Decimals
[Maths Operations with Decimals]
Decimals is a important topic in mathematics. decimals are not special numbers. They are part and parcel of everyday maths.
Lets understand what are decimals using an example:
Lets say, there is a person D purchased a book sold at 60 rupees 50 paise.
Now when D gave Rs.100 to book seller, then how much does the book seller needs to return post deducting the cost of book.
To represent the transaction and calculate the money that D gets back, decimals are used.
This blog includes
- How to represent a decimal number?
- How do you write numbers in decimal form?
- How to represent decimals on number line ?
- How to represent a fraction as a decimal value?
- How to represent fractions in decimal form where denominator is not a multiple of 10?
- How to convert Decimal to fraction in lowest terms?
- How to find the greater decimal?
- Addition and subtraction of decimals
- Multiplication and division of decimals
- Negative decimal numbers
- Word problems on decimals
============================
How do you write numbers in decimal form?
Ans:
Decimal number is represented with the symbol ( . ), it is called decimal point.
Example:
- 1 can be represented as 1.00
- 2 can be repsented as 2.00
- 25 can be represented as 25.00
Any value between 25 and 26 , should be written after putting a decimal point next to 25
How to represent decimals on number line ?
A Number line represents all numbers, positive, negative , zero, ration numbers, decimal numbers .
If the distance between 1 and 2 is divided into 10 equal parts...then each of these parts will be 0.1 distance apart.
Each of the part can be represented using decimal points as 1.1, 1.2 , 1.3, 1.4 , 1.5, 1.6 , 1.7, 1.8, 1.9 and 2
Similarly, the values between 1.1 and 1.2 can also represented as decimals.
1.11 ,1.12 , 1.13, 1.14, 1.15, 1.16, 1.17, 1.18, 1.19 and 1.2
Similarly
the values between 1.11 and 1.12 can also represented as decimals.
Suppose if the value is between the numbers 1 and 2, say, for example, 1.35
1.35 , can be represented as follows
1.35 is between 1 and 1.5 , depending the scale on number line, it needs to represented
How to represent a fraction as a decimal value?
Lets start with fraction values where the denominator is 10 and its multiples
1 / 10 can be written in decimal form as 0.1
7 / 10 can be written in decimal form as 0.7
It there is a fraction with one zero in the denominator, then the decimal point will be after one digit from left to right of the numerator.
Similarly
15 / 100 can be written in decimal form as 0.15
67 / 10 can be written in decimal form as 0.67
It there is a fraction with Two zeroes in the denominator, then the decimal point will be after Two digits from left to right of the numerator.
What if the numerator is a single digit number?
If numerator is single digit number then add a zero before the numerator value and put a decimal,
Example,
5 / 100 can be written in decimal form as 0.05
9 / 100 = 0.09
Note: The values 0.09 and just .09 are one and the same. They are different forms of representing a decimal value if there is no digit present left of decimal point.
Similarly
It there is a fraction with Three zeroes in the denominator, then the decimal point will be after Three digits from left to right of the numerator.
33 / 1000 = 0.033
168 / 1000 = 0.168
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How to represent fractions in decimal form where denominator is not a multiple of 10?
Suppose there is a fraction , 171 / 2 ,
There are two methods,
Method 1 :
To represent it in a decimal form , the denominator should be converted to multiple of 10 and decimal point needs to put appropriately.
171 / 2 =
Multiply both numerator and denominator with 5 so that the denominator value changes to multiple of 10
( 171 x 5 ) / (2 x 5 ) = 855 / 10 = 85.5
Method 2 :
Divide 171 by 2 by normal division method
2) 171 ( 85.5
16
11
10
1 0
1 0
0
===========================
How to convert Decimal to fraction in lowest terms?
To convert decimal to lowest fraction is easy and should be done as below
It follows method which is opposite to the method 1 seen above.
For example,
If there is a decimal value as 9. 55 .
To convert this into a fraction, we just have to divide the number with 100 as there are two digits after the decimal point.
9.55 = 955 / 100
Next the fraction obtained ( 955 / 100 ) can be simplified.
Going by the divisibility rules (link) , the numbers 955 and 100 is divisible by 5
955 / 100 = 191 / 20
191 / 20 is the simplified value of 955 / 100
So, 9.55 in fraction form = 191 / 20
This procedure is also called converting decimal as fraction in lowest terms or fraction in lowest form
==========================
Comparing decimals
Before understanding how to compare decimals, it is important to know following
0.4 = is same as .4 , as a zero to left of decimal point without any other digit does not change the value
also the form , 0.40 is same as 0.4 and .4 , as a zero at the end of the decimal number doesnot change the value of the number
So ,
0.4 = .4 = 0.40 = .40 ==All these four forms are equal to one another.
=======================
Decimal greater or less
How to find the greater decimal?
Suppose there are two decimal numbers., 96.36 and 95.98 , how to find largest decimal ?
Case1 : check for the part which comes before a decimal point - If one of the numbers have greater part then that decimal number is greater between the two , irrespective of the part after the decimal point.
In the above example, clearly 96 is greater than 95 , so ignore the part after the decimal point
Therefore 96.36 > 95.98
Case 2: Only when both numbers have same value for the part before decimal point, then check for the part after decimal point.
95.36 and 95.98 , Here parts before decimal point are equal for both the numbers, hence check for the part after decimal point.
0.98 > 0.36 , hence 95.98 > 95.36
Case 3: When the part after decimal point starts with zeros
95.36 and 95.098 -
As we know,
0.9 > 0.8 > 0.7 > 0.6 > 0.5 > 0.4 > 0.3 > 0.2 > 0.1 > 0.09 > 0.08 > 0.07 > 0.06 > 0.05 > 0.04 > 0.03 > 0.02 > 0.01 > 0.009 > 0.008 > .....and so on
The more the number of zeros immediately after decimal point, smaller the number.
So 0.36 > 0.098
Therefore, 95.36 > 95.098
===========================
Addition and subtraction of decimals
How to add decimal numbers ?
Add decimal numbers is simple and almost same as normal addition.
Example: Add 584.32 and 690.45 ?
584.32
+ 690.45
1274.77
But if one of the numbers has only 1 digit after decimal point, then zero should be added / assumed
at the second place.
Example: Add 54.36 and 60.4 ?
54.36
+ 60.40
114.76
how to subtract decimals
Example : Subtract 666.45 from 884.39 ?
884.39
- 666.45
217.94
Example : Subtract 3616.4594 from 6884.39 ?
As there are 4 digits after decimals in the first decimal number , the second number 6884.39 need to written as 6884.3900
6884.3900
- 3616.4594
3267.9306
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How to add decimal and fraction?
Consider adding, 2.506 with fraction 1 / 2
one cannot perform the addition if the numbers are in different forms. Hence we need either change decimal value to fraction or fraction into decimal , to make both numbers in same form.
Here it is difficult timing consuming to convert 2.506 into a fraction
But the fraction 1/ 2 can be converted into decimal value.
1/2 = 0.5 = 0.500
Now, Add both numbers
2.506
+ 0.500
3.106
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Multiplication and division of decimals
Multiplication of decimals:
1) Multiplication of decimals by 10, 100 and 1000
Multiplication of decimals by 10, 100 and 1000 is very easy.
If a decimal is multiplied by 10 then the decimal point moves 1 place to the right from the current position.
Example : 156.356 x 10 = 1563.56
If a decimal is multiplied by 100 then the decimal point moves 2 place to the right from the current position.
Example : 156.356 x 100 = 15635.6
If a decimal is multiplied by 1000 then the decimal point moves 3 place to the right from the current position.
Example : 156.356 x 1000 = 156356
2) Multiplication of decimals with whole numbers
Multiplication of decimals with whole numbers is like normal multiplication except that after performing multiplication a decimal point needs to be added appropriately ( we need to put a decimal point after same number of digits as in the decimal number given the question )
Example : 4.55 x 15 =
Multiply normally and put a decimal point after 2 digits from left end of the answer
4.55 x 15 = 68.25
Example :
2.55 x 150 =
Multiply normally and put a decimal point after 2 digits from left end of the answer
2.55 x 150 = 382.50
3)Multiplication of decimals with decimal numbers
While performing multiplication of decimals with decimals , we need to count the number of digits after which the decimal is placed in each number and put the decimal point in answer after counting the places as per the total obtained.
Example :
1) 10.26 x 12.36 = 126.8136
2) 0.56 x 0.09 = 0.0504
If there are not enough digits to put a decimal point, we need to add '0' s appropriately and place the decimal point
3) 0.50 x .001 = 0.00050
4) Multiplication of decimals with fractions
While performing multiplication of decimals with fractions is same as discussed above.
Example :
1) 45.25 x 2 / 5 = 90.50 / 5 = 18.10
OR
9.05
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Division of Decimals
1) Division of decimals with 10, 100 and 1000
Division of decimals by 10, 100 and 1000 is same as multiplication except that , in division the decimal point moves towards left, making the value of numerator smaller.
If a decimal is divided by 10 then the decimal point moves 1 place to the left from the current position.
Example : 96.756 / 10 = 9.6756
Divided by 10 means the value in the numerator is made into 10 equal parts, so numerator value decreases.
If a decimal is multiplied by 100 then the decimal point moves 2 place to the left from the current position.
Example : 96.756 / 100 = 0.96756
Divided by 100 means the value in the numerator is made into 100 equal parts, so numerator value decreases.
If a decimal is multiplied by 1000 then the decimal point moves 3 place to the left from the current position.
Example : 96.756 / 1000 = 0.096756
Divided by 1000 means the value in the numerator is made into 1000 equal parts, so numerator value decreases.
2) How to divide decimals by whole numbers ?
While performing division with whole numbers we need to perform normal division and put decimal point after counting the number of places after which decimal point is present in question and put decimal point in the answer after counting equal number of places.
Example:
1) 89.96 / 2 = 44.98
2) 89.96 / 20 = 4.498
3) Division of Decimals by decimals
While Dividing decimals by decimals ,
Step 1 : First divisor ( denominator) should be made as whole number by multiplying with a suitable number.
Step 2: Dividend ( numerator) also should be multiplied with same number.
This makes the division is easy.
Examples :
1) 543 / 0.3
To make 0.3 as whole number multiply numerator and denominator with 10
( 543 x 10 ) / ( 0.3 x 10 ) = 5430 / 3 = 1810
2) 0.543 / 0.3
= ( 0.543 x 10 ) / ( 0.3 x 10 )
= 5.43 / 3 = 1.81
3) 0.0891 / .011 =
To make 0.011 as whole number multiply numerator and denominator with 1000
(0.0891 x 1000) / (0.011 x 1000) = 89.1 / 11 = 8.1
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Negative decimal numbers
What are negative decimal numbers?
Can we have negative values in decimals?
The answer is Yes.
Decimals can also both postive or negative.
Adding negative decimals, subtracting negative decimals , multiplying negative decimals and dividing negative decimals are done in exactly same method as normal negative numbers ( not decimals).
1) Adding and subtracting negative decimals
Examples:
1) - 1.50 + (-0.65) = ?
- 1.50 - 0.65 = - 2.15
2) - 7.60 - (-9.65) = ?
- 7.60 + 9.65 = + 2.05
2) Multiplying and dividing negative decimal values
1) (-1.56 ) x ( -2.35) = + 3.666
2) (1.06 ) x ( -2.05) = - 2.173
3) - (1.21 ) / -1.1 = 1.1
4) - (9999.9 ) / 999.99 = - 10
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Problems on decimals
( including decimal word problems)
Question 1: D and H friends. D paid credit card bill of H amounting Rs. 1548.56 as D was short of funds. Next month H paid D credit card bill amounting to Rs.985.06. How much does H needs to pay to D to clear the help received by D.
Solution:
Given ,
D paid Rs. 1548.56 on behalf of H
H paid Rs. 985.06 on behalf of D
Amount D paid in excess is = 1548.56
- 985.06
563.50
So H needs to pay Rs. 563.50 to D to clear the help received from D
Question 2: solve the following addition 15.63 + 56.32 + 44.36 = ?
Solution: 15.63
+ 56.32
+ 44.36
116.31
Question 3:
Add 863.56 and 6947.23 and subtract 2315.35
Solution:
863.56
+ 6947.23
7810. 79
- 2315.35
5495.44
Question 4: Find the largest and smallest number among the following and find their difference
18.0612 , 18.09, 18.0053, 18.0072
Solution:
The part before the decimal point is equal for all the numbers so check the part after the decimal point.
Largest = 18.09
Smallest = 18.0053
Their difference in value is
18.0900
- 18.0053
0.0847
Question 5:
Solve (63.1 x 11.1) - ( 50 x 45.6)
Solution:
(63.1 x 11.1) - ( 5 x 45.6)
= 700.41 - 228
= 700.41 - 228.00
= 472.41