Multiplication of Decimal Numbers with a Power of 10

Introduction:

Have you ever thought about what exactly is power of 10 is? Here are some examples where we use the powers of 10.

1. In St. Paul School, there are 10,000 students. Mr. David informed John tomorrow there would be a fest. John went to the class and told this news secretly to 10 of his friends, his 10 friends told the same to their 10 friends and the chain continued.
2. Ellie performed in a singer’s contest and got her video. She sent the video to 10 of her friends and those 10 friends forwarded the video to 10 of their friends.
3. Rita used lens to enlarge the specimen 100 times.

In all the above examples, we are dealing with powers of 10.

Can you now give any other example of using the power of 10?

What is a product of decimal numbers with a power of 10?

Example:

Jason and Sarah each get 4 apples.

Now, if Jason and Sarah each get 2 apples.

Suppose we are going to half the number of apples to Jason and Sarah again.

What if, again, I want to give them half of the previous amount?

Product of Decimal Number and a Power of 10:

Cut the apple into 2 equal parts

Distribute 3 apples evenly equal to two persons:

Now, let’s distribute one and a half apples to Jason and Sarah.

Now distribute 1.5 apples to 10 people. How many apples do you need?

What if we have 100 kids, and we want to distribute 1.5 apples to each kid?

How many apples do you need for the same?

Distribute 1.5 apples to 1000 people.

Find the pattern in all the 3 cases.

When 1.5 apples were distributed to 1,000 people:

Pattern for the Power of 10:

1.5 x 1 = 1.5

1.5 x 10¹ =15

1.5 x 10² =150

1.5 x 10³ =1,500

In the pattern, we meant that

If 1.5 apple is distributed to 1 person, we need 1.5 apples.

If the same 1.5 apple is to be given to each person among 10¹ we will need 15 apples.

If 1.5 apple is to be given to each person among 10² we will need 150 apples.

If 1.5 apple is to be given to each person among 10³ we will need 1500 apples.

It is observed that:

A digit in one place is worth 10 times more when moved to the place on its left. Every time a number is multiplied by 10, the digits of the number shift to the left.

Let’s understand this better by solving some examples.

Example 1:

What pattern follows when we multiply any decimal number say 4.98 and a power of 10?  

Solution:

Hence,

Every time it seems as if the decimal is shifting towards the right.

Example 2:

Saniya went to the ice-cream parlor and found that each ice-cream costs 12.5 dollars. If she wants to give the same to 100 of her friends as a treat. How much money will she require?

Solution:

Cost of an ice-cream = 12.5 dollars

Cost of 100 ice-creams = 12.5 × 100 = 12.5 × 10²   = $1250

So, Saniya requires $1,250.

Example 3: Jose uses a microscope to observe specimens in science class. The microscope enlarges objects to 100 times their actual size. Find the size of each specimen as seen in the microscope.

Specimen	Actual length	Size seen in the microscope
A	0.008
B	0.011
C	0.0025
D	0.004

Solution:

As the microscope enlarges the image 100 times, i.e., 10² times. Hence, the decimal point will shift to its right 2 times. Therefore, the table will be as follows.

Specimen	Actual length	Size seen in the microscope (into 102 )
A	0.008	0.8
B	0.011	1.1
C	0.0025	0.25
D	0.004	0.4

Example 4: James wants to distribute dictionaries to 1000 of his students. If the cost of 1000 books is $14400, what is the price of each dictionary?

Solution: 

Let the cost of 1 dictionary be ‘d’.

Cost of 1000 (= dictionaries is = 14400 dollars

d x 10³ = 14400

Since the power of 10 is 3. To find the value of ‘d’ keep the decimal point before 3 digits of a number.

d = 14.4

So, the price of each dictionary is $14.4.

Example 5: During a fest, gifts were distributed to some people. If the cost of a single gift was $11.34 and Michel spent $113400. How many people were there?   

Solution:

Cost of single gift = $11.34

Let the total number of persons be ‘m’.

Amount spent = $113400

$11.34 $113400

The decimal point has moved to four digits right from the original place.

11.34 x 10⁴ = 113400

m= 10⁴ =10,000

So, there were 10,000 people in the fest who received gifts.

Exercise:

1. 1.275 × __ = 1275. Find the missing number.
2. Which of the following is correct?
   • 681 ×  = 6817
   • 42.3 × = 4230
   • 0.678 ×  = 6.78
   • 0.086 × = 0.00086
3. 26.014 × __= 26014. Find the missing number.
4. If 0.056 × = 56. Find .
5. Which one has the greatest value?
   • 0.1
   • 0.001
   • 0.01
   • 0.0001
6. Complete the chart.

	x 101	x 102	x 103
1.275
26.014
0.4

7. When multiplying by a power of 10, like 4.58 x 10³ how do you know you are moving the decimal in the correct direction?

(8-10) Find the missing numbers.

8. 0.629 x 10^? = 62.5

9. 10^? × 0.056 = 560

10. 94.6 x 10³ = ?

Concept Summary:

What We Have Learned:  

• Solve problems related to product of a decimal number and a power of 10.
• Perform multiplication on power of 10.  
• Know the pattern followed in multiplication for decimal numbers with power of 10.
• Use problems of product of decimals with power of 10 in day-to-day life.