Introduction to Decimals
You go to a shop and buy a notebook for Rs. 35.50. The shopkeeper gives you change of Rs. 4.50 from a fifty-rupee note. These numbers with a dot in between are called decimals. The dot is called the decimal point.
Decimals are a way of writing fractions that have denominators like 10, 100, or 1000 — without actually writing the fraction. Instead of writing 1/2 as a fraction, you can write it as 0.5. Instead of 3/4, you can write 0.75.
You see decimals every day — in prices (Rs. 99.99), in measurements (1.5 metres of cloth), in temperature (36.6 degrees body temperature), and even in cricket averages (45.28). Think about the distance markers on a highway. They show distances like 245.5 km, not 245 and 1/2 km. That is because decimals are shorter and easier to read than fractions.
Decimals extend the place value system you already know. In whole numbers, you have ones, tens, hundreds, and so on. Decimals continue this pattern to the right of the ones place, giving you tenths, hundredths, and thousandths. Each place is 10 times smaller than the one before it, just as each place to the left is 10 times bigger. In this chapter, you will learn what decimals are, how to read and write them, their place values, and how they connect to fractions. This topic is part of the Decimals chapter in Grade 6 Maths (NCERT/CBSE).
What is Introduction to Decimals - Grade 6 Maths (Decimals)?
Definition: A decimal number is a number that has a whole number part and a fractional part separated by a decimal point (.).
For example, in 23.75:
- 23 is the whole number part (to the left of the decimal point)
- 75 is the decimal part or fractional part (to the right of the decimal point)
- The dot (.) is the decimal point
Place value after the decimal point:
- The first place after the decimal point is the tenths place (1/10)
- The second place is the hundredths place (1/100)
- The third place is the thousandths place (1/1000)
So in 23.75:
- 2 is in the tens place (value = 20)
- 3 is in the ones place (value = 3)
- 7 is in the tenths place (value = 7/10 = 0.7)
- 5 is in the hundredths place (value = 5/100 = 0.05)
Reading decimals:
- 23.75 is read as "twenty-three point seven five" (NOT "twenty-three point seventy-five")
- Read each digit after the decimal point separately
- 0.3 is read as "zero point three"
- 4.08 is read as "four point zero eight"
Introduction to Decimals Formula
Converting a fraction to a decimal:
Divide the numerator by the denominator
Examples:
- 1/2 = 1 ÷ 2 = 0.5
- 3/4 = 3 ÷ 4 = 0.75
- 1/5 = 1 ÷ 5 = 0.2
Converting a decimal to a fraction:
Write the decimal digits over the place value, then simplify
Examples:
- 0.5 = 5/10 = 1/2
- 0.25 = 25/100 = 1/4
- 0.125 = 125/1000 = 1/8
Expanded form of a decimal:
34.56 = 30 + 4 + 5/10 + 6/100
Derivation and Proof
Decimals are built on the place value system that you already know for whole numbers.
In whole numbers, each place is 10 times the place to its right:
- Ones × 10 = Tens
- Tens × 10 = Hundreds
- Hundreds × 10 = Thousands
Now, what happens if we go to the right of the ones place? Each place becomes 1/10 of the previous place:
- Ones ÷ 10 = Tenths (1/10)
- Tenths ÷ 10 = Hundredths (1/100)
- Hundredths ÷ 10 = Thousandths (1/1000)
The decimal point separates the whole number places from the fractional places.
Thousands | Hundreds | Tens | Ones . Tenths | Hundredths | Thousandths
1000 | 100 | 10 | 1 . 1/10 | 1/100 | 1/1000
For the number 5.347:
- 5 is in ones place → 5 × 1 = 5
- 3 is in tenths place → 3 × 1/10 = 0.3
- 4 is in hundredths place → 4 × 1/100 = 0.04
- 7 is in thousandths place → 7 × 1/1000 = 0.007
- Total: 5 + 0.3 + 0.04 + 0.007 = 5.347
This is exactly how whole number place value works, just extended to the right of the decimal point.
Types and Properties
Type 1: Reading and writing decimals
- Write the number in words or write the word form as a number.
- Example: 12.09 = twelve point zero nine
Type 2: Identifying place value in decimals
- Find the place value of a particular digit in a decimal number.
- Example: In 3.486, the digit 8 is in the hundredths place.
Type 3: Expanded form of decimals
- Write a decimal in expanded form using place values.
- Example: 7.25 = 7 + 2/10 + 5/100
Type 4: Converting fractions to decimals
- Convert fractions with denominators 10, 100, 1000 to decimals directly.
- Example: 37/100 = 0.37
Type 5: Converting decimals to fractions
- Write the decimal as a fraction and simplify.
- Example: 0.6 = 6/10 = 3/5
Type 6: Representing decimals on a number line
- Divide the unit length into 10 equal parts for tenths.
- Example: 0.7 is 7 small divisions after 0 on a number line marked in tenths.
Type 7: Decimals in money and measurement
- Write amounts using decimals: Rs. 5 and 75 paise = Rs. 5.75
- Write lengths: 3 m 25 cm = 3.25 m
Solved Examples
Example 1: Example 1: Reading a Decimal Number
Problem: Read the following numbers: (a) 15.8 (b) 0.06 (c) 234.507
Solution:
- (a) 15.8 → Fifteen point eight
- (b) 0.06 → Zero point zero six
- (c) 234.507 → Two hundred thirty-four point five zero seven
Remember: Read each digit after the decimal point one by one.
Example 2: Example 2: Finding Place Value
Problem: In the number 46.385, find the place value of each digit.
Solution:
- 4 is in the tens place → value = 40
- 6 is in the ones place → value = 6
- 3 is in the tenths place → value = 3/10 = 0.3
- 8 is in the hundredths place → value = 8/100 = 0.08
- 5 is in the thousandths place → value = 5/1000 = 0.005
Answer: 46.385 = 40 + 6 + 0.3 + 0.08 + 0.005
Example 3: Example 3: Writing in Expanded Form
Problem: Write 83.29 in expanded form.
Solution:
- 8 is in tens place = 80
- 3 is in ones place = 3
- 2 is in tenths place = 2/10
- 9 is in hundredths place = 9/100
Answer: 83.29 = 80 + 3 + 2/10 + 9/100
Example 4: Example 4: Converting Fraction to Decimal
Problem: Convert the following fractions to decimals: (a) 7/10 (b) 43/100 (c) 9/1000
Solution:
- (a) 7/10 = 0.7 (7 in tenths place)
- (b) 43/100 = 0.43 (4 in tenths, 3 in hundredths)
- (c) 9/1000 = 0.009 (0 in tenths, 0 in hundredths, 9 in thousandths)
Tip: Count the number of zeros in the denominator. That tells you how many decimal places.
Example 5: Example 5: Converting Decimal to Fraction
Problem: Convert to fractions: (a) 0.8 (b) 0.45 (c) 0.125
Solution:
- (a) 0.8 = 8/10 = (8 ÷ 2)/(10 ÷ 2) = 4/5
- (b) 0.45 = 45/100 = (45 ÷ 5)/(100 ÷ 5) = 9/20
- (c) 0.125 = 125/1000 = (125 ÷ 125)/(1000 ÷ 125) = 1/8
Example 6: Example 6: Decimals in Money
Problem: Write in decimal form: (a) Rs. 8 and 50 paise (b) Rs. 12 and 5 paise (c) 325 paise in rupees
Solution:
Remember: 100 paise = Rs. 1. So 1 paisa = Rs. 1/100 = Rs. 0.01
- (a) Rs. 8 and 50 paise = Rs. 8 + Rs. 50/100 = Rs. 8.50
- (b) Rs. 12 and 5 paise = Rs. 12 + Rs. 5/100 = Rs. 12.05
- (c) 325 paise = 325/100 rupees = Rs. 3.25
Example 7: Example 7: Decimals in Measurement (Length)
Problem: Convert: (a) 5 m 35 cm to metres (b) 2 km 750 m to kilometres
Solution:
Given: 100 cm = 1 m, so 1 cm = 1/100 m = 0.01 m. Also, 1000 m = 1 km, so 1 m = 1/1000 km = 0.001 km.
- (a) 5 m 35 cm = 5 + 35/100 m = 5 + 0.35 = 5.35 m
- (b) 2 km 750 m = 2 + 750/1000 km = 2 + 0.750 = 2.750 km
Example 8: Example 8: Decimals in Measurement (Weight)
Problem: Express 3 kg 250 g in kilograms.
Solution:
Given: 1000 g = 1 kg, so 1 g = 1/1000 kg = 0.001 kg
- 3 kg 250 g = 3 + 250/1000 kg
- = 3 + 0.250 kg
- = 3.250 kg
Answer: 3 kg 250 g = 3.250 kg (or simply 3.25 kg)
Example 9: Example 9: Representing on a Number Line
Problem: Represent 0.3 and 0.7 on a number line.
Solution:
Step 1: Draw a number line from 0 to 1.
Step 2: Divide the space between 0 and 1 into 10 equal parts.
Step 3: Each part represents 0.1 (one tenth).
- 0.3 is at the 3rd mark from 0 (3 tenths)
- 0.7 is at the 7th mark from 0 (7 tenths)
0---0.1---0.2---0.3---0.4---0.5---0.6---0.7---0.8---0.9---1
Example 10: Example 10: Writing Decimals as Words and Back
Problem: (a) Write 0.205 in words. (b) Write "seven and four hundredths" as a decimal.
Solution:
- (a) 0.205 → Zero point two zero five
- In fraction form: 205/1000 = 205 thousandths
- (b) Seven and four hundredths → whole part = 7, hundredths = 04
- = 7.04
Real-World Applications
Decimals are used in almost every part of daily life:
- Money: All prices use decimals. Rs. 49.99 means 49 rupees and 99 paise. Banking transactions, bills, and receipts all use decimal notation. Even your pocket money can be expressed as a decimal — Rs. 50.00 means exactly fifty rupees with zero paise.
- Measurement: Height (5.4 feet), weight (52.5 kg), temperature (98.6°F), distance (3.7 km) — measurements use decimals to show values between whole numbers. Your height might be 4.5 feet today and 4.8 feet next year.
- Sports: A cricketer's batting average (45.67), a sprinter's time (9.58 seconds for 100m), or a gymnast's score (9.75) — all use decimals for precision. In the Olympics, the difference between gold and silver can be just 0.01 seconds.
- Fuel: Petrol is priced per litre as Rs. 106.31 and sold in amounts like 5.25 litres. The fuel gauge in a car shows decimal readings.
- Science: Scientists measure things very precisely using decimals — a cell might be 0.005 mm wide, or the speed of light is 299,792.458 km/s.
- Cooking: Recipes sometimes use decimals — 0.5 litres of milk, 1.25 kg of flour.
- Weather: Rainfall is measured in decimals like 2.5 mm or 12.7 mm. Temperature forecasts use decimals for accuracy.
- Electricity bills: Your home electricity meter shows readings in decimals. The bill is calculated based on these decimal readings.
Key Points to Remember
- A decimal number has a whole part and a fractional part separated by a decimal point (dot).
- The places to the right of the decimal point are tenths (1/10), hundredths (1/100), thousandths (1/1000), and so on.
- Each place value is 1/10th of the place to its left.
- Read digits after the decimal point one by one: 3.45 is "three point four five" (not "three point forty-five").
- To convert a fraction with denominator 10, 100, or 1000 to a decimal, count the zeros and place digits accordingly.
- To convert a decimal to a fraction, write the digits over the place value (10, 100, or 1000) and simplify.
- 100 paise = Re 1, so paise are hundredths of a rupee.
- 100 cm = 1 m, so cm are hundredths of a metre.
- 1000 g = 1 kg, so grams are thousandths of a kilogram.
- Adding zeros to the right of the last digit after the decimal point does NOT change the value: 0.5 = 0.50 = 0.500.
Practice Problems
- Write the place value of 6 in: (a) 23.64 (b) 0.006 (c) 6.13.
- Write in expanded form: (a) 9.07 (b) 56.423.
- Convert to decimals: (a) 3/10 (b) 57/100 (c) 8/1000.
- Convert to fractions in simplest form: (a) 0.4 (b) 0.75 (c) 0.008.
- Express in decimal form: Rs. 7 and 30 paise.
- Express 4 km 85 m in kilometres.
- Represent 1.4 and 1.9 on a number line.
- Write as a decimal: five hundred and six thousandths.
Frequently Asked Questions
Q1. What is a decimal?
A decimal is a number that uses a decimal point (dot) to separate the whole number part from the fractional part. For example, 3.75 has whole part 3 and fractional part 75 hundredths.
Q2. Why is 0.5 the same as 0.50?
Adding zeros to the right end of a decimal does not change its value. 0.5 = 5/10 = 50/100 = 0.50. The extra zero just means zero hundredths, which adds nothing to the value.
Q3. How do you read 4.07?
Four point zero seven. Read each digit after the decimal point separately. Do NOT say 'four point seven' because that would be 4.7, which is a different number.
Q4. What is the difference between 0.3 and 0.03?
0.3 means 3 tenths (3/10), while 0.03 means 3 hundredths (3/100). 0.3 is ten times bigger than 0.03. On a number line, 0.3 is much further from 0 than 0.03.
Q5. How are decimals related to fractions?
Every decimal can be written as a fraction with a denominator of 10, 100, 1000, etc. For example, 0.7 = 7/10 and 0.25 = 25/100 = 1/4. Decimals are just another way of writing these fractions.
Q6. Why do we use decimals instead of fractions?
Decimals are easier to compare, add, and subtract than fractions. It is simpler to see that 0.75 is greater than 0.5 than to compare 3/4 and 1/2. Calculators and computers also work with decimals more easily.
Q7. What is a decimal point?
A decimal point is the dot (.) placed between the whole number part and the fractional part of a decimal number. It tells you where the ones place ends and the tenths place begins.
Q8. How do you convert paise to rupees in decimal form?
Since 100 paise = 1 rupee, divide the paise by 100. For example, 75 paise = 75/100 = Rs. 0.75. And Rs. 3 and 45 paise = Rs. 3.45.
