Multiplication of Decimal Numbers

Reshma purchased 1.5 kg vegetable at the rate of 8.50 rupees per kg. How much money should she pay? Certainly it would be Rs.(8.50 × 1.50). Both 8.5 and 1.5 are decimal numbers.

So, we have come across a situation where we need to know how to multiply two decimals. Let us now learn the multiplication of two decimal numbers.

First we find 0.1 × 0.1.

Now, 0.1 = 110. So, 0.1 × 0.1 = 110 × 110 = 1 × 110 ×10 = 1100 = .

Let us see it's pictorial representation {.reveal(when="blank-0")} Let us see it's pictorial representation

To do this lets take another grid divided into 10 equal parts horizontally(shown above). Now move the second grid over the first grid to see how each 110th part of the first grid is divided into 10 equal parts or 110 of 110 or 110×110.

Instructions: Overlap the two grids while maintaining the given alignment.

How many small squares do you find in the above overlapped figure?

There are 100 small squares. The coloured square is one part out of 10 of the 110th part. That is, it represents 110 × 110 or 0.1 × 0.1.

How can we represent the square which has both the red and green colour?

The red and green coloured square represents one out of 100 or 0.01. Hence, 0.1 × 0.1 = .

Let us now find 0.2 × 0.3.

We have, 0.2 × 0.3 = 210 × 310

As we did for 110 × 110 , let us divide the square into 10 equal parts and take three parts out of it, to get 3/105/10.

Again divide each of these three equal parts into 10 equal parts and take two from each. We get 2/103/15 × 3/106/8.

Instructions: Overlap the two grids while maintaining the given alignment.

The intersected squares represent 210 × 310 or 0.2 × 0.3.

Since there are 6 intersected squares out of 100, so they also reprsent 0.06. Thus, 0.2 × 0.3 = .

Observe that 2 × 3 = 6 and the number of digits to the right of the decimal point in 0.06, is 2 (= 1 + 1).

Lets check whether this applies to 0.1 × 0.1 also. We have: 0.1 × 0.1 =

1 × 1 = 1 and the number of digits to the right of the decimal point in 0.01 is 2 (= 1 + 1).

Lets find 0.2 × 0.4 by applying these observations.

While finding 0.1 × 0.1 and 0.2 × 0.3, you might have noticed that first we multiplied them as whole numbers ignoring the decimal point. In 0.1 × 0.1, we found 0.1 × 0.1 or 1 × 1.

Similarly in 0.2 × 0.3 we found 0.2 × 0.3 or 22.510 × 3456.

Then, we count the number of digits starting from the rightmost digit and moved towards left. We then put the decimal point there. The number of digits to be counted is obtained by adding the number of digits to the right of the decimal point in the decimal numbers that are being multiplied.

Let us now find 1.2 × 2.5.

Multiply and .

We get 300. Both, in the individual decimal numbers 1.2 and 2.5, there is 123 digit(s) to the right of the decimal point.

So, count 1 + 1 = 2 digits from the rightmost digit (i.e., ) in 300 and move towards left.

We get 3.00 or .

While multiplying 2.5 and 1.25, you will first multiply 25 and 125.

For placing the decimal in the product obtained, you will count 1 + 2 = digits starting from the rightmost digit.

Thus, 2.5 × 1.25 =

The side of an equilateral triangle is 3.5 cm. Find its perimeter.

Instruction

The length of a rectangle is 7.1 cm and its breadth is 2.5 cm. What is the area of the rectangle?

Instruction

Let us see if we can find a pattern of multiplying numbers by 10 or 100 or 1000. Have a look at the table given below and fill in the blanks:

Observe the shift of the decimal point of the products in the table. Here, the numbers are multiplied by 10,100 and 1000.

In 1.76 × 10 = , the digits are same i.e., 1, 7 and 6. Do you observe this in other products also? Observe 1.76 and 17.6. To which side has the decimal point shifted, right or left?

The decimal point has shifted to the rightleft by place.

Note that 10 has one zero over 1 while 100 has two zeros over 1.

Answer the below questions

Instruction

Division of Decimal Numbers

Savita was preparing a design to decorate her classroom. She needed a few coloured strips of paper of length 1.9 cm each. She had a strip of coloured paper of length 9.5 cm. How many pieces of the required length will she get out of this strip? She thought it would be 9.51.9 cm. Is she correct? YesNo

Both 9.5 and 1.9 are numbers.

So we need to know the division of decimal numbers too!

Division by 10, 100 and 1000

Let us see if we can find a pattern for dividing numbers by 10, 100 or 1000. This may help us in dividing numbers by 10, 100 or 1000 in a shorter way.

In 31.5 and 3.15, the digits are same i.e., 3, 1, and 5 but the decimal point has shifted in the quotient. To which side? leftright and by how many digits?

The decimal point has shifted to the left by one place. Note that 10 has one zero over .

While dividing a number by 10, 100 or 1000, the digits of the number and the quotient are samedifferent but the decimal point in the quotient shifts to the left by as many places as there are zeros over 1.

Find:

Instruction

(i) Let us find 6.42. (Remember we also write it as 6.4 ÷ 2)

Instruction

Continue

(ii) Let us find 12.96 ÷ 4

Instruction

Continue

Note that here and in the next section, we have considered only those divisions in which, ignoring the decimal, the number would be completely divisible by another number to give remainder zero. Like in 19.5 ÷ 5, the number 195 when divided by 5, leaves a remainder of 012.

However, there are situations in which the number may not be completely divisible by another number, i.e., we may not get remainder zero. For example, 195 ÷ 7. We deal with such situations in later classes.

Continue

Instruction

Find the average of 4.2, 3.8 and 7.6

Solution:

The average of 4.2, 3.8 and 7.6 = 4.2+3.8+7.63 =

(i)Let us find 25.50.5

Instruction

Continue

Instruction

What do you observe? For 25.50.5, we find that there is one digit to the right of the decimal in 0.5. This could be converted to whole number by dividing by 10. Accordingly 25.5 was also converted to a fraction by dividing by 10.

Or, we say the decimal point was shifted by one place to the right in 0.5 to make it 555.

So, there was a shift of one decimal point to the right in 25.5 also to make it 255655.

Each side of a regular polygon is 2.5 cm in length. The perimeter of the polygon is 12.5cm. How many sides does the polygon have?

Instruction

A car covers a distance of 89.1 km in 2.2 hours. What is the average distance covered by it in 1 hour?

Instruction