PW Class 7 Maths Notes Chapter 3: A Peek Beyond the Point

Decimals help you represent quantities that are not always complete whole numbers. You may come across them while working with money, length, weight, volume, and many other everyday measurements. A Peek Beyond the Point builds on your earlier understanding of whole numbers and fractions by focusing on numbers beyond the decimal point.

PW Class 7 Maths Notes Chapter 3: A Peek Beyond the Point helps you understand decimal numbers, their place values, conversion between decimals and fractions, and addition and subtraction of decimals. You can also see how decimals are connected with common measurements and real-life quantities.

What Are Decimal Numbers?

A decimal number contains a whole-number part and a fractional part separated by a decimal point.

For example, consider:

12.34

Here:

• 12 is the whole-number part.

• .34 is the decimal part.

The decimal part represents quantities smaller than one whole using tenths, hundredths, thousandths, and further place values.

Understanding what the digits on both sides of the decimal point represent is important before you start performing calculations with decimal numbers.

Decimal Place Value System

Decimal numbers follow a place value system similar to whole numbers. However, the places to the right of the decimal point represent fractions of a whole.

Consider the number:

3.4823.482

Therefore:

3.482=3+4/10+8/100+2/1000

This expanded form helps you see exactly what each digit contributes to the value of a decimal number.

Converting Decimals and Fractions

Decimals and fractions are two different ways of representing parts of a whole. You should be able to move between these forms depending on the question.

How to Convert a Decimal into a Fraction

To convert a decimal into a fraction:

1. Count the number of digits after the decimal point.

2. Remove the decimal point and write the number as the numerator.

3. Use 10, 100, 1000, or another appropriate power of 10 as the denominator.

4. Simplify the fraction when possible.

For example:

0.6=6/10

Another example is:

0.75=75/100=3/4

The number of decimal places tells you which denominator to use.

How to Convert a Fraction into a Decimal

To convert a fraction into decimal form, divide the numerator by the denominator.

For example:

1/2=1÷2=0.5

Similarly:

5/8=5÷8=0.625

Practising both types of conversion can help you become more comfortable moving between fractions and decimals.

Addition and Subtraction of Decimals

When adding or subtracting decimal numbers, the most important step is to align the decimal points vertically. This ensures that digits with the same place value remain in the same column.

Keep these three rules in mind:

• Write the decimal points directly below one another.

• Add zeros when the numbers have unequal decimal places.

• Perform the addition or subtraction as you would with whole numbers.

For example:

12.450+7.305

Since the decimal places are already aligned, add the corresponding digits:

12.450+7.305=19.755

Keeping the decimal points aligned can prevent common place-value errors while calculating.

How Are Decimals Used in Everyday Life?

Decimals are useful when you need to represent quantities more precisely than whole numbers allow. You regularly encounter them while dealing with money and measurements.

Decimals in Money

Money often involves rupees and paise.

For example:

₹45.50

represents 45 rupees and 50 paise.

Similarly:

₹123.75

represents 123 rupees and 75 paise.

Decimals in Length

Decimals allow you to express a measurement containing a whole unit and part of another unit.

For example:

1.5 m=1 metre and 50 centimetres

Similarly:

3.25 km=3 kilometres and 250 metres

Decimals in Weight

A weight such as:

2.75 kg

can be understood as 2 kilograms and 750 grams.

Decimals in Volume

Similarly:

1.25 L 

represents 1 litre and 250 millilitres.

These examples show why decimals are useful when a quantity contains both a complete unit and a part of that unit.

Understanding Measurement and Decimals

Decimals are closely connected with measurement because units can be divided into smaller units. Knowing the relationship between these units can help you interpret decimal measurements correctly.

Length

Since:

1 metre=100 centimetres

we can write:

1.45 m=1 m+45 cm

Volume

Since:

1 litre=1000 millilitres

we get:

0.75 L=750 ml

Weight

Since:

1 kg=1000 grams

We can write:

2.25 kg=2 kg+250 g

Using rulers, measuring cups, weighing scales, and similar tools can also help you understand how decimal measurements represent parts of a unit.