Chapter focus
The decimal point separates whole-number places from fractional places. Decimal notation lets us represent values between whole numbers, compare measurements and perform calculations precisely.
NCERT CIET identifies decimal multiplication, HCF and LCM activities for this chapter. These original notes support practice alongside the assigned Ganita Prakash Part I.
1. Decimal place value
To the left of the decimal point, places are ones, tens, hundreds and so on. To the right, they are tenths, hundredths, thousandths and so on. Each place to the right is one-tenth of the place before it.
In 23.406, 2 means two tens, 3 means three ones, 4 means four tenths, 0 means no hundredths, and 6 means six thousandths. So 23.406 = 20 + 3 + 0.4 + 0.006.
Trailing zeroes do not change a decimal’s value: 0.5 = 0.50. Compare decimals by aligning decimal points or writing equal numbers of decimal places: 0.7 = 0.70 > 0.68.
2. Add and subtract decimals
Line up the decimal points so like place values are in the same column. Add or subtract as with whole numbers, then place the decimal point directly below the others.
Worked example: 12.75 − 8.6
- Write 8.6 as 8.60.
- Align: 12.75 − 8.60.
- Subtract: 4.15.
Estimate first: about 13 − 9 = 4, so 4.15 is reasonable.
3. Multiply decimals using place value or a grid
Multiply as whole numbers first, then count the total decimal places in both factors. The product must have that many decimal places.
Worked example: 0.3 × 0.4
Think of 0.3 as 3 tenths and 0.4 as 4 tenths. A 10-by-10 grid has 100 small squares; 3 tenths in one direction and 4 tenths in the other cover 3 × 4 = 12 small squares. The overlap is 12 hundredths, so 0.3 × 0.4 = 0.12.
Place-value check: 3 × 4 = 12, and two factors each have one decimal place, so the product has two decimal places.
Example: 2.4 × 0.35
24 × 35 = 840. There are three decimal places in total (one in 2.4 and two in 0.35), so the product is 0.840 = 0.84.
4. Find HCF and LCM
The highest common factor (HCF) is the greatest whole number that divides each given number. The least common multiple (LCM) is the smallest positive number that is a multiple of each.
Worked example: 18 and 30
Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 30 include 1, 2, 3, 5, 6, 10, 15, 30. The greatest common factor is 6.
Multiples of 18: 18, 36, 54, 72, 90, … . Multiples of 30: 30, 60, 90, … . The first common positive multiple is 90.
Use HCF to make equal groups as large as possible without leftovers; use LCM to find when repeating cycles coincide.
Common errors to avoid
- Line up decimal points for addition and subtraction, not the last digits.
- In multiplication, count decimal places in both factors together.
- A larger denominator does not automatically make a larger decimal or fraction.
- HCF is a factor; LCM is a multiple. Keep the terms distinct.
Practice: calculate and check
1. Write 7.035 in expanded form.
Answer: 7 + 0.03 + 0.005.
2. Calculate 4.8 + 0.75.
Answer: 4.80 + 0.75 = 5.55.
3. Multiply 0.6 × 0.08.
Answer: 6 × 8 = 48 with three decimal places, so 0.048.
4. Find the HCF of 24 and 36.
Answer: Common factors include 1, 2, 3, 4, 6 and 12; HCF = 12.
5. Two bells ring every 8 minutes and 12 minutes. If they ring together now, after how many minutes will they next ring together?
Answer: LCM(8, 12) = 24, so they next ring together after 24 minutes.
Chapter summary
Decimal place value extends the base-ten system to tenths, hundredths and smaller units. Align places for addition and subtraction; count decimal places when multiplying. HCF finds a greatest shared factor, while LCM finds the earliest shared multiple.
Open Ganita Prakash Part I on NCERT · NCERT CIET Grade 7 learning resources