Decimals
A number is known as a decimal number that contains a decimal point.
1.01 is a decimal number.In decimals, we will learn about decimal fractions, decimal numbers, converting decimal into fraction, converting fraction into decimal, decimal places, like decimals, unlike decimals, converting unlike decimals to like decimals, decimal and fractional expansion, comparing decimals, representation of decimal numbers on the number line, operations on decimals (addition of decimals, subtraction of decimals, multiplication of decimals and division of decimals), terminating decimals, non-terminating decimals, converting pure recovering decimal into a vulgar fraction, converting mixed recurring decimal into vulgar fraction, rounding off decimals to the required decimal place, H.C.F and L.C.M of decimals and significant figures.
To learn more about Decimals …………
What a Decimal Number is ………
Decimal Numbers: How to read and write a decimal number in words?
Decimal Fractions: How to read and write decimal fractions?
Decimal Places: How to read in words the place values of a decimal number?
Decimal and Fractional Expansion: How to expand the decimal numbers? How to expand the decimal in fraction?
Like and Unlike Decimals: Identification of like decimals and unlike decimals
Conversion of Unlike Decimals to Like Decimals: How to convert the unlike decimals in like decimals?
Arranging Decimal Numbers in Order ...
Comparing Decimals: Ordering decimal numbers How to arrange decimals in ascending order? How to arrange decimals in descending order? Operation of decimal numbers……….
Adding Decimals: How to find the sum of the decimal numbers? How to solve addition word problems?
Subtracting Decimals: How to find the difference between the decimal numbers? How to solve subtraction word problems?
Simplify Decimals Involving Addition and Subtraction Decimals: How to simplify decimals including addition and subtraction?
Multiplying Decimal by a Whole Number: How to find the product of a decimal by a whole number? How to solve word problems using multiplication of a decimal by a whole number?
Multiplying Decimal by a Decimal Number: How to find the product of a decimal by a decimal number? How to solve word problems using multiplication of a decimal by a decimal number?
Dividing Decimal by a Whole Number: How to find the quotient of a decimal by a whole number? How to solve word problems using division of a decimal by a whole number?
Dividing Decimal by a Decimal Number: How to find the quotient of a decimal by a decimal number? How to solve word problems using division of a decimal by a decimal number?
Simplification of Decimal: How to simplify decimals using the operation of decimal number?
Conversion of decimals…..
Converting Decimals to Fractions: How to convert a decimal number to a fraction?
Converting Fractions to Decimals: How to convert a fraction to a decimal number?
Rounding off big and small decimal numbers……
Rounding Decimals: How to round decimals to the nearest place value?
Rounding Decimals to the Nearest Whole Number: How to round decimals to the nearest whole number?
Rounding Decimals to the Nearest Tenths: How to round decimals to the nearest tenths place value?
Rounding Decimals to the Nearest Hundredths: How to round decimals to the nearest hundredths place value?
Round a Decimal: How to solve decimal problems rounding to its nearest places?
H.C.F. and L.C.M. of Decimals: How to find the highest common factor and the lowest common multiple of two or more than two decimal numbers?
Terminating Decimal: What is terminating decimals? How to identify terminating decimals?
Non-Terminating Decimal: What is a non-terminating decimal? How to identify non-terminating decimals?
Repeating or Recurring Decimal: What is repeating decimal? How to solve and identify repeating decimal?
Pure Recurring Decimal: What is pure recurring decimal? How to solve and identify pure recurring decimal?
Mixed Recurring Decimal: What is mixed recurring decimal? How to solve and identify mixed recurring decimal?
Conversion of Pure Recurring Decimal into Vulgar Fraction: How to express or convert pure recurring decimals into vulgar fractions?
Conversion of Mixed Recurring Decimals into Vulgar Fractions: How to express or convert mixed recurring decimals into vulgar fractions?
Decimal Numbers
Definition of decimal numbers:
We have learnt that the decimals are an extension of our number system. We also know that decimals can be considered as fractions whose denominators are 10, 100, 1000, etc. The numbers expressed in the decimal form are called decimal numbers or decimals.
For example: 5.1, 4.09, 13.83, etc.
A decimal has two parts:
(a) Whole number part
(b) Decimal part
These parts are separated by a dot ( . ) called the decimal point.
• The digits lying to the left of the decimal point form the whole number part. The places begin with ones, then tens, then hundreds, then thousands and so on.
• The decimal point together with the digits lying on the right of decimal point form the decimal part. The places begin with tenths, then hundredths, then thousandths and so on………
For example:
(i) In the decimal number 211.35; the whole number part is 211 and the decimal part is .35
It can be arranged in the place-value chart as:
(ii) In the decimal number 57.031; the whole number part is 57 and the decimal part is.031
(iii) In the decimal number 197.73; the whole number part is 197 and the decimal part is.73
Decimal Fractions
Definition of decimal fractions:
The fractions whose denominator (bottom number) is 10 or higher powers of 10, i.e., 100, 1000, 10,000 etc., are called decimal fractions.
For example; 7/10, 7/100, 7/1000, etc, are all decimal fractions
9/10 is a decimal fraction and it can also be written as 0.9
23/100 is a decimal fraction and it can also be written as 0.23
31/1000 is a decimal fraction and it can also be written as 0.031
Note: We can also write decimal fractions with a decimal point (without a denominator), that makes easier to solve math calculations like addition and multiplication on fractions.
For example in details we can write decimal fraction;
23/100 is a decimal fraction and it can also be written as 0.23
31/1000 is a decimal fraction and it can also be written as 0.031
Like and Unlike Decimals
Concept of like and unlike decimals:
Decimals having the same number of decimal places are called like decimals i.e. decimals having the same number of digits on the right of the decimal point are known as like decimals. Otherwise, decimals not having the same number of digits on the right of the decimal point are unlike decimals.
Examples on like and unlike decimals:
5.45, 17.04, 272.89, etc. are like decimals as all these decimal numbers are written up to 2 places of decimal.
7.5, 23.16, 31.054, etc. are unlike decimals. As in 7.5 has one decimal place. 23.16 has two decimal places. 31.054 has three decimal places
Note:
Note:
If we put any number of annexing zeroes on the right side of the extreme right digit of the decimal part of a number does not alter the value of the number. So, unlike decimals can always be converted into like decimals by annexing required number of zeros on the right side of the extreme right digit in the decimal part.
For example;
9.3, 17.45, 38.105 are unlike decimals. These decimals can be re-written as 9.300, 17.450, 38.105 so now, these are like decimals.
Suppose 0. 1 = 0. 10 = 0. 100 etc, 0.5 = 0.50 = 0.500 etc, and so on. That is by annexing zeros on the right side of the extreme right digit of the decimal part of a number does not alter the value of the number.
Unlike decimals may be converted into like decimals by annexing the requisite number of zeros on the right side of the extreme right digit in the decimal part.
For example;
9.3, 17.45, 38.105 are unlike decimals. These decimals can be re-written as 9.300, 17.450, 38.105 so now, these are like decimals.
Suppose 0. 1 = 0. 10 = 0. 100 etc, 0.5 = 0.50 = 0.500 etc, and so on. That is by annexing zeros on the right side of the extreme right digit of the decimal part of a number does not alter the value of the number.
Unlike decimals may be converted into like decimals by annexing the requisite number of zeros on the right side of the extreme right digit in the decimal part.
Comparing Decimals
In comparing decimals, we will learn how to compare the two decimal numbers and also arranging the decimals in ascending or descending order.
One decimal number is either greater than or less than or equal to the other decimal number.
The following steps will help us to compare the decimal numbers:
Step I: Obtain the decimal numbers.
Step II: Compare the whole parts of the numbers. The whole number part with greater number will be greater. If the whole number parts are equal, then go to next step.
Step III: Compare the last left digits of the decimal parts of two numbers with the greater left digit at the last will be greater. If the left digits at the last of decimal parts are equal, then compare the next digits and so on.
Note: Now we will follow the steps and try to solve the various types of questions on comparing decimals accordingly. While comparing the two decimal numbers, convert each of the decimal numbers into like decimals and then solve.
Three different types of questions based on comparing decimals:
A. First compare the whole number part of the decimal number. Decimal with the greater whole number is greater.
1. Compare 23.14 and 8.67
Solution:
In 23.14 the whole number part is 23 and in 8.67 the whole number part is 8.
But 23 > 8
Therefore, 23.14 > 8.67
B. If the whole number part is the same, then compare the digit at the tenths place. The decimal with the greater tenths digit is greater.
2. Compare 53.47 and 53.81.
Solution:
In 53.47 and 53.81, the whole number part is the same, i.e., 53.
In 53.47, the decimal part is .47 and the digit in the tenths place is 4.
In 53.81, the decimal part is .81 and the digit in the tenths place is 8.
But 8 > 4
Therefore, 53.81 > 53.47
C. If the whole number part and the digit in the tenths place are the same, then compare the digit at the hundredths place and so on.
3. Compare 81.39 and 81.37.
Solution:
In 81.39 and 81.37, the whole number part is the same, i.e., 81.
In 81.39 and 81.37, the decimal part in the tenths place is the same, i.e., 3
In 81.39, the decimal part is .39 and the digit in the hundredths place is 9.
In 81.37, the decimal part is .37 and the digit in the hundredths place is 7.
But 9 > 7
Therefore, 81.39 > 81.37
Worked-out examples on comparing decimals and arranging decimals:
1. Which is greater of 58.23 and 49.35?
Solution:
The given decimals have distinct whole number parts, so we compare whole number parts only.
In 58.23, the whole number part is 58.
In 49.35, the whole number part is 49.
But 58 > 49
Therefore, 58.23 > 49.35
Solution:
The given decimals have distinct whole number parts, so we compare whole number parts only.
In 58.23, the whole number part is 58.
In 49.35, the whole number part is 49.
But 58 > 49
Therefore, 58.23 > 49.35
2. Write the following decimals in ascending order:
5.64, 2.54, 3.05, 0.259 and 8.32
Solution:
To convert the given decimal numbers into like decimals, we get
5.640, 2.540, 3.050, 0.259 and 8.320
Therefore, 0.259 < 2.540 < 3.050 < 5.640 < 8.320
Hence, the given decimals in ascending order are:
0.259, 2.54, 3.05, 5.64 and 8.32
Solution:
To convert the given decimal numbers into like decimals, we get
5.640, 2.540, 3.050, 0.259 and 8.320
Therefore, 0.259 < 2.540 < 3.050 < 5.640 < 8.320
Hence, the given decimals in ascending order are:
0.259, 2.54, 3.05, 5.64 and 8.32
3. Arrange the following decimals in descending order.
8.14, 5.96, 0.863, 6.4, 3.81 and 0.5
Solution:
By converting each of the decimal number to like decimals we get
8.140, 5.960, 0.863, 6.400, 3.810 and 0.500
Therefore, 8.140 > 6.400 > 5.960 > 3.810 > 0.863 > 0.500
Hence, the given decimals in descending order are:
8.14, 6.4, 5.96, 3.81, 0.863, 0.5
Decimal Places
The number of digits contained in the decimal part of a given decimal number gives the number of decimal places.
Examples:
1. The number 5.76 has 2 decimal places.
2. The number 0.315 has 3 decimal places.
3. The number 86.261 has 3 decimal places.
4. The number 912.67 has 2 decimal places.
To understand about the explanation of a decimal numbers we must first know about the Place Value.
For example; let us take a decimal number 42.37
• The number is read as forty two and thirty seven hundredths or forty two point three seven.
• In the decimal number the whole number part is 42 and the decimal part is 0.37.
• In the whole number part 42
The place of 2 is ones or unit and its place value is 2 × 1 = 2.
The place of 4 is tens and its place value is 4 × 10 = 40.
• In the decimal part 0.37
The place of 3 is tenths and its place value is 3 × 1/10 = 3/10 = 0.3.
The place of 7 is hundredths and its place value is 7 × 1/100 = 7/100 = 0.07.
We can also write the number in the expanded form as 42.37 = 40 + 2 + 0.3 + 0.07
= 4 10 + 2 1 + 3/10 + 7/100
Note:
The expansion form of a decimal number is shown in three ways; we can do it in either of the ways.
Conversion of
Unlike Decimals to Like Decimals
In conversion of unlike decimals to like decimals follow the steps of the method.Note:
If we put a number of zeros to the extreme right of decimal, the value of the decimal remains the same.
0.8 = 0.80 = 0.800
0.8 = 8/10 and 0.80 = 8
Thus to convert unlike decimals to like decimals, we follow the same method.
Examples on conversion of unlike decimals to like decimals:
1. Convert the decimal numbers 5.42, 11.6 and 212.075 into like decimals.
Solution:
We observe that in the given decimals 5.42, 11.6 and 212.075; the maximum number of decimal places is three.
The decimal 212.075 has the maximum number of decimal places, i.e., 3. So, we convert each of the other decimal numbers into the one having three places of decimal.
So, 5.42 is written as 5.420,
11.6 is written as 11.600
212.075 is already having three decimal places.
Therefore, 5.420, 11.600 and 212.075 are expressed as like decimals.
2. Covert the following unlike decimals 1.72, 26.361, 3.35 and 0.9 into like decimals.
Solution:
We observe that in the given decimals 1.72, 26.361, 3.35 and 0.9 the maximum number of decimal places is three.
The decimal 26.361 has the maximum number of decimal places, i.e., 3. So, we convert each of the other decimal numbers into the one having three places of decimal.
So, 1.72 is written as 1.720,
26.361 is already having three decimal places,
3.35 is written as 3.350,
0.9 is written as 0.900
Therefore, all the decimal numbers 1.720, 26.361, 3.350 and 0.900 are converted to like decimals.
3. (i) Are the following decimals 9.5, 18.235 and 20.0254 are like or unlike decimals.
(ii) If, the decimals are unlike then convert it into like decimals.
Solution:
(i) The following decimals 9.5, 18.235 and 20.0254 are unlike decimals.
(ii) We observe that in the given decimals 9.5, 18.235 and 20.0254; the maximum number of decimal places is four.
The decimal 20.0254 has the maximum number of decimal places, i.e., 4. So, we convert each of the other decimal numbers into the one having four places of decimal.
So, 9.5 is written as 9.5000,
18.235 is written as 18.2350
20.0254 is already having four decimal places.
Therefore, 9.5000, 18.2350 and 20.0254 are the conversion to like decimals.
Step I: Find the decimal number having the maximum number of decimal places, say (n).
Step II: Now, convert each of the decimal numbers to ‘n’ places of decimals.
Decimal and Fractional Expansion
Let us observe the decimal and fractional expansion in the place-value chart in case of decimal numbers is represented as follows:
To expand the given decimal number, arrange it in the place value chart and expand. The explanation will help us to understand both the decimal expansion and the fractional expansion.
1. Write the decimal and fractional expansion of 284.361.
Solution:
In decimal expansion:
2 × 100 + 8 × 10 + 4 × 1 + 3 × 1/10 + 6 × 1/100 + 1 × 1/1000
200 + 80 + 4 + 3/10 + 6/100 + 1/1000
200 + 80 + 4 + 0.3 + 0.06 + 0.001
In fractional expansion:
2 × 100 + 8 × 10 + 4 × 1 + 3 × 1/10 + 6 × 1/100 + 1 × 1/1000
200 + 80 + 4 + 3/10 + 6/100 + 1/1000
Repeating or Recurring Decimal
Definition of Recurring Decimal:
Recurring decimal is also known as repeating decimal. In a decimal, a digit or a sequence of digits in the decimal part keeps repeating itself infinitely. Such decimals are called non-terminating repeating decimals or recurring decimals.
More information about Repeating or Recurring Decimal:
● The repeating decimal can be expressed by putting a bar over the digit or on the repeated digits that repeat itself.
● Also, dot can be put over the first and the last digit of the repeated block.
A few worked-out problems are explained here with step-by-step explanation to understand how to calculate or find repeating or recurring decimal.
Examples of Recurring Decimal:
(a) 17/45
= 0.3777…….
For, 17/45 when 17 is divided by 45, the quotient is 0.3777.... and the digit 7 is repeating.
For instance, 0.377777777777......... can also be written as 0.37
Alternatively, we can write it by placing a dot above the repeating digit 7 in the quotient.
Therefore, 17/45 is a repeating decimal.
(b) 1/3
= 0.333.........
For, 1/3 when 1 is divided by 3, the quotient is 0.333..... and the digit 3 is repeating.
For instance, 0.33333333......... can also be written as 0.3
Alternatively, we can write it by placing a dot above the repeating digit 3 in the quotient.
Therefore, 1/3 is a repeating decimal.
(c) 2/11
= 0.1818
For, 2/11 when 2 is divided by 11, the quotient is 0.1818..... and the digits 18 are repeating.
For instance, 0.181818......... can also be written as 0.18
Alternatively, we can write it by placing a dot above the repeating digits 18 in the quotient.
Therefore, 2/11 is a repeating decimal.
Terminating Decimal
While expressing a fraction in the decimal form when we perform division we notice that the division is complete after a certain number of steps i.e., we get the remainder zero. The quotient obtained as decimal is called the terminating decimal. Such a decimal has a finite number of terms after the decimal point.
Worked-out examples on terminating decimal:
1. Express 17/8 in the decimal form.
Solution:
Since, the remainder is zero.
Therefore, 17/8 is terminating and 2.125 is a terminating decimal.
2. Express 1/4 in the decimal form.
Solution:
= 0.25
Since, the remainder is zero.
Therefore, 1/4 is terminating and 0.25 is a terminating decimal.
3. Express 3/5 in the decimal form.
Solution:
= 1.125
Since, the remainder is zero.
Therefore, 27/24 is terminating and 1.125 is a terminating decimal.
4. Express 3/5 in the decimal form.
Solution:
= 0.6
Since, the remainder is zero.
Therefore, 3/5 is terminating and 0.6 is a terminating decimal.
Non-Terminating Decimal
Definition of Non-terminating Decimal:
While expressing a fraction in the decimal form, when we perform division we get some remainder. If the division process does not end i.e. we do not get the remainder equal to zero; then such decimal is known as non-terminating decimal.
Note:
In some cases, a digit or a block of digits repeats itself in the decimal part. Such decimals are called non-terminating repeating decimals or pure recurring decimals. These decimal numbers are represented by putting a bar on the repeated part.
Example of Non-terminating Decimal:
(a) 2.666... is a non-terminating repeating decimal and can be expressed as 2.6.
(b) 0.141414 ... is a non-terminating repeating decimal and can be expressed as 0.14.
(b) 0.141414 ... is a non-terminating repeating decimal and can be expressed as 0.14.
Calculating Non Terminating Decimals:
Using long division method, we will observe the steps in calculating 5/3.
Therefore, 1.666... is a non-terminating repeating decimal and can be expressed as 1.6.
• In some cases at least one of the digits after the decimal point is not repeated and some digit/digits are repeated, such decimals are called mixed recurring decimals.
Examples of mixed recurring decimals are:
(a) 3.1444... = 3.14
(b) 8.12333... = 8.123
(c) 7.3656565... = 7.365
(b) 8.12333... = 8.123
(c) 7.3656565... = 7.365
Solved examples on non-terminating decimal:
Find the decimal representation of 16/45.
Solution:
Using long division method, we get
Therefore, 0.3555... = 0.35 and is a mixed recurring decimal.
Converting Decimals to Fractions
In converting decimals to fractions, we know that a decimal can always be converted into a fraction by using the following steps:
Step I: Obtain the decimal.
Step II: Remove the decimal points from the given decimal and take as numerator.
Step III: At the same time write in the denominator, as many zero or zeros to the right of 1(one) (For example 10, 100 or 1000 etc.) as there are number of digit or digits in the decimal part. And then simplify it.
The problem will help us to understand how to convert decimal into fraction.
In 0.7 we will change the decimal to fraction.
First we will write the decimal without the decimal point as the numerator.
Now in the denominator, write 1 followed by one zeros as there are 1 digit in the decimal part of the decimal number.
= 7/10
Therefore, we observe that 0.7 (decimal) is converted to 7/10 (fraction).
Worked-out examples on converting decimals to fractions:
1. Convert each of the following into fractions.
(i) 3.91
Solution:
3.91
Write the given decimal number without the decimal point as numerator.
In the denominator, write 1 followed by two zeros as there are 2 digits in the decimal part of the decimal number.
= 391/100
(ii) 2.017
Solution:
2.017
= 2.017/1
= 2.017 × 1000/1 × 1000 → In the denominator, write 1 followed by three zeros as there are 3 digits in the decimal part of the decimal number.
= 2017/1000
2. Convert 0.0035 into fraction in the simplest form.
Solution:
0.0035
Write the given decimal number without the decimal point as numerator.
In the denominator, write 1 followed by four zeros to the right of 1 (one) as there are 4 decimal places in the given decimal number.
Now we will reduce the fraction 35/10000 and obtained to its lowest term or the simplest form.
= 7/2000
3. Express the following decimals as fractions in lowest form:
(i) 0.05
Solution:
0.05
= 5/100 → Write the given decimal number without the decimal point as numerator.
In the denominator, write 1 followed by two zeros to the right of 1 (one) as there are 2 decimal places in the given decimal number.
= 5/100 ÷ 5/5 → Reduce the fraction obtained to its lowest term.
= 1/20
(ii) 3.75
Solution:
3.75
= 375/100 → Write the given decimal number without the decimal point as numerator.
In the denominator, write 1 followed by two zeros to the right of 1 (one) as there are 2 decimal places in the given decimal number.
= 375/100 ÷ 25/25 → Reduce the fraction obtained to its simplest form.
= 15/4
(iii) 0.004
Solution:
0.004
= 4/1000 → Write the given decimal number without the decimal point as numerator.
In the denominator, write 1 followed by three zeros to the right of 1 (one) as there are 3 decimal places in the given decimal number.
= 4/1000 ÷ 4/4 → Reduce the fraction obtained to its lowest term.
= 1/250
(iv) 5.066
Solution:
5.066
= 5066/1000 → Write the given decimal number without the decimal point as numerator.
In the denominator, write 1 followed by three zeros to the right of 1 (one) as there are 3 decimal places in the given decimal number.
= 5066/1000 ÷ 2/2 → Reduce the fraction obtained to its simplest form.
= 2533/500
Converting Fractions to Decimals
In converting fractions to decimals, we know that decimals are fractions with denominators 10, 100, 1000 etc. In order to convert other fractions into decimals, we follow the following steps:
Step I: Convert the fraction into an equivalent fraction with denominator 10 or 100 or 1000 if it is not so.
Step II: Take the given fraction’s numerator. Then mark the decimal point after one place or two places or three places from right towards left if the given fraction’s denominator is 10 or 100 or 1000 respectively.
Note that; insert zeroes at the left of the numerator if the numerator has fewer digits.
The problem will help us to understand how to convert fraction into decimal.
In 351/100 we will change the fraction to decimal.
First write the numerator and then divide the numerator by denominator and complete the division.
Put the decimal point such that the number of digits in the decimal part is the same as the number of zeros in the denominator.
Let us check the division of decimal by showing a complete step by step decimal divide.
We know that when the number obtained by dividing by the denominator is the decimal form of the fraction.
There can be two situations in converting fractions to decimals:
• When division stops after a certain number of steps as the remainder becomes zero.
• When division continues as there is a remainder after every step.
Here, we will discuss when the division is complete.
Explanation on the method using a step-by-step example:
• Divide the numerator by denominator and complete the division.
• If a non-zero remainder is left, then put the decimal point in the dividend and the quotient.
• Now, put zero to the right of dividend and to the right of remainder.
• Divide as in case of whole number by repeating the above process until the remainder becomes zero.
1. Convert 233/100 into decimal.
Solution:
2. Express each of the following as decimals.
(i) 15/2
Solution:
15/2
= (15 × 5)/(2 × 5)
= 75/10
= 7.5
(Making the denominator 10 or higher power of 10)
(ii) 19/25
Solution:
19/25
= (19 × 4)/(25 × 4)
= 76/100
= 0.76
(iii) 7/50
Solution:
7/50 = (7 × 2)/(50 × 2) = 14/100 = 0.14
Note:
Conversion of fractions into decimals when denominator cannot be converted to 10 or higher power of 10 will be done in division of decimals.
Examples on conversion of fractions into decimal numbers:
Express the following fractions as decimals:
1. 3/10
Solution:
Using the above method, we have
3/10
= 0.3
2. 1479/1000
Solution:
1479/1000
= 1.479
3. 71/2
Solution:
71/2
= 7 + 1/2
= 7 + (5 × 1)/(5 × 2)
= 7 + 5/10
= 7 + 0.5
=7.5
4. 91/4
Solution:
91/4
= 9 + 1/4
= 9 + (25 × 1)/(25 × 4)
= 9 + 25/100
= 9 + 0.25
= 9.25
5. 121/8
Solution:
121/8
= 12 + 1/8
= 12 + (125 × 1)/(125 × 8)
= 12 + 125/1000
= 12 + 0.125
= 12.125
Pure Recurring Decimal
Definition of Pure recurring decimal:
A decimal in which all the digits in the decimal part are repeated is called a pure recurring decimal.
A few solved problems are explained step-by-step with detailed explanation.
Worked-out example of pure recurring decimal:
(a) 5/3
= 1.666........
For, 5/3 when 5 is divided by 3, the quotient is 1.666.... and the digit 6 is repeating.
For instance, 1.666666 ......... can also be written as 1.6
Alternatively, we can write it by placing a dot above the repeating digit 6 in the quotient.
Therefore, 5/3 is a pure recurring decimal.
(b) 1/37
= 0.027027........
For, 1/37 when 1 is divided by 37, the quotient is 0.027027.... and the digits 027 are repeating.
For instance, 0.027027........ can also be written as 0.027
Alternatively, we can write it by placing a dot above the repeating digits 027 in the quotient.
Therefore, 1/37 is a pure recurring decimal.
(c) 9/37
= 0.243243......
For, 9/37 when 9 is divided by 37, the quotient is 0.243243....and the digits 243 are repeating.
For instance, 0.243243....... can also be written as 0.243
Alternatively, we can write it by placing a dot above the repeating digits 243 in the quotient.
Therefore, 9/37 is a pure recurring decimal.
Mixed Recurring Decimal
Definition of Mixed recurring decimal:
A decimal in which at least one of the digits after the decimal point is non-repeated and some digits are repeated is called a mixed recurring decimal.
A few solved problems are explained step-by-step with detailed explanation.
Worked-out example on mixed recurring decimal:
(a) 5/18
= 0.2777….
For, 5/18 when 5 is divided by 18, the quotient is 0.2777... and the digit 7 is repeating.
For instance, 0.277777777777......... can also be written as 0.27
Alternatively, we can write it by placing a dot above the repeating digit 7 in the quotient.
Therefore, 5/18 is a mixed recurring decimal.
(b) 5/6
= 0.833……..
For, 5/6 when 5 is divided by 6, the quotient is 0.8333... and the digit 3 is repeating.
For instance, 0.833......... can also be written as 0.83
Alternatively, we can write it by placing a dot above the repeating digit 3 in the quotient.
Therefore, 5/6 is a mixed recurring decimal.
(c) 111/900
= 0.12333........
For, 111/900 when 111 is divided by 900, the quotient is 0.1233... and the digit 3 is repeating.
For instance, 0.1233......... can also be written as 0.123
Alternatively, we can write it by placing a dot above the repeating digit 3 in the quotient.
Therefore, 111/900 is a mixed recurring decimal.
