000, move three places: 425.

Dividing decimals is similar to dividing whole numbers, but with placement of the decimal point according to the divisor. To divide a number by 10, 100, or 1000, move the decimal point to the left by one, two, or three places respectively. For example, 3.9 ÷ 10 = 0.39 or 1325 ÷ 4 = 331.25.

• Splitting 3.9 m ribbon into 10 pieces gives 0.39 m each.
• Sharing 29 m ribbon between 2 people: Each gets 14.5 m.
• 9.5 kg sugar divided into 4 bags: Each has 2.375 kg.
• 0.06 ÷ 5 = 0.012.

For division where either dividend or divisor is a decimal, first make the divisor a whole number by multiplying both numbers by a suitable power of 10. For example, to divide 4.68 by 1.3, multiply both by 10 and use 46.8 ÷ 13. Sometimes, the quotient repeats forever, like 10 ÷ 3 = 3.333..., which is called a recurring decimal.

Usually, when dividing a whole number by another whole number, the quotient is smaller. But with decimals, it's possible for the quotient to be larger than the original number — for example, 128 ÷ 0.4 = 320.

• 2/5 = 0.4, 5/8 = 0.625, 13/4 = 3.25, 4/50 = 0.08
• To convert decimal to fraction, count the decimal places and use 10, 100, 1000 as denominator.

Earth takes 365.2422 days to complete one revolution around the Sun. Because a year is counted as 365 days, the lost time is made up by adding a leap day every four years. This shows how decimals are practically used to keep calendar years in sync with the seasons. More rules are used for leap years in centuries and millennia for accurate matching.

• Decimal operations follow similar rules as whole numbers with careful placement of the decimal point.
• To multiply decimals: multiply as whole numbers, count all decimal digits and place decimal point accordingly.
• Divide by moving decimal as per zeros in the divisor or use long division.
• Some decimal results are repeating or infinite (non-terminating).
• Convert between decimals and fractions as needed for different problems.

Practice using decimals in price calculations, area or length conversion, and solving puzzles like Hidato. Make sure to pay attention to how and where you place the decimal point after calculations, and check your answer for accuracy. Use decimal concepts in real-life contexts such as measuring, shopping, and understanding scientific facts like calendars and leap years.

Class 7 Maths Chapter 4 Notes – Another Peek Beyond the Point: Key Revision Concepts

Strengthen your understanding of decimals with these comprehensive Class 7 Maths Chapter 4 notes. All key concepts, rules, and solved examples from "Another Peek Beyond the Point" are covered for easy revision and quick reference. These notes explain decimal operations in simple steps and show practical uses too.

Use these revision notes to confidently solve word problems, convert between decimals and fractions, and avoid common calculation mistakes. Regular practice with these clear pointers will help master the fundamentals of decimal multiplication and division for exams and daily life.