1. Chapter at a Glance
A pencil is 12.5 cm long. A bag of rice weighs 2.75 kg. Your temperature may be 98.6 °F. Numbers with a decimal point let us measure quantities that lie between whole numbers. In this chapter you will see why decimals were invented, understand tenths, hundredths and thousandths, place decimals on the number line, compare them, add and subtract them, and use them in money and measurement.
- The need for units smaller than one: tenths and hundredths.
- Decimal place value: tenths, hundredths, thousandths.
- Decimals as fractions with denominators 10, 100, 1000.
- Decimals on the number line; comparing and ordering decimals.
- Adding and subtracting decimals.
- Decimals in money, length, weight and capacity.
2. Why Do We Need Decimals?
Suppose you measure a pencil with a ruler marked only in centimetres. It goes past 12 cm but does not reach 13 cm. To be exact, we divide each centimetre into 10 equal parts (millimetres). If the pencil ends at the 5th small mark after 12, its length is 12 and 5 tenths cm, written 12.5 cm.
- 1 unit divided into 10 equal parts gives tenths: 1/10 = 0.1.
- 1 tenth divided into 10 equal parts gives hundredths: 1/100 = 0.01.
- 1 hundredth divided into 10 equal parts gives thousandths: 1/1000 = 0.001.
The decimal point separates the whole-number part from the parts smaller than one.
3. Decimal Place Value
Our place-value system simply continues to the right of the ones place. Each place is one-tenth of the place on its left.
| Hundreds | Tens | Ones | . | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|---|
| 100 | 10 | 1 | . | 1/10 | 1/100 | 1/1000 |
| 3 | 4 | 7 | . | 2 | 0 | 5 |
347.205 = 3 hundreds + 4 tens + 7 ones + 2 tenths + 0 hundredths + 5 thousandths = 300 + 40 + 7 + 2/10 + 5/1000. It is read "three hundred forty-seven point two zero five".
Write as a decimal: 6 tens + 3 ones + 4 tenths + 9 hundredths.
Solution: 60 + 3 + 0.4 + 0.09 = 63.49.
Decimals as fractions
- 0.7 = 7/10; 0.07 = 7/100; 0.007 = 7/1000.
- 2.35 = 235/100 = 47/20 (simplest form).
- Fractions with denominator 10, 100 or 1000 convert directly: 9/100 = 0.09.
- Other fractions can sometimes be changed to such denominators: 3/4 = 75/100 = 0.75; 2/5 = 4/10 = 0.4.
4. Decimals on the Number Line
To mark 2.7, divide the gap between 2 and 3 into 10 equal parts; count 7 parts from 2. To mark 2.75, divide the gap between 2.7 and 2.8 further into 10 parts and count 5 more.
5. Comparing Decimals
- Compare the whole-number parts first; the larger whole part gives the larger number.
- If the whole parts are equal, compare the tenths digits, then the hundredths, and so on.
- It helps to make the number of decimal places equal by adding zeros at the end: 3.4 = 3.40 = 3.400.
Which is greater: 0.6 or 0.58?
Solution: Write 0.6 as 0.60. Compare 0.60 and 0.58: tenths 6 > 5. So 0.6 > 0.58, even though 58 looks bigger than 6.
"More digits means bigger" is true for whole numbers but not for decimals. 0.5 is greater than 0.4999. Always line up the decimal points and compare place by place.
6. Adding and Subtracting Decimals
- Write the numbers one below the other with the decimal points in a straight line.
- Fill empty places with zeros so both numbers have the same number of decimal places.
- Add or subtract as with whole numbers, carrying or borrowing across the decimal point as usual.
- Place the decimal point in the answer directly below the others.
Sita bought vegetables for ₹48.75 and fruits for ₹126.50. She paid ₹200. How much change did she get?
Solution: Total = 48.75 + 126.50 = ₹175.25. Change = 200.00 − 175.25 = ₹24.75.
7. Decimals in Measurement and Money
| Relation | As a decimal | Example |
|---|---|---|
| 1 mm = 1/10 cm | 0.1 cm | 45 mm = 4.5 cm |
| 1 cm = 1/100 m | 0.01 m | 235 cm = 2.35 m |
| 1 m = 1/1000 km | 0.001 km | 2,750 m = 2.75 km |
| 1 g = 1/1000 kg | 0.001 kg | 650 g = 0.65 kg |
| 1 mL = 1/1000 L | 0.001 L | 1,250 mL = 1.25 L |
| 1 paisa = 1/100 rupee | ₹0.01 | ₹7 and 5 paise = ₹7.05 |
Convert 3 km 45 m into kilometres.
Solution: 45 m = 45/1000 km = 0.045 km. So 3 km 45 m = 3.045 km (not 3.45 km).
A pinch of history
The decimal place-value system for whole numbers was developed in India. Writing fractions of a unit in the same place-value way (decimal fractions) was developed further by mathematicians in the Arab world and later in Europe. Today, decimals are used everywhere — in money, science, sport timing and computers.
8. Solved Problems (NCERT Style)
Write 205.068 in expanded form using fractions.
Solution: 205.068 = 200 + 5 + 0/10 + 6/100 + 8/1000. The zeros in the tens and tenths places still matter — they keep the other digits in their correct places.
Convert 7/25 and 9/8 into decimals.
Solution: 7/25 = 28/100 = 0.28 (multiply top and bottom by 4). 9/8 = 1 1/8 and 1/8 = 125/1000, so 9/8 = 1.125.
Arrange from shortest to longest: 1.2 m, 108 cm, 1.15 m, 1,250 mm.
Solution: Convert all to metres: 1.2 m, 1.08 m, 1.15 m, 1.25 m. Order: 108 cm < 1.15 m < 1.2 m < 1,250 mm.
A notebook costs ₹38.50 and a geometry box costs ₹112.75. How much more does the geometry box cost, and what do both cost together?
Solution: Difference = 112.75 − 38.50 = ₹74.25. Total = 112.75 + 38.50 = ₹151.25.
A bag contains 2.5 kg of sugar. 750 g is used for sweets and 0.35 kg for tea. How much sugar is left?
Solution: 750 g = 0.75 kg. Used = 0.75 + 0.35 = 1.10 kg. Left = 2.50 − 1.10 = 1.4 kg (= 1,400 g).
Which of 4.09, 3.91, 4.1 and 3.99 is closest to 4?
Solution: Distances from 4 are 0.09, 0.09, 0.1 and 0.01. So 3.99 is closest.
In a 100 m race, Asha finished in 14.08 s and Babita in 13.9 s. Who was faster and by how much?
Solution: 13.90 < 14.08, so Babita took less time and was faster, by 14.08 − 13.90 = 0.18 s.
I am a decimal number between 6 and 7. My tenths digit is the same as my ones digit. My hundredths digit is 1 less than my tenths digit, and I have no thousandths. Who am I?
Solution: Between 6 and 7 means the ones digit is 6. The tenths digit is also 6, and the hundredths digit is 6 − 1 = 5. The number is 6.65. Check: 6 < 6.65 < 7 ✔.
9. Master Formula Table
| Idea | Rule | Example |
|---|---|---|
| Tenth | 1/10 = 0.1 | 3 tenths = 0.3 |
| Hundredth | 1/100 = 0.01 | 27 hundredths = 0.27 |
| Thousandth | 1/1000 = 0.001 | 9 thousandths = 0.009 |
| Decimal → fraction | digits over 10, 100, 1000 | 0.36 = 36/100 = 9/25 |
| Fraction → decimal | make denominator 10, 100, 1000 | 3/20 = 15/100 = 0.15 |
| Trailing zeros | do not change value | 5.2 = 5.20 |
| Add/subtract | line up decimal points | 4.5 + 0.75 = 5.25 |
| cm to m | ÷ 100 | 85 cm = 0.85 m |
| g to kg | ÷ 1000 | 75 g = 0.075 kg |
10. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Need | Measure between whole numbers | Real examples |
| Place value | Each place = 1/10 of left place | Expanded form |
| Number line | Split units into tenths, hundredths | Marking decimals |
| Comparing | Equalise places, compare left to right | Ordering lists |
| Operations | Align decimal points | Money problems |
| Measurement | Unit conversions use decimals | km–m, kg–g, L–mL |
Chart: four ways of writing the same amount
| Words | Fraction | Decimal | Measurement |
|---|---|---|---|
| Three-quarters | 3/4 | 0.75 | 75 cm of a metre |
| One-fifth | 1/5 | 0.2 | 200 g of a kilogram |
| One and a half | 3/2 | 1.5 | 1.5 L = 1500 mL |
11. Mind Map
- Tenths, hundredths
- Thousandths
- Decimal point
- Place-value table
- Fractions
- Number line
- Equalise places
- Left to right
- Add, subtract
- Money, length, weight
12. Tips and Tricks
- Add zeros at the end of decimals to make the same number of places — it never changes the value.
- For unit conversions, count the zeros: 1000 g in 1 kg means move the decimal point 3 places.
- Estimate first: 19.8 + 4.95 ≈ 20 + 5 = 25.
- Read 0.05 as "five hundredths" — this avoids confusing it with 0.5.
- Thinking 0.45 > 0.5 because 45 > 5.
- Writing 3 km 45 m as 3.45 km (correct: 3.045 km).
- Not lining up decimal points while adding.
- Placing the decimal point incorrectly in the final answer.
13. Important One-liners
- 0.1 = 1/10, 0.01 = 1/100, 0.001 = 1/1000.
- Each decimal place is one-tenth of the place to its left.
- Zeros at the end of a decimal do not change its value.
- Compare decimals place by place from the left.
- Line up decimal points before adding or subtracting.
- 1 cm = 0.01 m and 1 g = 0.001 kg.
- 1 paisa = ₹0.01.
- 3/4 = 0.75 and 1/2 = 0.5.
- A decimal like 2.35 can be written as the fraction 235/100.
14. Exam-pattern Questions with Detailed Answers
Write 7/100 as a decimal.
- 0.7
- 0.07
- 0.007
- 7.00
Show answer
7 hundredths is written with 7 in the hundredths place: 0.07.
Answer: (B) 0.07
Arrange in ascending order: 0.8, 0.75, 0.805, 0.08.
Show answer
Write with 3 places: 0.800, 0.750, 0.805, 0.080. Ascending: 0.080, 0.750, 0.800, 0.805.
Answer: 0.08 < 0.75 < 0.8 < 0.805
Express 1,250 g in kilograms.
Show answer
1,250 ÷ 1000 = 1.25 kg.
Answer: 1.25 kg
Find 34.5 + 7.86 − 12.375.
Show answer
34.500 + 7.860 = 42.360. Then 42.360 − 12.375 = 29.985.
Answer: 29.985
Write 4.36 as a fraction in simplest form.
Show answer
4.36 = 436/100. HCF(436, 100) = 4, giving 109/25.
Answer: 109/25 (= 4 9/25)
Ravi ran 2.4 km on Monday, 3.08 km on Tuesday and 1.75 km on Wednesday. How far did he run in all?
- 7.23 km
- 6.95 km
- 7.32 km
- 8.03 km
Show answer
2.40 + 3.08 + 1.75 = 7.23 km.
Answer: (A) 7.23 km
Which decimal lies exactly halfway between 3.6 and 3.7?
Show answer
3.6 = 3.60 and 3.7 = 3.70. Halfway is 3.65.
Answer: 3.65
A ribbon 5 m long is cut into pieces of 1.25 m, 85 cm and 1.4 m. How much ribbon is left?
Show answer
Used = 1.25 + 0.85 + 1.40 = 3.50 m. Left = 5.00 − 3.50 = 1.50 m.
Answer: 1.5 m
Write three decimals between 0.4 and 0.41.
Show answer
Write 0.4 = 0.400 and 0.41 = 0.410. Any decimal from 0.401 to 0.409 lies between them, e.g. 0.401, 0.405, 0.409.
Answer: For example 0.401, 0.405, 0.409
A water tank holds 2 kL. 1,375 L were used on Monday and 0.48 kL on Tuesday. How many litres are left?
Show answer
2 kL = 2000 L; 0.48 kL = 480 L. Used = 1375 + 480 = 1855 L. Left = 2000 − 1855 = 145 L.
Answer: 145 L
Using each of the digits 4, 0, 7 and 2 once and one decimal point, form the greatest number less than 10 and the smallest number greater than 1.
Show answer
Less than 10 → one digit before the point; make it as large as possible: 7.420.
Greater than 1 → the whole part must be at least 1; the smallest choice is 2, followed by the smallest arrangement: 2.047.
Answer: 7.42 and 2.047
Two friends measured a table: one said 1.2 m, the other said 118 cm. Who said the longer length, and by how much?
Show answer
1.2 m = 120 cm, which is 2 cm more than 118 cm.
Answer: 1.2 m is longer by 2 cm (0.02 m)
15. Quick Revision Summary
- Decimals extend place value: tenths (0.1), hundredths (0.01), thousandths (0.001).
- Decimals are fractions with denominators 10, 100, 1000.
- On the number line, divide each unit into 10, then each tenth into 10.
- Compare by equalising decimal places and checking from the left.
- Add/subtract with decimal points lined up.
- Use decimals for money (paise), length (cm, m, km), weight (g, kg) and capacity (mL, L).