A Peek Beyond the Point — Class 7 maths Chapter 3 Explanation (2026-27)

A PEEK BEYOND THE POINT3

3.1 The Need for Smaller Units

Sonu’s mother was fixing a toy. She was trying to join two pieces with the help of a screw. Sonu was watching his mother with great curiosity. His mother was unable to join the pieces. Sonu asked why. His mother said that the screw was not of the right size.She brought another screw from the box and was able to fix the toy. The two screws looked the same to Sonu. But when he observed them closely, he saw they were of slightly different lengths.Sonu was fascinated by how such a small difference in lengths could matter so much. He was curious to know the difference in lengths. He was also curious to know how little the difference was because the screws looked nearly the same.In the following figure, screws are placed above a scale. Measure them and write their length in the space provided.

Between 2 cm and 3 cm_______________

More than 2 12

_______________

cm but less than3 cm

_______________

2 710 cm

Which scale helped you measure the length of the screws accurately? Why?

What is the meaning of 2 710 cm (the length of the first screw)?

As seen on the ruler, the unit length between two consecutive

numbers is divided into 10 equal parts. To get the length 2 710 cm, we

go from 0 to 2 and then take seven parts of 110. The length of the screw

is 2 cm and 710 cm. Similarly, we can make sense of the length 3 210 cm.

We read 2 710 cm as two and seven-tenth centimeters, and 3 210 cm

as three and two-tenth centimeters.

Can you explain why the unit was divided into smaller parts to measure the screws?

Measure the following objects using a scale and write their measurements in centimeters (as shown earlier for the lengths of the screws): pen, sharpener, and any other object of your choice.

Write the measurements of the objects shown in the picture:

As seen here, when exact measures are required we can make use of smaller units of measurement.

3.2 A Tenth Part

The length of the pencil shown in the figure below is 3 410 units, which can

also be read as 3 units and four one-tenths, i.e., (3 × 1) + (4 × 110) units.

110 + 110 + 110 + 110 + 110 + 110 + 110 + 110 + 110 + 110

= 10 times 110 = 10 × 110 = 1 unit

This length is the same as 34 one-tenths units because 10 one-tenths units make one unit.

34 × 1

10 = 34

10 = 10

10 + 10

10 + 4

10 (34 one-tenths)

= 1 + 1 + 1 + 4

10 (3 and 4 one-tenths)

A few numbers with fractional units are shown below along with how to read them.

4 1

10 ‘four and one-tenth’

10 ‘four one-tenths’ or ‘four-tenths’

10 ‘forty-one one-tenths’ or ‘forty-one tenths’

41 1

10 ‘forty-one and one-tenth’

For the objects shown below, write their lengths in two ways and read them aloud. An example is given for the USB cable. (Note that the unit length used in each diagram is not the same).

The length of the USB cable is 4 and 8

10 units or 48

10 units.

Arrange these lengths in increasing order:

(a) 9

10(b) 1 7

10 (c) 130

10(d) 13 1

(e) 10 5

10(f) 7 6

10(g) 6 7

10(h) 410

Arrange the following lengths in increasing order: 4 110, 410, 4110, 41 110.

Sonu is measuring some of his body parts. The length of Sonu’s lower

arm is 2 710 units, and that of his upper arm is 3 610 units. What is the

total length of his arm?

To get the total length, let us see the lower and upper arm length as 2 units and 7 one-tenths, and 3 units and 6 one-tenths, respectively.So, there are (2 + 3) units and (7 + 6) one-tenths. Together, they make 5 units and 13 one-tenths. But 13 one-tenths is 1 unit and 3 one-tenths. So, the total length is 6 units and 3 one-tenths.

(a) (2 + 3) + (

710 + 610)

= (2 + 3) + (

1310)

= 5 + 1310

= 5 + 1010 + 310 = 5 + 1 + 310

= 6 + 310

= 6 310

(b) 2 710

+ 3 610

= 5 1310

= 6 310

Or, both the lengths can be converted to tenths and then added:

2710 + 3610 = 63106310 is the same as 60 one-tenths (

6010) and 3 one-tenths (

310), which is

equal to 6 units and 3 one-tenths, i.e., 6 310.

The lengths of the body parts of a honeybee are given. Find its total length.

Head Thoraxabdomen

Head: 2 310 units

Thorax: 5 410 units

Abdomen: 7 510 units

The length of Shylaja’s hand is 12 410 units,

and her palm is 6 710 units, as shown in the picture. What is the length

of the longest (middle) finger?

The length of the finger can be found by

evaluating (12 + 410) – (6 + 710). This can be

12 410 units

done in different ways. For example,

6 710 units

(a) 12 + 410 – 6 – 710

= (12 – 6) + (

410 – 710)

= 6 – 310

Discuss what is being done here and why.

= 5 + 1 – 310

= 5 + 1010 – 310

= 5 + 710 = 5 710

(b) 12 410 11 1410

– 6 710 – 6 710

= 5 710

As in the case of counting numbers, it is convenient to start subtraction from the tenths. We cannot remove 7 one-tenths from 4

one-tenths. So we split a unit from 12 and convert it to 10 one-tenths. Now, the number has 11 units and 14 one-tenths. We subtract 7 one-tenths from 14 one-tenths and then subtract 6 units from 11 units.

Try computing the difference by converting both lengths to tenths.

A Celestial Pearl Danio’s length is 2 410 cm, and the length of a Philippine

Goby is 910 cm. What is the difference in their lengths?

How big are these fish compared to your finger?

Celestial Pearl DanioPhilippine Goby

Observe the given sequences of numbers. Identify the change after each term and extend the pattern:

(a) 4, 4 310 , 4 610 , _________, _________, _________, _________

(b) 8 210 , 8 710 , 9 210 , _________, _________, _________, _________

(d) 5 710 , 5 310 , _________, _________, _________, _________

(e) 13 510 , 13, 12 510 , _________, _________, _________, _________

(f) 11 510 , 10 410 , 9 310 , _________, _________, _________, _________

3.3 A Hundredth Part

The length of a sheet of paper was 8

910 units, which can also be said as

8 units and 9 one-tenths. It is folded in half along its length. What is its length now?

We can say that its

length is between 4 410

units and 4 510 units. But

we cannot state its exact measurement, since there are no markings. Earlier, we split a unit into 10 one-tenths to measure smaller lengths. We can do something similar and split each one-tenth into 10 parts.

What is the length of this smaller part? How many such smaller parts make a unit length?

As shown in the figure below, each one-tenth has 10 smaller parts, and there are 10 one-tenths in a unit; therefore, there will be 100 smaller parts in a unit. Therefore, one part’s length will

be 1100 of a unit.

Returning to our question, what is the length of the folded paper?

We can see that it ends at 4 410 5100, read as 4 units and 4 one-tenths

and 5 one-hundredths.

How many one-hundredths make one-tenth? Can we also say that the length is 4 units and 45 one-hundredths?

Math Talk

Observe the figure below. Notice the markings and the corresponding lengths written in the boxes when measured from 0. Fill the lengths in the empty boxes.

1 310110

012

210

201001100

130100

99100

The length of the wire in the first picture is given in three different ways. Can you see how they denote the same length?

1 110 4100 One and one-tenth and four-hundredths

1 14100 One and fourteen-hundredths

114100 One Hundred and Fourteen-hundredths

For the lengths shown below write the measurements and read out the measures in words.

In each group, identify the longest and the shortest lengths. Mark each length on the scale.

(a) 310 , 3100 , 33100

(b) 3 110 , 3010 , 1 310

(d) 3 610 , 3 6100 , 3 610 6100

(e) 810 2100 , 9100 , 1 8100

(f) 7 310 5100 , 7 510 , 7 41100

(g) 6510 15100 , 5 87100 , 5 7100

What will be the sum of 15 310 4100 and 2 610 8100 ?

This can be solved in different ways. Some are shown below.

(a) Method 1

(15 + 2) + (

310 + 610) + (

4100 + 8100)

10 hundredths is the same as 1 tenth.

= 17 + 910 + 12100

= 17 + 910 + 110 + 2100

= 17 + 1010 + 2100

= 18 2100 .

15 310 4100

(b) Method 2

+ 2 610 8100

= 17 910 12100

= 17 1010 2100

= 18 2100

Are both these methods different?

Observe the addition done below for 483 + 268. Do you see any similarities between the methods shown above?

Math Talk

(400 + 80 + 3) + (200 + 60 + 8) = (400 + 200) + (80 + 60) + (3 + 8) = 600 + 140 + 11 = 600 + 150 + 1 = 700 + 50 + 1 = 751

One can also find the sum 15 310 4100 + 2 610 8100 by converting to hundredths, as follows.

34100 + 68100)

= 17 + 102100

100 hundredths is same as 1 unit.

= 17 + 1 + 2100

= 18 2100

1534100 ) + (

268100)

(d) (

= 1802100

15 is the same as 1500 hundredths and 2 is the same as 200 hundredths.

= 1800100 + 2100

= 18 2100

What is the difference: 25 910 – 6 410 7100 ?

One way to solve this is as follows:

25 810 10100

25 910

– 6 410 7100

= 19 410 3100

Math Talk

Solve this by converting to hundredths.

What is the difference 15 310 4100 –2 610 8100 ?

One way to solve this is as follows:

15 310 410015 210 1410014 1210 14100

– 2 610 8100– 2 610 8100– 2 610 8100

= 12 610 6100

Observe the subtraction done below for 653 – 268. Do you see any similarities with the methods shown above?

Math Talk

(600 + 50 + 3) – (200 + 60 + 8) = (600 – 200) + (50 – 60) + (3 – 8) = (600 – 200) + (40 – 60) + (13 – 8) = (600 – 200) + (40 – 60) + 5 = (500 – 200) + (140 – 60) + 5 = 300 + 80 + 5 = 385

Figure it OutFind the sums and differences:

(a) 310 + 3 4100(b) 9 510 7100 + 2 110 3100

(e) 8 6100 – 5 3100(f) 12 6100 2100 – 910 9100

3.4 Decimal Place Value

You may have noticed that whenever we need to measure something more accurately, we split a part into 10 (smaller) equal parts ― we split a unit into 10 one-tenths and then split each one-tenth into 10, one-hundredths and then we use these smaller parts to measure.

Can we not split a unit into 4 equal parts, 5 equal parts, 8 equal parts, or any other number of equal parts instead?

Yes, we can. The example below compares how the same length is represented when the unit is split into 10 equal parts and when the unit is split into 4 equal parts.If an even more precise measure is needed, each quarter can further be split into four equal parts. Each

part then measures 116 of a unit, i.e.,

16 such parts make 1 unit.

Then why split a unit into 10 parts every time?

The reason is the special role that 10 plays in the Indian place value system. For a whole number written in the Indian place value system — for example, 281 — the place value of 2 is hundreds (100), that of 8 is tens (10), and that of 1 is one (1). Each place value is 10 times bigger than the one immediately to its right. Equivalently, each place value is 10 times smaller than the one immediately to its left:10 ones make 1 ten,10 tens make 1 hundred,10 hundreds make 1 thousand, and so on.

× 10

10,0001000100101

÷ 10

In order to extend this system of writing numbers to quantities smaller than one, we divide one into 10 equal parts. What does this give? It gives one-tenth. Further dividing it into 10 parts gives one-hundredth, and so on.

× 10

10,0001000100101110

1100

÷ 10

Can we extend this further?

What will the fraction be when 1100 is split into 10 equal parts?

It will be 11000 , i.e., a thousand such parts make up a unit.

Just as when we extend to the left of 10,000, we get bigger place

values at each step, we can also extend to the right of 11000 , getting smaller place values at each step.

× 10× 10× 10× 10× 10× 10× 10× 10× 10

1,00,00010,0001000100101110

1100

11000 110,000

÷ 10÷ 10÷ 10÷ 10÷ 10÷ 10÷ 10÷ 10÷ 10

This way of writing numbers is called the “decimal system” since it is based on the number 10; “decem” means ten in Latin, which in turn is cognate to the Sanskrit daśha meaning 10, with similar words for 10 occurring across many Indian languages including Odia, Konkani, Marathi, Gujarati, Hindi, Kashmiri, Bodo, and Assamese. We shall learn about other ways of writing numbers in later grades.

How Big?

We already know that a hundred 10s make 1000, and a hundred 100s make 10000.

We can ask similar questions about fractional parts:

(a) How many thousandths make one unit?(b) How many thousandths make one tenth?(c) How many thousandths make one hundredth?(d) How many tenths make one ten?(e) How many hundredths make one ten?

Make a few more questions of this kind and answer them.

Notation, Writing and Reading of Numbers

We have been writing numbers in a particular way, say 456, instead of writing them as 4 × 100 (4 hundreds) + 5 × 10 (5 tens) + 6 × 1 (6 ones). Similarly, can we skip writing tenths and hundredths?

10) ?

Math

10 be written as 42 (skipping the 1

Can the quantity 4 2

10 in 2 × 1

Talk

If yes, how would we know if 42 means 4 tens and 2 units or it means 4 units and 2 tenths?

Similarly, 705 could mean:

(a) 7 hundreds, and 0 tens and 5 ones (700 + 0 + 5)

(b) 7 tens and 0 units and 5 tenths (70 + 0 + 5

10)

100)

10 + 5

Since these are different quantities, we need to have distinct ways of writing them.

To identify the place value where integers end and the fractional parts start, we use a point or period (‘.’) as a separator, called a decimal point.

The above quantities in decimal notation are then:

QuantityDecimal Notation

7 hundreds and 5 ones

(700 + 0 + 5)705

7 tens and 5 tenths (70 + 0 + 5

10)70.5

7 units and 5 hundredths

(

7 + 0 + 5

100)

7.05

These numbers, when shown through place value, are as follows:

Decimal

numberHundredsTensUnitsTenthsHundredths

7057 × 1000 × 105 × 1

70.57 × 100 × 15 × 1

7.057 × 10 × 1

105 × 1

× 10 × 1 × 1

10 × 1

number of

number of one-hundredths

tens

number of one-tenths

number of

units

Thus decimal notation is a natural extension of the Indian place

value system to numbers also having fractional parts. Just as 705 means

7 × 100 + 5 × 1, the number 70.5 means 7 × 10 + 5 × 110 , and 7.05 means

7 × 1 +5 × 1100.

We have seen how to write numbers using the decimal point (‘.’). But how do we read/say these numbers?

We know that 705 is read as seven hundred and five.70.5 is read as seventy point five, short for seventy and five-tenths.7.05 is read as seven point zero five, short for seven and five hundredths.0.274 is read as zero point two seven four. We don’t read it as zero point two hundred and seventy four as 0.274 means 2 one-tenths and 7 one-hundredths and 4 one-thousandths.

Make a place value table similar to the one above. Write each quantity in decimal form and in terms of place value, and read the number:

(a) 2 ones, 3 tenths and 5 hundredths

(b) 1 ten and 5 tenths

(d) 1 hundred, 1 one and 1 hundredth

(e) 8100 and 910

(f) 5100

(g) 110

(h) 2 1100 , 4 110 and 7 71000In the chapter on large numbers, we learned how to write 23 hundreds.23 hundreds = 23 × 100 = 2000 + 300 = 2300.

ThousandsHundredsTensUnits

2300

Similarly, 23 tens would be:23 tens = 23 × 10 = 200 + 30 = 230.

ThousandsHundredsTensUnits

230

How can we write 234 tenths in decimal form?

234 tenths = 23410

= 20010 + 3010 + 410

= 20 + 3 + 410= 23.4.

HundredsTensUnitsTenthsHundredths

234

Write these quantities in decimal form: (a) 234 hundredths, (b) 105 tenths.

3.5 Units of Measurement

Length Conversion

We have been using a scale to measure length for a few years. We already know that 1 cm = 10 mm (millimeters).

How many cm is 1 mm?

1 mm = 110 cm = 0.1 cm (i.e., one-tenth of a cm).

How many cm is (a) 5 mm? (b) 12 mm?

5 mm = 510 cm = 0.5 cm

12 mm = 10 mm + 2 mm

= 1 cm + 210 cm

= 1.2 cm.

How many mm is 5.6 cm? Since each cm has 10 mm, 5.6 cm (5 cm + 0.6 cm) is 56 mm.

Fill in the blanks below (mm <–> cm)

12 mm = 1.2 cm56 mm = 5.6 cm70 mm = _______

________ = 0.9 cm134 mm =__________________ = 203.6 cm

The illustration below shows how small some things are! Try taking an approximate measurement of each.

1.5 mm

1 mm

0.5 mm

• The three blue stripes represent the typical relative sizes of pen strokes: fine stroke, medium stroke, and bold stroke.

• A human hair is about 0.1 mm in thickness.

• The thickness of a newspaper can range from 0.05 to 0.08 mm.

• Mustard seeds have a thickness of 1 – 2 mm.

• The smallest ant species discovered so far, Carabera Bruni, has a total length of 0.8 – 1 mm. They are found in Sri Lanka and China.

• The smallest land snail species discovered so far, Acmella Nana, has a shell diameter of 0.7 mm. They are found in Malaysia.

We also know that 1 m = 100 cm. Based on this, we can say that

1 cm = 1100 m = 0.01 m.

How many m is (a) 10 cm? (b) 15 cm?

10 cm = 110 m = 0.1 m

Since each cm is one-hundredth of a meter, 15 cm can be written as

15 cm = 15100 m

= 10100 m + 5100 m

= 110 m + 5100 m

= 0.15 m.

Fill in the blanks below (cm <–> m):

36 cm = _______50 cm = ______________ = 0.89 m

4 cm = _______325 cm = _______________ = 2.07 m

How many mm does 1 meter have?

Can we write 1 mm = 11000 m?

Here, we have some more interesting facts about small things in nature!

1 cm

1 cm0

• The egg of a hummingbird typically is 1.3 cm long and 0.9 cm wide.

• The Philippine Goby is about 0.9 cm long. It can be found in the Philippines and other Southeast Asian countries.

• The smallest known jellyfish, Irukandji, has a belly size of 0.5 – 2.5 cm. Its tentacles can be as long as 1 m. They are found in Australia. Its venom can be fatal to humans.

• The Wolfi octopus, also known as the Star-sucker Pygmy Octopus, is the smallest known octopus in the world. Their typical size is around 1 – 2.5 cm and they weigh less than 1 gm. They are found in the Pacific Ocean.

Weight Conversion

Let us look at kilograms (kg). We know that 1 kg = 1000 gram (g). We can say that

1 g = 11000 kg = 0.001 kg.

How many kilograms is 5 g?

5 g = 51000 kg = 0.005 kg.

How many kilograms is 10 g?

10 g = 101000 kg = 1100 kg = 0.010 kg.

As each gram is one-thousandth of a kg, 254 g can be written as

254 g = 2541000 kg

2001000 + 501000 + 41000) kg

= (

210 + 5100 + 41000) kg

= (

= 0.254 kg.

Fill in the blanks below (g <–> kg)

465 g = _______68 g = _________1560 g = ________

704 g = _______________ = 0.56 kg_______ = 2.5 kg

Look at the picture below showing different quantities of rice. Starting from the 1g heap, subsequent heaps can be found that are 10 times heavier than the previous heap/packets. The combined weight of rice in this picture is 11.111 kg.

Also,

1 gram = 1000 milligrams (mg). So, 1 mg = 11000 g = 0.001 g.

Rupee ─ Paise conversion

You may have heard of ‘paisa’. 100 paise is equal to 1 rupee. As we have coins and notes for rupees, coins for paise were also used commonly until recently. There were coins for 1 paisa, 2 paise, 3 paise, 5 paise, 10 paise, 20 paise, 25 paise, and 50 paise. All denominations of 25 paise and less were removed from use in the year 2011. But we still see paise in bills, account statements, etc.

1 rupee = 100 paise

1 paisa = 1100 rupee = 0.01 rupee

As each paisa is one-hundredth of a rupee,

75 paise = 75100 rupee

= (

70100 + 5100) rupee

= (

710 + 5100) rupee

= 0.75 rupee.

Fill in the blanks below (rupee <–> paise)

10 p = __________________p = ₹ 0.05________p = ₹ 0.36

_________ = ₹ 0.5099 p = _________250 p = _________

During the 1970s, a masala dosa cost just 50 paise, one could buy a banana for 20-25 paise, a handful of peppermints were available for 2 paise or 3 paise, and a kg of rice cost ₹2.45.

Discuss with adults at home/school the prices of different products and services during their childhood. Try to find old coins and stamps.

TryThis

3.6 Locating and Comparing Decimals

Let us consider the decimal number 1.4. It is equal to 1 unit and 4 tenths. This means that the unit between 1 and 2 is divided into 10

equal parts, and 4 such parts are taken. Hence, 1.4 lies between 1 and 2. Draw the number line and divide the unit between 1 and 2 into 10 equal parts. Take the fourth part, and we have 1.4 on the number line.

11.11.21.31.41.51.61.71.81.92

Name all the divisions between 1 and 1.1 on the number line.

1.04

1.11.21.3

Identify and write the decimal numbers against the letters.

BCD

55.15.35.4

There is Zero Dilemma!

Sonu says that 0.2 can also be written as 0.20, 0.200; Zara thinks that putting zeros on the right side may alter the value of the decimal number. What do you think?

We can figure this out by looking at the quantities these numbers represent using place value.

Decimal numberUnitsTenthsHundredthsThousandths

0.202

0.20020

0.2000200

0.02002

0.0020002

We can see that 0.2, 0.20, and 0.200 are all equal as they represent the same quantity, i.e., 2 tenths. But 0.2, 0.02, and 0.002 are different.

Can you tell which of these is the smallest and which is the largest?

Which of these are the same: 4.5, 4.05, 0.405, 4.050, 4.50, 4.005, 04.50?

Observe the number lines in Figure (a) below. At each level, a particular segment of the number line is magnified to locate the number 4.185.

Identify the decimal number in the last number line in Figure (b) denoted by ‘?’.

010

4.2

4.1

4.184.1854.19

(a)(b)

Make such number lines for the decimal numbers: (a) 9.876 (b) 0.407.

In the number line shown below, what decimal numbers do the boxes labelled ‘a’, ‘b’, and ‘c’ denote?

bac

The box with ‘b’ corresponds to the decimal number 7.5; are you able to see how? There are 5 units between 5 and 10, divided into 10 equal parts. Hence, every 2 divisions make a unit, and so every division

is 12 unit. What numbers do ‘a’ and ‘c’ denote?

Using similar reasoning find out the decimal numbers in the boxes below.

Which is larger: 6.456 or 6.465?

8.1

To answer this, we can use the number line to locate both decimal numbers and show which is larger.This can also be done by comparing the corresponding digits at each place value, as we do with whole numbers. This comparison is visualised step by step below. Note that the visualisation below is not to scale.

4.3

4.8

fgh

Both numbers have 6 units.

6.456

6.465

Both numbers have 6 units and 4 tenths.

6.456

66.4

6.465

Both numbers have 6 units and 4 tenths, but the first number has only 5 hundredths, whereas the second number has 6 hundredths.

6.45

6.456

66.4

6.465

6.46

We start by comparing the most significant digits (digits with the highest place value) of the two numbers. If the digits are the same, we compare the next smaller place value. We keep going till we find a position where the digits are not equal. The number with the larger digit at this position is the greater of the two.

Why can we stop comparing at this point? Can we be sure that whatever digits are there after this will not affect our conclusion?

Math Talk

Which decimal number is greater?

(a) 1.23 or 1.32

(b) 3.81 or 13.800

Closest Decimals

Consider the decimal numbers 0.9, 1.1, 1.01, and 1.11. Identify the decimal number that is closest to 1.Let us compare the decimal numbers. Arranging these in ascending order, we get 0.9 < 1 < 1.01 < 1.1 < 1.11. Among the neighbours of 1, 1.01 is 1/100 away from 1 whereas 0.9 is 10/100 away from 1. Therefore, 1.01 is closest to 1.

Which of the above is closest to 1.09?

Which among these is closest to 4: 3.56, 3.65, 3.099?

Which among these is closest to 1: 0.8, 0.69, 1.08?

In each case below use the digits 4, 1, 8, 2, and 5 exactly once and try to make a decimal number as close as possible to 25.

Math Talk

3.7 Addition and Subtraction of Decimals

Priya requires 2.7 m of cloth for her skirt, and Shylaja requires 3.5m for her kurti. What is the total quantity of cloth needed?

We have to find the sum of 2.7m + 3.5m.

Earlier, we saw how to add 2 710 + 3 510 (also shown below). Can you

carry out the same addition using decimal notation? It is shown below.

Share your observations.The total quantity of cloth needed is 6.2 m.

2 710

2.7 + 3.5

+ 3 510

= 5 1210

= 6.2

= 6 210

How much longer is Shylaja’s cloth compared to Priya’s?

We have to find the difference of 3.5m – 2.7m. Again, observe how

the differences 3 510 – 2 710 and 3.5m – 2.7m are computed.

2 1

3.5

3 510

2 1510

– 2.7

– 2 710

= 0 810 = 0.8

As you can see, the standard procedure for adding and subtracting whole numbers can be used to add and subtract decimals.A detailed view of the underlying place value calculation is shown below for the sum 75.345 + 86.691. Its compact form is shown next to it.

1 × 1

1 × 11 × 101 × 100

7 × 10 5 × 1 3 × 1

10 4 × 1

100 5 × 11000

11 1 75.345 +86.691

8 × 10 6 × 1 6 × 1

10 9 × 1

100 1 × 11000

=162.036

16 × 10 12 × 1 10 × 1

10 13 × 1

100 6 × 11000

1 × 100

Write the detailed place value computation for 84.691 – 77.345, and its compact form.

TryThis

Figure it Out

1. Find the sums

(a) 5.3 + 2.6(b) 18 + 8.8

(e) 29.19 + 9.91(f) 0.934 + 0.6

(g) 0.75 + 0.03(h) 6.236 + 0.487

2. Find the differences

(a) 5.6 – 2.3(b) 18 – 8.8

(e) 17 – 0.05(f) 34.505 – 18.1

(g) 9.9 – 9.09(h) 6.236 – 0.487

Decimal Sequences

Observe this sequence of decimal numbers and identify the change after each term.4.4, 4.8. 5.2, 5.6, 6.0, …We can see that 0.4 is being added to a term to get the next term.

Continue this sequence and write the next 3 terms.

Similarly, identify the change and write the next 3 terms for each sequence given below. Try to do this computation mentally.

(a) 4.4, 4.45, 4.5, …(b) 25.75, 26.25, 26.75, …

(e) 8.5, 9.4, 10.3, …(f) 5, 4.95, 4.90, …

(g) 12.45, 11.95, 11.45, …(h) 36.5, 33, 29.5, …

Make your own sequences and challenge your classmates to extend the pattern.

Estimating Sums and Differences

Sonu has observed sums and differences of decimal numbers and says, “If we add two decimal numbers, then the sum will always be greater than the sum of their whole number parts. Also, the sum will always be less than 2 more than the sum of their whole number parts.” Let us use an example to understand what his claim means: If the two numbers to be added are 25.936 and 8.202, the claim is that their sum will be greater than 25 + 8 (whole number parts) and will be less than 25 + 1 + 8 + 1.

What do you think about this claim? Verify if this is true for these numbers. Will it work for any 2 decimal numbers?

Math Talk

What about for the sum of 25.93603259 and 8.202?

Similarly, come up with a way to narrow down the range of whole numbers within which the difference of two decimal numbers will lie.

TryThis

Note to the Teacher: Estimating the result before computing may help in identifying if a mistake happens with the calculation.

3.8 More on the Decimal System

Decimal and Measurement Disasters

Decimal point and unit conversion mistakes may seem minor sometimes but they can lead to serious problems. Here are some actual incidents in which such errors caused major issues.

• In 2013, the finance ofÏce of Amsterdam City Council (Netherlands) mistakenly sent out €188 million in housing benefits instead of the intended €1.8 million due to a programming error that processed payments in euro cents instead of euros. (1 euro-cent = 1/100 euro).

• In 1983, a decimal error nearly caused a disaster for an Air Canada Boeing 767. The ground staff miscalculated the fuel, loading 22,300 pounds instead of kilograms—about half of what was needed (1 pound ~ 0.453 kg). The plane ran out of fuel mid-air, forcing the pilots to make an emergency landing at an abandoned airfield. Fortunately, everyone survived.

Several incidents have occurred due to incorrect reading of decimal numbers while giving medication. For example, reading 0.05 mg as 0.5 mg can lead to using a medicine 10 times more than the prescribed quantity. It is therefore important to pay attention to units and the location of the decimal point.

Deceptive Decimal Notation

Sarayu gets a message: “The bus will reach the station 4.5 hours post noon.” When will the bus reach the station: 4:05 p.m., 4:50 p.m., 4:25 p.m.? None of these! Here, 0.5 hours means splitting an hour into 10 equal parts and taking 5 parts out of it. Each part will be 6 minutes (60 minutes/10) long. 5 such parts make 30 minutes. So, the bus will reach the station at 4:30.Here is a short-story of a decimal mishap: A girl measures the width of an opening as 2 ft 5 inches but conveys to the carpenter to make a door 2.5 ft wide. The carpenter makes a door of width 2 ft 6 inches (since 1 ft = 12 inches, 0.5 ft = 6 inches), and it wouldn’t close fully.

I said the door’s width should be 2.5 ft.

It is 2.5 ft! You can verify.

If you watch cricket, you might have noticed decimal-looking numbers like ‘Overs left: 5.5’. Does this mean 5 overs and 5 balls or 5

overs and 3 balls? Here, 5.5 overs means 5 5

6 overs (as 1 over = 6 balls), i.e., 5 overs and 5 balls.

Math

Where else can we see such ‘non-decimals’ with a decimal-like notation?

Talk

A Pinch of History – Decimal Notation Over Time

Decimal fractions (i.e., fractions with denominators like 1

10 , 1100 , 11000 , and so on) are used in the works of a number of ancient Indian

astronomers and mathematicians, including in the important 8th century works of Śhrīdharāchārya on arithmetic and algebra. Decimal notation, in essentially its modern form, was described in detail in Kitāb al-Fuṣūl fī al-Ḥisāb al Hindī (The Book of Chapters on Indian Arithmetic) by Abūl Ḥassan al-Uqlīdisī, an Arab mathematician, in around 950 CE. He represented the number 0.059375 as 0ˈ 059375.

In the 15th century, to separate whole numbers from fractional parts, a number of different notations were used:

• a vertical mark on the last digit of the whole number part (as

shown above),

• use of different colours and

• a numerical superscript giving the number of fractional decimal

places (0.36 would be written as 36

In the 16th century, John Napier, a Scottish mathematician, and Christopher Clavius, a German mathematician, used the point/period (‘.’) to separate the whole number and the fractional parts, while François Viète, a French mathematician, used the comma (‘,’) instead.

Currently, several countries use the comma to separate the integer part and the fractional part. In these countries, the number 1,000.5 is written as 1 000,5 (space as a thousand separator). But the decimal point has endured as the most popular notation for writing numbers having fractional parts in the Indian place value system.

Figure it Out

1. Convert the following fractions into decimals:

(a) 5100(b) 161000(c) 1210(d) 2541000

2. Convert the following decimals into a sum of tenths, hundredths

and thousandths:

(a) 0.34(b) 1.02(c) 0.8(d) 0.362

3. What decimal number does each letter represent in the number

line below?

6.66.56.4

bac

4. Arrange the following quantities in descending order:

(a) 11.01, 1.011, 1.101, 11.10, 1.01

(b) 2.567, 2.675, 2.768, 2.499, 2.698

(d) 33.13 m, 33.31 m, 33.133 m, 33.331 m, 33.313 m

5. Using the digits 1, 4, 0, 8, and 6 make:

(a) the decimal number closest to 30

(b) the smallest possible decimal number between 100 and 1000.

6. Will a decimal number with more digits be greater than a decimal

number with fewer digits?

7. Mahi purchases 0.25 kg of beans, 0.3 kg of carrots, 0.5 kg of potatoes,

0.2 kg of capsicums, and 0.05 kg of ginger. Calculate the total weight of the items she bought.

8. Pinto supplies 3.79 L, 4.2 L, and 4.25 L of milk to a milk dairy in

the first three days. In 6 days, he supplies 25 litres of milk. Find the total quantity of milk supplied to the dairy in the last three days.

9. Tinku weighed 35.75 kg in January and 34.50 kg in February. Has

he gained or lost weight? How much is the change?

10. Extend the pattern: 5.5, 6.4, 6.39, 7.29, 7.28, 6.18, 6.17, ____, _____

11. How many millimeters make 1 kilometer?

12. Indian Railways offers optional travel insurance for passengers

who book e-tickets. It costs 45 paise per passenger. If 1 lakh people opt for insurance in a day, what is the total insurance fee paid?

13. Which is greater?

(a) 10

1000 or 1

10 ?

(b) One-hundredth or 90 thousandths?

14. Write the decimal forms of the quantities mentioned (an example is given):

(a) 87 ones, 5 tenths and 60 hundredths = 88.10

(b) 12 tens and 12 tenths

(d) 25 tens, 25 ones, 25 tenths, and 25 hundredths

TryThis

15. Using each digit 0 – 9 not more than once, fill the boxes below so that the sum is closest to 10.5:

16. Write the following fractions in decimal form:

(a) 12(b) 32

(e) 15 (f) 45

SUMMARY

• We can split a unit into smaller parts to get more exact/accurate measurements. • We extended the Indian place value system and saw that

»1 unit = 10 one-tenths, »1 tenth = 10 one-hundredths, »1 hundredth = 10 one-thousandths, »10 one-hundredths = 1 tenth, »100 one-hundredths = 1 unit.• A decimal point (‘.’) is used in the Indian place value system to separate the whole number part of a number from its fractional part.• We also learnt how to compare decimal numbers, locate them on the number line, and perform addition and subtraction on them.

Chapter – 3

Page No. 47

Write the measurements of the objects shown in the picture.

Ans:

Eraser → 2 410 cm, Pencil → 4 510 cm → 4 12 cm, Chalk → 1 410 𝑐𝑚

Page No. 49

Arrange the lengths in increasing order:

(a) 𝟗

𝟏𝟎 (b) 1 𝟕

𝟏𝟎 (c) 𝟏𝟑𝟎

𝟏𝟎 (d) 13 𝟏

(e) 10 𝟓

𝟏𝟎 (f) 7 𝟔

𝟏𝟎 (g) 6 𝟕

𝟏𝟎 (h) 𝟒

Ans:

10, 9

10 , 17

10 , 67

10 , 76

10 , 10 5

10 , 130

10 , 13 1

Page No. 50

Arrange the following lengths in increasing order: 41

10 ,

10 ,41

10 , 411

Ans:

Increasing order: 4

10 , 41

10 = 4 1

10 , 411

Page No. 51

The lengths of the body parts of a honeybee are given. Find its total

length.

Head = 2 310 units

Thorax = 5 410 units

Abdomen = 7 510 units

Ans: 15 2

10 units.

Page No. 52

A Celestial Pearl Danio’s length is 𝟐𝟒

𝟏𝟎 cm, and the length of a

Philippine Goby is 𝟗

𝟏𝟎 cm. What is the difference in their lengths?

Ans: 1 5

10 cm

Observe the given sequences of numbers. Identify the change after each term and extend the pattern:

(a) 4, 4 𝟑

𝟏𝟎, 4 𝟔

𝟏𝟎, _______, _______, _______, _______

𝟏𝟎, 8 𝟕

𝟏𝟎, 9 𝟐

(b) 8 𝟐

𝟏𝟎, _______, _______, _______, _______

(c) 7 𝟔

𝟏𝟎, 8 𝟕

𝟏𝟎, _______, _______, _______, _______

(d) 5 𝟕

𝟏𝟎, 5 𝟑

𝟏𝟎, _______, _______, _______, _______

(e) 13 𝟓

𝟏𝟎, 13, 12 𝟓

𝟏𝟎, _______, _______, _______, _______

(f) 11 𝟓

𝟏𝟎, 10 𝟒

𝟏𝟎, 9 𝟑

𝟏𝟎, _______, _______, _______, _______

Ans:

(a) 4, 4 310 , 4 610 , 4 910 , 5 210 , 5 510 , 5 810 → increment of 310 each time.

(b) 8 210 , 8 710 , 9 210 , 9 710 , 10 210 , 10 710 , 11 210 →increment of 510 each time.

(d) 5 710 , 5 310 , 4 910 , 4 510 , 4 110 , 3 710 →subtracting: 410 each time

(e) 13 510 , 13 , 12 510 , 12, 11 510 , 11, 10 510 →decreasing by 510

(f) 11 510 , 10 410 , 9 310 , 8 210 , 7 110 , 6 → decreasing by 1 110

Page No. 53

How many one‑hundredths make one‑tenth? Can we also say that

the length is 4 units and 45 one‑hundredths?

Ans: 1

10 = 10

So, 10 one‑hundredths make one‑tenth.

Yes, we can write it 4 + 45

100 = 4 units and 45 one‑hundredths.

Page No. 54

Observe the figure below. Notice the markings and the

corresponding lengths written in the boxes when measured from 0.

Fill the lengths in the empty boxes.

Ans:

For the lengths shown below write the measurements and read out

the measures in words.

Ans:

Measurement −5 37100

In words: Five and thirtyseven-hundredths.

Measurement −15 3100

In words: Fifteen and three-hundredths.

Page No. 55

Measurement −7 52100

In words: Seven and fiftytwo-hundredths.

Measurement −9 80100

In words: Nine and eighty-hundredths.

In each group, identify the longest and the shortest lengths. Mark

each length on the scale.

Ans:

(a) Longest = 33

100 Shortest = 3

(b) Longest = 3 1

10 Shortest = 13

100 Shortest = 4

(d) Longest = 3 6

10 6

100 Shortest = 3 6100

Page No. 56

(e) Longest = 18

100 Shortest = 9

(f) Longest = 7 5

100 Shortest = 7 3

10 5

(g) Longest = 65

10 15

100 Shortest = 5 7

Page No. 58

Figure it Out

Find the sums and differences

(a) 3

10 + 3 4

Ans: 3 34

(b) 9 5

10 7

10 3

100 + 21

Ans: 117

10 4

10 6

100 + 143

Ans: 30

(d) 77

100 – 44

Ans: 33

(e) 86

100 – 53

Ans: 33

(f) 12 6

10 2

100 – 9

10 9

Ans: 11 63

Page No. 63

Make a place value table similar to the one above. Write each quantity in decimal form and in terms of place value, and read the number:

Ans:

Hundredths

Thousandth

Hundreds

Decimal

Read as

Tenths

Form

Ones

Tens

Question

– – 2 3 5 – 2 + 310 +5100

2 ones, 3

Two point

tenths and 5

three five (or

= 2.35

hundredths

two and

thirty-five

hundredths)

– 1 – 5 – – 10 + 510 = 10.5 Ten point

1 ten and 5

tenths

five (or ten

and five

tenths)

– – 4 – 6 – 4 + 0 + 6100

4 ones and

Four point

zero six (or

= 4.06

hundredths

four and six

hundredths)

1 – 1 – 1 – 100 + 0 + 1 + 0 + 1100

1 hundred,

One hundred

1 one and 1

one point

hundredth

zero one (or

=101.01

one hundred

one and one

hundredth)

8100 and 910 – – – 9 8 – 910 +8100

Zero point

nine eight

= 0.98

(or ninety-

eight

hundredths)

100 – – – 5 – 5100 = 0.05 Zero point

zero five (or

five

hundredths)

10 – – – 1 – – 110 = 0.1 Zero point

one (or one

tenth)

Two point

10, 41

– 2

2.01

10,

zero one

and 77

and

4 + 110 = 4.1

1000

Four point

7 + 010 + 0100 + 71000= 7.007

one

and

Seven point

zero zero

seven

13.117 Thirteen

point one

one seven.

Page No. 64

Write these quantities in decimal form:

(a) 234 hundredths

Ans: 2.34

(b) 105 tenths.

Ans: 10.5

Fill in the blanks below (mm < – > cm)

Ans:

12 mm= 1.2 cm 56 mm = 5.6 cm 70 mm = 7.0 cm

9 mm = 0.9 cm 134 mm = 13.4 cm 2036 mm = 203.6 cm

Page No. 66

Fill in blanks (cm <-> m):

36 cm = 0.36 m 50 cm = 0.5 m 89 cm = 0.89 m

4 cm = 0.04 m 325 cm = 3.25 m 207 cm = 2.07 m

How many mm does 1 meter have?

Ans: 1m = 1000 mm

Can we write 1 mm = 𝟏

𝟏𝟎𝟎𝟎 m?

Ans: Yes

Page No. 68

Fill in the blanks below (g < – > kg)

Ans:

465 g = 0.465 kg 68 g = 0.068 kg 1560 g = 1.56 kg

704 g = 0.704 kg 560 g = 0.56 kg 2500 g = 2.5 kg

Page No. 69

Fill in the blanks below (rupee < – > paise)

10 p = ₹0.10 5 p = ₹0.05 36 p = ₹0.36

50 p = ₹0.50 99 p = ₹0.99 250 p = ₹2.50

Page No. 70

Name all the divisions between 1 and 1.1 on the number line.

Ans: The small divisions between 1 and 1.1 on the number line (if the sub-

division is of 10 parts) are: 1.01, 1.02, 1.03, 1.04, 1.05, 1.06, 1.07, 1.08, 1.09

Identify and write the decimal numbers against the letters.

Ans:

Page No. 71

Can you tell which of these is the smallest and which is the largest?

Ans:

Decimal Number Units Tenths Hundredths Thousandths

0.2 0 2 0 0

0.20 0 2 0 0

0.200 0 2 0 0

0.02 0 0 2 0

0.002 0 0 0 2

0.2 = 0.20 = 0.200 is the largest.

0.002 is the smallest.

Which of these are the same: 4.5, 4.05, 0.405, 4.050, 4.50, 4.005, 04.50? Ans: 4.5, 4.50, and 04.50 are equal.

4.05 and 4.050 are equal.

Identify the decimal number in the last number line in Figure (b)

denoted by ‘?’

Ans:

Make such number lines for the decimal numbers: (a) 9.876 (b) 0.407.

Ans: (a) (b)

In the number line shown below, what decimal numbers do the boxes labelled ‘a’, ‘b’, and ‘c’ denote?

Ans:

Page No. 72

Using similar reasoning find out the decimal numbers in the boxes

below.

Ans:

Page No. 73

Which decimal number is greater? (a) 1.23 or 1.32 (b) 3.81 or 13.800 (c) 1.009 or 1.090

Ans:

(a) 1.32 is larger.

(b) 13.800 is larger.

Consider the decimal numbers 0.9, 1.1, 1.01 and 1.11

Which of the above is closest to 1.09? Ans: 1.1 is closest to 1.09.

Which among these is closest to 4: 3.56, 3.65, 3.099?

Ans: 3.65 is closest to 4.

Which among these is closest to 1: 0.8, 0.69, 1.08?

Ans: 1.08 is closest to 1.

In each case below use the digits 4, 1, 8, 2, and 5 exactly once and try

to make a decimal number as close as possible to 25.

Ans:

Page No. 75

Figure it Out

1. Find the sums

(a) 5.3 + 2.6 (b) 18 + 8.8 (c) 2.15 + 5.26 (d) 9.01 + 9.10 (e) 29.19 + 9.91 (f) 0.934 + 0.6 (g) 0.75 + 0.03 (h) 6.236 + 0.487

Ans:

(a) 7.9 (b) 26.8 (c) 7.41 (d) 18.11 (e) 39.10 (f) 1.534 (g) 0.78 (h) 6.723

2. Find the differences

(a) 5.6 – 2.3 (b) 18 – 8.8 (c) 10.4 – 4.5 (d) 17 – 16.198 (e) 17 – 0.05 (f) 34.505 – 18.1 (g) 9.9 – 9.09 (h) 6.236 – 0.487

Ans:

(a) 3.3 (b) 9.2

(e) 16.95 (f) 16.405 (g) 0.81 (h) 5.749

Continue this sequence 4.4, 4.8. 5.2, 5.6, 6.0, … and write the next 3 terms.

Ans: 4.4, 4.8. 5.2, 5.6, 6.0, 6.4, 6.8, 7.2

Page No. 76

Similarly, identify the change and write the next 3 terms for each

sequence given below.

Try to do this computation mentally.

(a) 4.4, 4.45, 4.5, … (b) 25.75, 26.25, 26.75, …

(c) 10.56, 10.67, 10.78, … (d) 13.5, 16, 18.5, …

(e) 8.5, 9.4, 10.3, … (f) 5, 4.95, 4.90, …

(g) 12.45, 11.95, 11.45, … (h) 36.5, 33, 29.5, …

Ans: (a) 4.4, 4.45, 4.5,4.55, 4.6, 4.65 → increment of + 0.05

(b) 25.75, 26.25, 26.75, 27.25, 27.75, 28.25 → add 0.5

(d) 13.5, 16, 18.5, 21.0, 23.5, 26.0 → Add 2.5

(e) 8.5, 9.4, 10.3, 11.2, 12.1, 13.0 → Add 0.9

(f) 5, 4.95, 4.90, 4.85, 4.80, 4.75 → Subtract 0.05

(g) 12.45, 11.95, 11.45, 10.95, 10.45, 9.95 → Subtract 0.50

(h) 36.5, 33, 29.5, 26.0, 22.5, 19.0 → Subtract 3.5

Page No. 78

Figure it out

1. Convert the following fractions into decimals:

(a) 𝟓

𝟏𝟎𝟎 (b) 𝟏𝟔

𝟏𝟎𝟎𝟎 (c) 𝟏𝟐

𝟏𝟎 (d) 𝟐𝟓𝟒

𝟏𝟎𝟎𝟎

Ans:

(a) 5

100 = 0.05 (b) 16

1000 = 0.016 (c) 12

10 = 1.2 (d) 254

1000 = 0.254

Page No. 79

2. Convert the following decimals into a sum of tenths, hundredths

and thousandths:

(a) 0.34 (b) 1.02 (c) 0.8 (d) 0.362

Ans:

(a) 3

10 + 4

100 (b) 100

100 + 2

100 = 1 + 2

10 (d) 3

10 + 6

100 + 2

1000

3. What decimal number does each letter represent in the number

line below?

Ans:

4. Arrange the following quantities in descending order:

(a) 11.01, 1.011, 1.101, 11.10, 1.01

→ Descending order: 11.10, 11.01, 1.101, 1.011, 1.01.

(b) 2.567, 2.675, 2.768, 2.499, 2.698

→ Descending order: 2.768, 2.698, 2.675, 2.567, 2.499.

(c) 4.678 g, 4.595 g, 4.600 g, 4.656 g, 4.666 g

→Descending order: 4.678, 4.666, 4.656, 4.600, 4.595.

(d) 33.13 m, 33.31 m, 33.133 m, 33.331 m, 33.313 m

→ Descending order: 33.331, 33.313, 33.31, 33.133, 33.13.

5. Using the digits 1, 4, 0, 8, and 6 make: (a) the decimal number

closest to 30 (b) the smallest possible decimal number between

100 and 1000.

Ans: Using the digits 1, 4, 0, 8, and 6, we can make:

(a) 40.168

(b) 104.68

6. Will a decimal number with more digits be greater than a

decimal number with fewer digits?

Ans: No. It is not necessary. For example, 2.05 (2 digits after decimal) vs 2.5 (1 digit after decimal). 2.5 >2.05

7. Mahi purchases 0.25 kg of beans, 0.3 kg of carrots, 0.5 kg of potatoes,

0.2 kg of capsicums, and 0.05 kg of ginger. Calculate the total

weight of the items she bought.

Ans: 1.3 kg

8. Pinto supplies 3.79 L, 4.2 L, and 4.25 L of milk to a milk dairy in the

first three days. In 6 days, he supplies 25 liters of milk. Find the

total quantity of milk supplied to the dairy in the last three days.

Ans: 12.76 L

9. Tinku weighed 35.75 kg in January and 34.50 kg in February. Has

he gained or lost weight? How much is the change?

Ans: He has lost weight. The change in the weight = 1.25 kg

10. Extend the pattern: 5.5, 6.4, 6.39, 7.29, 7.28, 8.18, 8.17, ____, _____

Ans: 9.07, 9.06

11. How many millimeters make 1 kilometer?

Ans: 1000000 mm

12. Indian Railways offers optional travel insurance for passengers

who book e-tickets. It costs 45 paise per passenger. If 1 lakh

people opt for insurance in a day, what is the total insurance fee

paid?

Ans: The total insurance fee paid for 1 lakh passengers is ₹45,000.

13. Which is greater?

(a) 𝟏𝟎

𝟏𝟎𝟎𝟎 or 𝟏

𝟏𝟎?

(b) One-hundredth or 90 thousandths?

(c) One-thousandth or 90 hundredths?

Ans:

(a) 1

10 > 1

(b) 90 thousandths > One-hundredth

Page No. 80

14. Write the decimal forms of the quantities mentioned (an

example is given):

(a) 87 ones, 5 tenths and 60 hundredths

(b) 12 tens and 12 tenths

(c) 10 tens, 10 ones, 10 tenths, and 10 hundredths

(d) 25 tens, 25 ones, 25 tenths, and 25 hundredths

Ans:

(a) 88.10 121.2

(b) 111.1 277.75

15. Using each digit 0 – 9 not more than once, fill the boxes below so

that the sum is closest to 10.5:

Ans:

16. Write the following fractions in decimal form:

(a) 1

2 (b) 3

2 (c) 1

(d) 3

4 (e) 1

5 (f) 4

Ans:

(a) 0.5 (b) 1.5 (c) 0.25

(d) 0.75 (e) 0.2 (f) 0.8