(i) Decimal multiplication through models (ii) Multiplication of Decimal Numbers by 10, 100 and 1000

**Operations on Decimal Numbers**

Already we are familiar with decimal
numbers. We know how to represent a decimal number as a fraction and the place
values of digits. Now, let us learn the operations on decimal numbers.

__Multiplication of Decimal Numbers__

Mathan wants to buy a shirt material
which costs ₹ 75.50 per metre. He needs 1.5 metre to stitch a shirt. How much does
he have to pay? Here we need to multiply 75.50 and 1.5. In real life, there are
many situations where we need to multiply decimal numbers.

** **

__(i) Decimal multiplication
through models__

Let us try to understand decimal multiplication
using grid model.

Let us find 0.1 × 0.1.

0.1= 1/10 . Therefore, 0.1× 0.1= 1/10×
1/10

That is, 1/10 ^{th} of 1/10 .

Shade horizontally 1/10 by blue colour (Fig. 1.7).
Shade vertically 1/10 by green colour (Fig. 1.8).

Then 1/10 ^{th} of 1/10 is the common portion,
that is 1/100 ^{th} .

Therefore, 1/10 × 1/10 = 1/100 = 0.01

Hence, 0.1 × 0.1 = 0.01.

__Example 1.16 __

Find 0.3 × 0.4

**Solution**

First shade 4 rows of the grid in blue
colour to represent 0.4. Shade 3 columns of the grid in green colour to represent
0.3 of 0.4. Now 12 squares represents the common portion. This represents 12 hundredth
or 0.12. Hence 0.3 × 0.4 = 0.12.

**Note**

The number of decimal digits
in 0.12 is two. So, we can conclude that the number of decimal digits in the product
of two decimal numbers is equal to the sum of decimal digits that are multiplied.

__Area model__

We have already learnt about the area
model in the addition and subtraction of decimal numbers. In the same way we are
going to multiply the decimal numbers. Now we shall see an example.

__Example 1.17__

** **Multiply 3.2 × 2.1

**Solution**

Let us try to represent the product of
decimal numbers (3.2 × 2.1) as the area of a rectangle. Let us consider a rectangular
portion as shown in Fig. 1.9.

The rectangular portion is split into
3 wholes and 2 tenth along it’s length (*Fig. 1.10*).

Since, 3 wholes and 2 tenth is multiplied
with 2.1, we split the same area into 2 wholes and 1 tenth along its breadth (Fig.
1.11).

Here, each row contains 3 wholes and
2 tenth. Each column contains 2 wholes and 1 tenth. The entire area model represents
6 wholes, 7 tenth and 2 hundredth.

Therfore, 3.2 × 2.1 = 6.72

**Think**

How are the products 2.1
× 3.2 and 21 × 32 alike? How are they different.

**Solution: **

2.1 × 3.2 = 6.72 and 21 × 32 = 672.

In both the cases the digits ambers are the same. But the place
value differs.

**Try these**

**(1) Shade the grid to multiply
0.3 × 0.6.**

**Solution:**

3 rows of Yellow represent 0.3, 6 columns of Red colour
represent 0.6 Double shaded 18 squares of orange colour represent.

0.30 × 0.6 = 0.18

**(2) Use the area model
to multiply 1.2 × 2.5**

**Solution:**

Here each row contains 1 whole and 2 tenths. Each column
contains 2 wholes and 5 tenths. The entire area model represents 2 wholes 9
tenths and 10 hundredths ( = 1 tenths). So 1.2 × 2.5 = 3.

__Example 1.18 __

Multiply the following

(i) 2.3 × 1.4

(ii) 5.6 × 3.2

**Solution**

(i) 2.3 × 1.4

First, let us multiply 23 × 14

23×14=322

Now 2.3 × 1.4 = 3.22

(ii) 5.6 × 3.2

First, let us multiply 56 × 32

56 × 32 =1792

Now 5.6 × 3.2 = 17.92

__Example 1.19__

Latha purchased a churidhar
material of 3.75 *m*** **at the rate of ₹ 62.50 per** **metre. Find the amount to
be paid.

**Solution**

Cost of churidhar
material
= ₹ 62.50 per metre

Length of churidhar material = 3.75 *m*

Amount to be paid = 3.75 × 62.50

= ₹ 234.3750

= ₹ 234.38 (rounded to two decimals).

__Example 1.20 __

The length and breadth of
a rectangle** **is 23.5 *cm* and 1.5
*cm* respectively. Find the area of the rectangle.

**Solution**

Area of a rectangle = *l* × b *sq. units*

Here, *l* =23.5 *cm*, *b*=1.5 *cm*

Area of the rectangle = 23.5 × 1.5

= 35.25 *sq.cm*.

** **

__(ii) Multiplication of
Decimal Numbers by 10, 100 and 1000__

We have studied about conversion of decimals
into fractions in the first term. Consider 45.6 and 4.56.

Expressing these decimal numbers into
fractions, we get

45.6 = 40 +
5 +
6/10 =
45 +
6/10 =
456/10

Now, 4.56 =
4 +
5/10 +
6/100=
456/100

Comparing the two fractions, we see that
if there is one digit after the decimal point, then the denominator is 10 and if
there are two digits after the decimal point, then the denominator is 100.

Let us see what happens if a decimal
number is multiplied by 10, 100 and 1000.

Observe the following table and complete
it.

**Try these**

Can you observe any pattern in the above
table? There is a pattern in the shift of decimal points of the products in the
table. In 2.35 × 10 = 23.5 the digits are the same, that is 2, 3, 5. Observe 2.35
and 23.5. To which side has the decimal point been shifted, right or left? The decimal
point is shifted to the right by one place. Also note that 10 is one zero followed
by 1.

In 2.35 × 100 = 235.0, observe 2.35 and
235. To which side and by how many digits has the decimal point been shifted? The
decimal point has been shifted to the right by two places. Note that 100 is two
zeros followed by 1.

Similarly in 2.35 × 1000 = 2350.0, we
see that the decimal point has been shifted to the right by three places by adding
one more digit 0 to the number 235. Note that 100 is three zeros followed by 1.

So we conclude that when a decimal number
is multiplied by 10, 100 or 1000, the digits in the product are same as in the decimal
number but the decimal point in the product is shifted to the right by as many places
as there are zeros followed by 1.

Based on these observations we can now
say,

0.02 × 10 = 0.2; 0.02 × 100 = 2 and 0.02
× 1000 = 20.

Can you find the following?

2.76×10=?

2.76 × 100 = ?

2.76 × 1000 = ?

**Try these **

**Find **

(i) 9.13×10 **= 91.3**

(ii) 9.13×100 **= 913**

(iii) 9.13×1000 **= 9130**

__Example 1.21__

** **Find the value of the following

(i) 3.26×10

(ii) 3.26×100

(iii) 3.26×1000

(iv) 7.01×10

(v) 7.01×100

(vi) 7.01×1000

**Solution**

(i) 3.26 × 10 = 32.6

(ii) 3.26 × 100 = 326.0

(iii) 3.26 × 1000 = 3260.0

(iv) 7.01 × 10 = 70.1

(v) 7.01 × 100 = 701.0

(vi) 7.01 × 1000 = 7010.0

__Example 1.22 __

A concessional entrance ticket
for students to visit a zoo is ₹ 12.50. How much has to be paid for 20 tickets?

**Solution**

Cost of one ticket = ₹ 12.50

Amount to be paid for 20 students = 12.50
× 20 = ₹ 250.00

We have already discussed about the multiplication
of decimal numbers by 10, 100 and 1000. In the same way we can find patterns for
multiplying decimal numbers by 0.1, 0.01 and 0.001. Observe the following.

From the above multiplication we can
conclude that, when multiplying by

• 0.1, the decimal point moves one place
left.

• 0.01, the decimal point moves two
places left.

• 0.001, the decimal point moves three
places left.

Zeros may be needed after the decimal point as they are needed before the decimal point of numbers when multiplied 0.1, 0.01 and 0.0001.

**Try this**

**Complete the following table :**

Tags : Number System | Term 3 Chapter 1 | 7th Maths , 7th Maths : Term 3 Unit 1 : Number System

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7th Maths : Term 3 Unit 1 : Number System : Multiplication of Decimal Numbers | Number System | Term 3 Chapter 1 | 7th Maths

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