Decimals Class 6 Notes Maths Chapter 8

Introduction

Decimals are an essential part of mathematics used to represent numbers that are not whole. A decimal number has two parts: a whole number part and a fractional part separated by a decimal point. Decimals are used in everyday life for money, measurements, data and many practical applications.

Example: If you have 1 rupee and 50 paise, you write it as ₹1.50. Here 1 is the whole number (rupee) and 50 is the fractional part (paise) after the decimal point. The decimal point separates the units place and the tenths place.

Comparing Decimals

To compare two decimals, follow these ideas in order: compare their whole number parts; if these are equal, compare digits in the tenths place; if still equal, compare the hundredths place, then thousandths place, and so on.

Converting between Fractions and Decimals

Converting a fraction to a decimal: Change the denominator to 10, 100, 1000, etc., by multiplying the numerator and denominator by the same number, then divide.

Example: Convert

to a decimal.

Multiply the numerator and denominator by a number that makes the denominator 10, 100, 1000, ...

So,

in decimal form.

Converting a decimal to a fraction: Place the decimal number over 10, 100, 1000, ... depending on the number of digits after the decimal point, then simplify.

Example: Convert 8.9 to a fraction.

8.9 = 8 + 0.9 = 8 + 9/10 = 89/10. As a mixed number, it is 8 9/10.

Visual and Place-Value Methods to Compare

Example: Compare 0.07 and 0.1.

• Think of a square divided into 100 equal parts.
• For 0.07: shade 7 of the 100 parts.
• For 0.1: shade 10 of the 100 parts (because 0.1 = 1/10 = 10/100).
• Since 10 parts out of 100 > 7 parts out of 100, 0.1 > 0.07.

• 0.07 =
  
• 0.1 =
  

Example: Which is greater: 0.3 or 0.25?

Ans: Imagine two chocolate bars divided into 100 equal pieces.

First bar (0.25): you have 25 pieces out of 100. This is

Second bar (0.3): you have 30 pieces out of 100. This is

Since 30 pieces > 25 pieces, 0.3 > 0.25.

Comparing Decimals with the Same Whole Number

• Example: Compare 45.78 and 45.7.
• Write both numbers so that decimal places align: 45.78 and 45.70.
• Whole parts are equal (45), tenths places are equal (7), compare hundredths: 8 (in 45.78) and 0 (in 45.70).
• Since 8 > 0, 45.78 > 45.7.
• 45.78 shown as
  
• 45.7 shown as
  

Always compare digit by digit from left to right starting just after the decimal point until a difference appears.

Example: Compare 64.24 and 64.205.

Ans: Compare whole parts: 64 = 64.

Compare tenths: 2 = 2.

Compare hundredths: 4 (in 64.24) and 0 (in 64.205).

Since 4 > 0, 64.24 > 64.205.

Using Decimals

Money

Decimals are used to represent parts of a rupee.

• 100 paise = ₹1
• 1 paise = ₹0.01

To convert paise to rupees, divide by 100.

Example: If 1 kg of turmeric costs ₹51, then 500 g (which is half a kg) costs ₹25.50. Here ₹25.50 means ₹25 and 50 paise.

Q1: Convert 165 paise to rupees.

Ans:

165 paise = 165 × (1/100) rupees.

165 × 0.01 = ₹1.65 (that is, ₹1 and 65 paise).

Q2: Convert 350 paise to rupees.

Ans:

350 paise = 350 × (1/100) rupees.

350 × 0.01 = ₹3.50 (that is, ₹3 and 50 paise).

Length

When a measurement does not fall exactly on a whole unit, decimals express the fractional part.

• 1 km = 1000 m
• 1 m = 100 cm
• 1 cm = 10 mm

To convert a measurement to decimal form, express the smaller unit as a fraction of the larger unit.

Q1: Convert 5 km 75 m into decimal kilometres.

Ans:

1 m = 1/1000 km.

75 m = 75 × (1/1000) km = 0.075 km.

5 km + 0.075 km = 5.075 km.

Q2: Convert 61 km 25 m into decimal kilometres.

Ans:

25 m = 25 × (1/1000) km = 0.025 km.

61 km + 0.025 km = 61.025 km.

Weight

Weights are often given in kilograms and grams. Use decimals to write grams as parts of a kilogram.

• 1 kg = 1000 g
• 1 g = 1/1000 kg

To convert grams to kilograms, divide grams by 1000.

Q1: Convert 250 g to kg.

Ans:

250 g = 250 × (1/1000) kg = 250/1000 kg = 0.250 kg.

Q2: Represent 5 kg 396 g in decimal form (kg).

Ans:

396 g = 396 × (1/1000) kg = 0.396 kg.

5 kg + 0.396 kg = 5.396 kg.

Addition of Numbers with Decimals

Steps for addition of decimals:

• Line up the decimal points: Place numbers one below the other so the decimal points are in the same column. Add zeros to the right where necessary.
• Add as whole numbers: Add digits from right to left as you would for whole numbers.
• Place the decimal point: In the result, place the decimal point directly below the others.

Example: Add 0.56 + 9 + 6.287.
Write numbers with decimals aligned and add zeros where needed: 0.560 + 9.000 + 6.287.The sum is 15.847.

Q1: The total weight of a box containing 14 kg 750 g of mangoes and 5 kg 80 g of apples is 22 kg 200 g. How much is the weight of the empty box?

Ans:

Weight of mangoes = 14 kg 750 g.

Weight of apples = 5 kg 80 g.

Total weight of fruits = 14 kg 750 g + 5 kg 80 g = 19 kg 830 g.

Total weight of box + fruits = 22 kg 200 g.

Empty box weight = total weight - weight of fruits.

22 kg 200 g - 19 kg 830 g = 2 kg 370 g.

Therefore the empty box weighs 2 kg 370 g.

Q2: Add 27.076 + 0.55 + 0.004.

Ans:

Write with aligned decimals: 27.076 + 0.550 + 0.004.

Add the numbers: 27.076 + 0.550 + 0.004 = 27.630.

Q3: Nasreen bought 3 m 20 cm cloth for her shirt and 2 m 5 cm cloth for a skirt. Find the total cloth bought by her.

Ans:

Convert to metres: 3 m 20 cm = 3.20 m.

2 m 5 cm = 2.05 m.

Total cloth = 3.20 m + 2.05 m = 5.25 m = 5 m 25 cm.

Subtraction of Decimals

Steps for subtraction of decimals:

• Line up the decimal points: Place numbers vertically with decimals aligned. Add zeros to the right to make equal length.
• Subtract as whole numbers: Subtract digits from right to left, borrowing when needed.
• Place the decimal point: Put the decimal point in the result directly below the others.

Example: Subtract 6 - 2.25.

Write 6.00 - 2.25 and subtract: 6.00 - 2.25 = 3.75.

Q1: Ganesh purchased a book worth Rs.156.65 from a bookseller and he gave him Rs.500 note. How much balance did he get back?

Ans:

Total given = Rs. 500.00.

Cost = Rs. 156.65.

Balance = 500.00 - 156.65.

500.00 - 156.65 = Rs. 343.35.

Q2: Subtract 27.56 from 52.1.

Ans:

Write with aligned decimals: 52.10 - 27.56.

52.10 - 27.56 = 24.54.