Chapter 4: Another Peek Beyond the Point
Decimals in depth: multiplying and dividing them by extending the place-value rules you already know.
Original notes — ready CBSE (NCERT, English Medium) • Class 7 • Mathematics • Book: Ganita Prakash — Part II
Chapter 4: Another Peek Beyond the Point — EG's own notes, in simple words
Book: Ganita Prakash — Part II (CBSE, Class 7) | Chapter 4 | Medium: English
Diagram (labels are printed on the image): Decimal place value for 27.53: Tens, Ones, decimal point, Tenths, Hundredths, with moving left multiplying by 10 and moving right dividing by 10. Drawn for this chapter; it replaces the old polygon diagram, which did not belong to a decimals chapter.
Measuring and sharing with decimals. A general picture, not a book scene.
What the chapter is about
Decimals are the natural extension of the Indian place-value system past the Ones place: Tenths, Hundredths, Thousandths, and so on. You already multiply and divide fractions; this chapter shows that decimal multiplication and division are the same ideas, carried through place value. If you understand why the decimal point moves, you will never need to memorise a separate rule for each case.
4.1 A quick recap of decimals
The number 27.53 means 2 Tens + 7 Ones + 5 Tenths + 3 Hundredths. Each step left multiplies the place value by 10; each step right divides it by 10. That is why 27.53 is also 2753 hundredths, and why trailing zeroes after the decimal point do not change a value: 27.50 = 27.5, because 50 hundredths equal 5 tenths. Keep the place-value chart in mind for everything that follows — every decimal rule in this chapter is a place-value rule in disguise.
4.2 Decimal multiplication
To multiply decimals, first multiply them as counting numbers, ignoring the decimal points. Then place the decimal point in the product so that the number of decimal digits in the product equals the total number of decimal digits in the two factors. Example from the book's method: 5.8 × 1.24. Multiply 58 × 124 = 7192. The factors have 1 + 2 = 3 decimal digits, so the product has 3 decimal digits: 7.192. Check by size: 5.8 is near 6 and 1.24 is near 1, so the product should be near 6 — and 7.192 is plausible, while 71.92 or 0.7192 would not be. Always estimate first; a misplaced decimal point is the commonest decimal error.
4.3 Decimal division
Decimal division uses the same long-division procedure as counting numbers, with regrouping continuing past the Ones place into Tenths, Hundredths, Thousandths, and so on. When the Ones are regrouped into Tenths, a decimal point is placed in the quotient. Example: 123 ÷ 10 = 12.3 — dividing by 10 shifts every digit one place right. For a fuller example: 7.2 ÷ 3 = 2.4, because 72 tenths ÷ 3 = 24 tenths = 2.4. When dividing by a decimal, first make the divisor a counting number by multiplying both numbers by the same power of 10: 4.8 ÷ 0.6 = 48 ÷ 6 = 8, since multiplying both by 10 does not change the quotient.
What you should learn
- Read any decimal through its place-value chart: Tens, Ones, Tenths, Hundredths, Thousandths.
- Multiply decimals by multiplying as counting numbers and counting total decimal digits.
- Divide decimals by long division with regrouping past the Ones place.
- Divide by a decimal by first making the divisor a counting number.
- Estimate every decimal answer first, and recognise non-terminating quotients.
Key Terms
- Tenths / Hundredths / Thousandths — The first, second and third places after the decimal point.
- Decimal digits — The digits after the decimal point, counted for placing the product's point.
- Regrouping — Carrying a remainder into the next smaller place during division.
- Non-terminating quotient — A quotient whose decimal digits never end.
Remember
- Product's decimal digits = sum of the factors' decimal digits.
- Dividing by 10, 100, 1000 shifts digits right by 1, 2, 3 places.
- To divide by a decimal, multiply divisor and dividend by the same power of 10.
- Place the decimal point in the quotient when Ones regroup into Tenths.
- Estimate first: a wrongly placed point is the easiest decimal mistake to catch.
Chart / Table: Decimal multiplication and division
| Operation | Method | Example |
|---|---|---|
| Multiply decimals | Multiply as counting numbers; decimal digits add | 5.8 × 1.24 = 7.192 |
| Divide by 10 / 100 | Shift digits right by 1 / 2 places | 123 ÷ 10 = 12.3 |
| Divide decimals | Long division, regroup past Ones | 7.2 ÷ 3 = 2.4 |
| Divide by a decimal | Multiply both by a power of 10 first | 4.8 ÷ 0.6 = 48 ÷ 6 = 8 |
| Non-terminating | Remainder never reaches zero | 1 ÷ 3 = 0.333… |
Practice set — try it yourself
- Write 27.53 in expanded place-value form.
Hint: Name each digit's place. - Find 3.6 × 2.5 by the counting-numbers method.
Hint: Multiply 36 × 25 first, then count decimal digits. - Find 9.6 ÷ 4.
Hint: Think in tenths. - Find 5.4 ÷ 0.9 by making the divisor a counting number.
Hint: Multiply both numbers by the same power of 10. - Estimate 48.2 × 0.51 before calculating. Is your exact answer near the estimate?
Hint: Round each factor to a friendly number first.
Questions with Answers
- What does each digit in 27.53 stand for?
2 = 2 Tens, 7 = 7 Ones, 5 = 5 Tenths, 3 = 3 Hundredths. Together: 20 + 7 + 0.5 + 0.03 = 27.53. - Find 5.8 × 1.24 step by step, and check by estimation.
Multiply as counting numbers: 58 × 124 = 7192. Decimal digits: 1 + 2 = 3, so the product is 7.192. Estimate: 5.8 ≈ 6 and 1.24 ≈ 1, so the product should be near 6 — 7.192 is plausible. - Find 123 ÷ 10 and 123 ÷ 100. What is the pattern?
123 ÷ 10 = 12.3 and 123 ÷ 100 = 1.23. Dividing by 10 shifts the point one place left; dividing by 100 shifts it two places left. - Find 7.2 ÷ 3 and explain the decimal point in the quotient.
72 tenths ÷ 3 = 24 tenths = 2.4. The point appears in the quotient when the Ones are regrouped into Tenths during the division. - Find 4.8 ÷ 0.6. Why is it valid to change it to 48 ÷ 6?
Multiplying both dividend and divisor by 10 does not change the quotient (both are scaled equally), and 0.6 × 10 = 6 is a counting number. So 4.8 ÷ 0.6 = 48 ÷ 6 = 8.
Extension questions with answers — NCERT exercise types
- Find 2.45 × 3.2 and 0.245 × 32. Why are the products related?
2.45 × 3.2: 245 × 32 = 7840, with 2 + 1 = 3 decimal digits → 7.840 = 7.84. 0.245 × 32: 245 × 32 = 7840, with 3 + 0 = 3 decimal digits → 7.840 = 7.84. Same digits, same decimal count — the point lands in the same place because the total decimal digits are equal. - Divide 15.6 by 0.12. Show the conversion step.
Multiply both by 100: 15.6 × 100 = 1560, 0.12 × 100 = 12. Then 1560 ÷ 12 = 130. So 15.6 ÷ 0.12 = 130. Check: 130 × 0.12 = 15.6. Correct. - Is 1 ÷ 7 terminating or non-terminating? Find its first four decimal digits.
Non-terminating: 1 ÷ 7 = 0.142857… The first four decimal digits are 1, 4, 2, 8 (0.1428…), and the block 142857 repeats. - A ribbon 8.4 m long is cut into pieces of 0.7 m. How many pieces? Solve by division.
8.4 ÷ 0.7 = 84 ÷ 7 = 12 pieces. Check: 12 × 0.7 = 8.4. Correct. - Find the missing digit: 3.__ × 4 = 13.6. Explain by place value.
13.6 ÷ 4 = 3.4, so the missing digit is 4. Check: 3.4 × 4 = 13.6. In tenths: 34 tenths × 4 = 136 tenths = 13.6. - Hidato-style reasoning: in a decimal puzzle, the product of two decimals has 4 decimal digits. What can you say about the factors' decimal digits?
Their decimal digits add to 4 — for example 2 + 2 (like 1.25 × 3.15 = 3.9375), or 1 + 3, or 0 + 4. Counting digits is the quickest consistency check on any decimal multiplication.
- Decimals extend place value past Ones: Tenths, Hundredths, Thousandths.
- Multiply as counting numbers; product's decimal digits = sum of factors' digits.
- Divide by long division past the Ones; make a decimal divisor whole first.
- Dividing by 10/100 shifts the point left; multiplying by 10/100 shifts it right.
- Some quotients never terminate (e.g. 1 ÷ 3 = 0.333…).
How to use this page: Read the summary, say the Key Terms in your own words, then try the practice set without looking. Use the Questions with Answers to check yourself afterwards. Ask your teacher or parent to check your answers.