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Class IX - maths: Numbers System
One Word Answer Questions:
Q) What are rational numbers?
A)  

Q) Define BODMAS rule?
A)  

Q) What are Real Numbers?
A)  

Q) Define Non-terminating Decimal?
A)  

Q) What is Non-terminating non-recurring?
A)  

Q) Every point on a number line represents?
A)  

Q) Every point on a number line represents?
A)  

Q) If a/b and c/d are any two rational numbers such that ____ then (a/b ×c/d) is also a rational number.
A)  

Q) To round a decimal to the nearest whole number analyse the digit at the _____ decimal place.
A)  

Short Answer Questions:
Q) Find six Rational Numbers between 3 and 4?
A)

Q) Determine whether the following rational numbers are integers or not? (i) 3/5 (ii) 8/4 (iiI)-14/2 (iv)48/-8 (v)500/-10.
A)

Q) Covert the following unlike decimals 1.72, 26.361, 3.35 and 0.9 into like decimals?
A)

Q) Simplify using BODMAS/PEMDAS rule? (a) 8 - 4.2% 6 + 0.3×0.4 (b) 7.6 - [3 + 0.5 of (3.1 - 2.3×1.02)].
A)

Q) Find the quotient of Divide 96.075 by 5.3?
A)

Q) What is the formula for the area of rectangle?
A)

Q) if n is a natural number, then √n is?
A)

Q) The numbers of the form a/b, or a number which can be expressed in the form a/b, where 'a' and 'b' are integers and _____, are called rational numbers.
A)

Q) (a - b)3/(a - b)2 = ?
A)

Long Answer Questions:
Q) Represent 10 and -10 on the number line?
A)

Q) Write the decimal and fractional expansion of 284.361 and 365.897?
A)

Q) Explain briefly about Properties of division of rational numbers?
A)

Q) Find the H.C.F. and the L.C.M. of 0.9,2.5,1.18,3.6,0.45 and 0.83?
A)

Q) Simplify the following? (a) {(0.9 - 0.6)²}/{(0.9)² - 2(0.9)(0.6) + (0.6)²} (b)[(8.65)2 - (4.35)2]/(8.65 - 4.35).
A)

Q) If the cost of 10 books is Rs 200, find the cost of 2 books.
A)

Q) The smallest rational numbers by which 1/3 should be multiplied so that its decimal expansion terminates after one place of decimal, is ?
A)

Q) The number of consecutive zeros in 23×33×53×7 is
A)

Q) Simplify using BODMAS rule, 25 - 25 ÷ 5 + 10 × 3 =?
A)

Q) If the cost of 5 pens is Rs 105.5, find the cost of 1 pen?
A)




Contents
  • Rational Numbers
  • BODMAS Rules
  • Worked-out problems for solving BODMAS rules - involving integers
  • Decimal Division
  • Decimal and Fractional Expansion
  • Examples on representation of rational numbers on the number line
  • Properties of division of rational numbers
  • Non-Terminating Decimal
  • Decimals-HCF and LCM
  • Examples on H.C.F. and L.C.M. of decimals
  • Rounding Decimals to the Nearest Whole Number

Formulae
  • The numbers of the form a/b, or a number which can be expressed in the form a/b, where 'a' and 'b' are integers and b ≠ 0, are called rational numbers.
  • numbersystems1
  • The positive (+ve) rational numbers will be represented by points on the number line lying to the right side of O and negative (-ve) rational numbers.
  • Property 1 (Closure Property of division of rational numbers):
    If a/b and c/d are any two rational numbers such that c/d ≠ 0 then (a/b ÷ c/d) is also a rational number.
  • Property 2 (Property of 1 of division of rational numbers):
    For every rational number a/b we have:
    (a/b ÷ 1) = a/b
  • Property 3:
    For every nonzero rational number a/b, we have:
    {a/b ÷ a/b} = 1
  • [(a - b)3]/[a2 - 2(a)(b) + b2]
    = (a - b)3/(a - b)2
    = (a - b)
  • [a3 - b3]/[a2 - 2ab + b2]
    = [(a - b) (a2 + ab + b32)]/[(a - b)2]
    = (a2 + ab + b2)/(a - b)

Rational Numbers

What are rational numbers?

  • The numbers of the form a/b, or a number which can be expressed in the form a/b, where 'a' and 'b' are integers and b ≠ 0, are called rational numbers.
  • In other words, a rational number is any number that can be expressed as the quotient of two integers with the condition that the divisor is not zero.
  • For examples; each of the numbers 2/3, 5/8, -3/14, -11/-5, 7/-9, 7/-15 and -6/-11 is a rational number.
  • Numerator and denominator: If a/b is a rational number, then the integer a is known as its numerator and the integer b is called the denominator.

Is every rational number an integer?

  • Every integer is a rational number but a rational number need not be an integer.
  • We know that 1 = 1/1, 2 = 2/1, 3 = 3/1, 4 = 4/1 and so on .... .
  • also, -1 = -1/1, -2 = -2/1, -3 = -3/1, -4 = -4/1 and so on .... .
  • In other words, any integer a can be written as a = a/1, which is a rational number.
  • Thus, every integer is a rational number.
    Clearly, 3/2,-5/3, etc. are rational numbers but they are not integers.
    Hence, every integer is a rational number but a rational number need not be an integer.

Let us determine whether the following rational numbers are integers or not:

    (i) 2/5
    2/5 is not an integer. Since we cannot express 2/5 without a fractional or decimal component

    (ii) 8/4
    8/4 is an integer. Since if we simplify 8/4 to its lowest term we get 2/1 = 2, which is an integer.

    (iii) -5/-5
    -5/-5 is an integer. Since if we simplify -5/-5 to its lowest term we get 1/1 = 1, which is an integer.

    (iv) -15/2
    -15/2 is not an integer. Since we cannot express -15/2 without a fractional or decimal component

    (v) -32/8
    -32/8 is an integer. Since if we simplify -32/8 to its lowest term we get -4, which is an integer.

    (vi) 49/-9
    49/-9 is not an integer. Since we cannot express 49/-9 without a fractional or decimal component

    (vii) -75/-20
    -75/-20 is not an integer. Since if we simplify -75/-20 to its lowest term we get 15/4 and we cannot express 15/4 without a fractional or decimal component

    (viii) 500/-10
    500/-10 is an integer. Since if we simplify 500/-10 to its lowest term we get 50/-1 = -50, which is an integer.

So, from the above explanation we conclude that every rational number is not an integer.


Easy and simple way to remember BODMAS rule!!

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Note:

    (i) Start Divide/Multiply from left side to right side since they perform equally.
    (ii) Start Add/Subtract from left side to right side since they perform equally.

Steps to simplify the order of operation using BODMAS rule:

First part of an equation is start solving inside the 'Brackets'.

    For Example: (6 + 4) × 5
    First solve inside 'brackets' 6 + 4 = 10, then 10 × 5 = 50.

Next solve the mathematical 'Of'.

    For Example: 3 of 4 + 9
    First solve 'of' 3 × 4 = 12, then 12 + 9 = 21.

Next, the part of the equation is to calculate 'Division' and 'Multiplication'.
We know that, when division and multiplication follow one another, then their order in that part of the equation is solved from left side to right side.

For Example: 15 ÷ 3 × 1 ÷ 5,

    'Multiplication' and 'Division' perform equally, so calculate from left to right side. First solve 15 ÷ 3 = 5, then 5 × 1 = 5, then 5 ÷ 5 = 1.

    In the last part of the equation is to calculate 'Addition' and 'Subtraction'. We know that, when addition and subtraction follow one another, then their order in that part of the equation is solved from left side to right side.

For Example; 7 + 19 - 11 + 13

'Addition' and 'Subtraction' perform equally, so calculate from left to right side. First solve 7 + 19 = 26, then 26 - 11 = 15 and then 15 + 13 = 28.

These are simple rules need to be followed for simplifying or calculating using BODMAS rule.

In brief, after we perform "B" and "O", start from left side to right side by solving any"D" or "M" as we find them. Then start from left side to right side solving any "A" or "S"as we find them


BODMAS Rules - Involving Integers

Follow the order of operation as:

1. Bracket → Solve inside the Brackets before Of, Multiply, Divide, Add or Subtract.

For example:

7 × (15 + 5)
= 7 × 20
= 140


2. Of → Then, solve Of part (Powers, Roots, etc.,) before Multiply, Divide, Add or Subtract.

For example:

6 + 3 of 7 - 5
= 6 + 3 × 7 - 5
= 6 + 21 - 5
= 27 - 5
= 22


3. Division/Multiplication → Then, calculate Multiply or Divide before Add or Subtract start from left to right.

For example:

20 + 21 ÷ 3 × 2
= 20 + 7 × 2
= 20 + 14
= 34


4. Addition/Subtraction → At last Add or Subtract start from left to right.

17 + (8 - 5) × 5
= 17 + 3 × 5
= 17 + 15
= 32


Worked-out problems for solving BODMAS rules - involving integers:

Simplify using BODMAS rule:

(a) 25 - 48 ÷ 6 + 12 × 2

Solution:

25 - 48 ÷ 6 + 12 × 2
= 25 - 8 + 12 × 2, (Simplifying 'division' 48 ÷ 6 = 8)
= 25 - 8 + 24, (Simplifying 'multiplication' 12 × 2 = 24)
= 17 + 24, (Simplifying 'subtraction' 25 - 8 = 17)
= 41, (Simplifying 'addition' 17 + 24 = 41)

Answer: 41

(b) 78 - [5 + 3 of (25 - 2 × 10)]

Solution:

78 - [5 + 3 of (25 - 2 × 10)]
= 78 - [5 + 3 of (25 - 20)], (Simplifying 'multiplication' 2 × 10 = 20)
= 78 - [5 + 3 of 5], (Simplifying 'subtraction' 25 - 20 = 5)
= 78 - [5 + 3 × 5], (Simplifying 'of')
= 78 - [5 + 15], (Simplifying 'multiplication' 3 × 5 = 15)
= 78 - 20, (Simplifying 'addition' 5 + 15 = 20)
= 58, (Simplifying 'subtraction' 78 - 20 = 58)

Answer: 58

(c) 52 - 4 of (17 - 12) + 4 × 7

Solution:

52 - 4 of (17 - 12) + 4 × 7
= 52 - 4 of 5 + 4 × 7, (Simplifying 'parenthesis' 17 - 12 = 5)
= 52 - 4 × 5 + 4 × 7, (Simplifying 'of')
= 52 - 20 + 4 × 7, (Simplifying 'multiplication' 4 × 5 = 20)
= 52 - 20 + 28, (Simplifying 'multiplication' 4 × 7 = 28)
= 32 + 28, (Simplifying 'subtraction' 52 - 20 = 32)
= 60, (Simplifying 'addition' 32 + 28 = 60)

Answer: 60


BODMAS/PEMDAS Rules - Involving Decimals

Follow the order of operation as:

1. Bracket → Solve inside the Brackets/parenthesis before Of, Multiply, Divide, Add or Subtract.

For example:

1.6 × (0.5 + 0.3)
= 1.6 × 0.8
= 1.28


2. Of → Then, solve Of part (Exponent, Powers, Roots, etc.,) before Multiply, Divide, Add or Subtract.

For example:

3.2 + 0.5 of 1.6 - 0.1
= 3.2 + 0.5 × 1.6 - 0.1
= 3.2 + 0.8 - 0.1
= 4 - 0.1
= 3.9

3. Division/Multiplication → Then, calculate Multiply or Divide before Add or Subtract start from left to right.

For example:

1.2 + 1.5 ÷ 0.3 × 0.3
= 1.2 + 5 × 0.3
= 1.2 + 1.5
= 2.7


4. Addition/Subtraction → At last Add or Subtract start from left to right.

For example:

1.7 + (4.2 - 0.5) × 0.1
= 1.7 + 3.7 × 0.1
= 1.7 + 0.37
= 2.07


Worked-out problems for solving BODMAS/PEMDAS rules - involving decimals:

Simplify using BODMAS/PEMDAS rule:

(a) 8 - 4.2 ÷ 6 + 0.3 × 0.4

Solution:

8 - 4.2 ÷ 6 + 0.3 × 0.4
= 8 - 0.7 + 0.3 × 0.4, (Simplifying 'division' 4.2 ÷ 6 = 0.7)
= 8 - 0.7 + 0.12, (Simplifying 'multiplication' 0.3 × 0.4 = 0.12)
= 7.3 + 0.12, (Simplifying 'subtraction' 8 - 0.7 = 7.3)
= 7.42, (Simplifying 'addition' 7.3 + 0.12 = 7.42)

Answer: 7.42

(b) 7.6 - [3 + 0.5 of (3.1 - 2.3 × 1.02)]

Solution:

7.6 - [3 + 0.5 of (3.1 - 2.3 × 1.02)]
= 7.6 - [3 + 0.5 of (3.1 - 2.346)], (Simplifying 2.3 × 1.02 = 2.346)
= 7.6 - [3 + 0.5 of 0.754], (Simplifying 'subtraction' 3.1 - 2.346 = 0.754)
= 7.6 - [3 + 0.5 × 0.754], (Simplifying 'of')
= 7.6 - [3 + 0.377], (Simplifying 'multiplication' 0.5 × 0.754 = 0.377)
= 7.6 - 3.377, (Simplifying 'addition' 3 + 0.377 = 3.377)
= 4.223, (Simplifying 'subtraction' 7.6 - 3.377 = 4.223)

Answer: 4.223

(c) 12.8 - 0.4 of (7.2 - 3.7) + 2.4 × 3.02

Solution:

12.8 - 0.4 of (7.2 - 3.7) + 2.4 × 3.02
= 12.8 - 0.4 of 3.5 + 2.4 × 3.02, (Simplifying 7.2 - 3.7 = 3.5)
= 12.8 - 0.4 × 3.5 + 2.4 × 3.02, (Simplifying 'of')
= 12.8 - 1.4 + 2.4 × 3.02, (Simplifying 'multiplication' 0.4 × 3.5 = 1.4)
= 12.8 - 1.4 + 7.248, (Simplifying 'multiplication' 2.4 × 3.02 = 7.248)
= 11.4 + 7.248, (Simplifying 'subtraction' 12.8 - 1.4 = 11.4)
= 18.648, (Simplifying 'addition' 11.4 + 7.248 = 18.648)

Answer: 18.648


Conversion of Unlike Decimals to Like Decimals

Conversion of unlike decimals to like decimals follow the steps of the method.

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.

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 = 80/100 = 8/10 and 0.800 = 800/1000 = 8/10
  • 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.


Decimal Division

Dividing decimal by a decimal number is just same like division as usual.

How to divide a decimal by a decimal number?

To divide a decimal by a decimal number follow the below steps:

  • Convert the divisor into a whole number by multiplying the dividend and divisor by the suitable power of 10.
  • Now, divide the new dividend by the whole number as discussed earlier.

Worked-out problems to find the quotient of a decimal by a decimal number

Read the above explanation step-by-step and try to understand the examples on division of decimals.

1. Find the quotient of:

(i) Divide 96.075 by 6.3

Solution:

Since, the divisor has 1 decimal place.
Therefore, multiply the dividend and divisor by 10
i.e., (96.075 × 10)/(6.3 × 10) = 960.75/63
Now, divide 960.75 by 63
i.e., 960.75 ÷ 63
Divide the decimal number without the decimal point,
so we have 96075 ÷ 63

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Since, 960.75 has 2 decimal places
Therefore, 960.75 ÷ 63 will also have 2 decimal places
Therefore, 960.75 ÷ 63 = 15.25

(ii) Divide 24.629 by 1.1

Solution:

Since, the divisor has 1 decimal place.
Therefore, multiply the dividend and divisor by 10
i.e., (24.629 × 10)/(1.1 × 10) = 246.29/11
Now, divide 246.29 by 11
i.e., 246.29 ÷ 11
Divide the decimal number without the decimal point,
so we have 24629 ÷ 11

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Since, 246.29 has 2 decimal places
Therefore, 246.29 ÷ 11 will also have 2 decimal places
Therefore, 960.75 ÷ 63 = 22.39


Word problems on division of decimal by a decimal number:

2. The length of a rectangle is 1.5 m and its area is 14.295. Find its breadth.

Solution:

Length of a rectangle is 1.5
Area of a rectangle is 14.295
Therefore, breadth of the rectangle = Area/Length = 14.295/1.5
= (14.295 × 10)/(1.5 × 10)
= 142.95/15
= 9.53 m


3. If the cost of 9 books is Rs206.55, find the cost of 1 book.

Solution:

Number of books = 9
Cost of 9 books = Rs 206.55
Therefore, cost of 1 book = Rs (206.55 ÷ 9) = Rs 22.95


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:

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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:

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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


Representation of Rational Numbers on the Number Line

  • In representation of rational numbers on the number line are discussed here. We know how to represent integers on the number line.To represent the integers on the number line, we need to draw a line and take a point O on it. Call it 0 (zero).
  • Set of equal distances on the right as well as on the left of O. Such a distance is known as a unit length. Let A, B, C, D, etc. be the points of division on the right of 'O' and A',B', C', D', etc. be the points of division on the left of 'O'. If we take OA = 1 unit, then clearly, the point A, B, C, D, etc. represent the integers 1, 2, 3, 4, etc. respectively and the point A', B', C', D', etc. represent the integers -1, -2, -3, -4, etc. respectively.

Note: The point O represents integer 0.

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  • Thus, we may represent any integer by a point on the number line. Clearly, every positive integer lies to the right of O and every negative integer lies to the left of O.
  • We can represent rational numbers on the number line in the same way as we have learnt to represent integers on the number line. In order to represent rational numbers on the number line, first we need to draw a straight line and mark a point O on it to represent the rational number zero. The positive (+ve) rational numbers will be represented by points on the number line lying to the right side of O and negative (-ve) rational numbers.

  • If we mark a point A on the line to the right of O to represent 1, then OA = 1 unit. Similarly, if we choose a point A' on the line to the left of O to represent -1, then OA' = 1 unit.

Consider the following examples on representation of rational numbers on the number line;

1. Represent 1/2 and -1/2 on the number line.

Solution:

Draw a line. Take a point O on it. Let the point O represent 0. Set off unit lengths OA to the right side of O and OA' to the left side of O.

Then, A represents the integer 1 and A' represents the integer -1.

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Now, divide the segment OA into two equal parts. Let P be the mid-point of segment OA and OP be the first part out of these two parts. Thus, OP = PA = 12. Since, O represents 0 and A represents 1, therefore P represents the rational number 12.

Again, divide OA' into two equal parts. Let OP' be the first part out of these two parts. Thus, OP' = PA' = -12. Since, O represents 0 and A' represents -1, therefore P' represents the rational number -12.


2. Represent 2/3 and -2/3 on the number line.

Solution:

Draw a line. Take a point O on it. Let it represent 0. From the point O set off unit distances OA to the right side of O and OA' to the left side of O respectively.

Divide OA into three equal parts. Let OP be the segment showing 2 parts out of 3. Then the point P represents the rational number 23.

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Again, divide OA' into three equal parts. Let OP' be the segment consisting of 2 parts out of these 3 parts. Then, the point P' represents the rational number -23.


3. Represent 13/5 and -13/5 on the number line.

Solution:

Draw a line. Take a point O on it. Let it represent 0.

Now, 135 = 235 = 2 + 35

From O, set off unit distances OA, AB and BC to the right of O. Clearly, the points A, B and C represent the integers 1, 2 and 3 respectively. Now, take 2 units OA and AB, and divide the third unit BC into 5 equal parts. Take 3 parts out of these 5 parts to reach at a point P. Then the point P represents the rational number 135.

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Again, from the point O, set off unit distances to the left. Let these segments be OA', A' B', B' C', etc. Then, clearly the points A', B' and C' represent the integers -1, -2, -3 respectively.

Now, = -135 = -(2 + 35)

Take 2 full unit lengths to the left of O. Divide the third unit B' C' into 5 equal parts. Take 3 parts out of these 5 parts to reach a point P'.

Then, the point P' represents the rational number -135.

Thus, we can represent every rational number by a point on the number line.


Properties of division of rational numbers

Property 1 (Closure Property of division of rational numbers):

If a/b and c/d are any two rational numbers such that c/d ≠ 0 then (a/b ÷ c/d) is also a rational number.

For example:

(i) 2/3 and 4/9 are any two rational numbers and clearly 4/9 ≠ 0 then
(2/3 ÷ 4/9) = (2/3 × 9/4) = (2 × 9)/(3 × 4) = 18/12 = 3/2 is also a rational number.

(ii) 6/7 and -6/7 are any two rational numbers and clearly -5/7 ≠ 0 then
(6/7 ÷ -6/7) = (6/7 × 7/-6) = (6 × 7)/(7 × -6) = 42/-42 = 1/-1 is also a rational number.

Property 2 (Property of 1 of division of rational numbers):

For every rational number a/b we have:
(a/b ÷ 1) = a/b

For example:

(i) 5/8 ÷ 1 = 5/8
(ii) -4/9 ÷ 1 = -4/9
(iii) 5/-2 ÷ 1 = 5/-2

Property 3:

For every nonzero rational number a/b, we have:
{a/b ÷ a/b} = 1

For example:

(i) 4/7 ÷ 4/7 = 4/7 × 7/4 = (4 × 7)/(7 × 4) = 28/28 = 1
(ii) -3/11 ÷ -3/11 = -3/11 × 11/-3 = (-3 × 11)/(11 × -3) = -33/-33 = 1
(iii) 3/-2 ÷ 3/-2 = 3/-2 × -2/3 = (3 × -2)/(-2 × 3) = -6/-6 = 1


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.


Calculating Non Terminating Decimals:

Using long division method, we will observe the steps in calculating 5/3.

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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

Solved examples on non-terminating decimal:

Find the decimal representation of 16/45.

Solution:

Using long division method, we get numbersystems11
Therefore, 0.3555... = 0.35 and is a mixed recurring decimal.


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:

numbersystems12

(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

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;

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


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:

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.


Terminating Decimal

Worked-out examples on terminating decimal:


1. Express 17/8 in the decimal form.

Solution:

numbersystems13

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:

numbersystems14

= 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:

numbersystems15

= 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:

numbersystems16

= 0.6
Since, the remainder is zero.
Therefore, 3/5 is terminating and 0.6 is a terminating 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.

numbersystems17

= 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

numbersystems18

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


Decimals-HCF and LCM

Steps to solve H.C.F. and L.C.M. of decimals:

Step I: Convert each of the decimals to like decimals.

Step II: Remove the decimal point and find the highest common factor and least common multiple as usual.

Step III: In the answer (highest common factor /least common multiple), put the decimal point as there are a number of decimal places in the like decimals.

Now we will follow the step-by-step explanation on how to calculate the highest common factor and the least common multiple of decimals.


Worked-out examples on H.C.F. and L.C.M. of decimals

1. Find the H.C.F. and the L.C.M. of 1.20 and 22.5

Solution:

Given, 1.20 and 22.5
Converting each of the following decimals into like decimals we get;
1.20 and 22.50
Now, expressing each of the numbers without the decimals as the product of primes we get
120 = 2 × 2 × 2 × 3 × 5 = 23 × 3 × 5
2250 = 2 × 3 × 3 × 5 × 5 × 5 = 2 × 32 × 53

Now, H.C.F. of 120 and 2250 = 2 × 3 × 5 = 30
Therefore, the H.C.F. of 1.20 and 22.5 = 0.30 (taking 2 decimal places)

L.C.M. of 120 and 2250 = 23 × 32 × 53 = 9000
Therefore, L.C.M. of 1.20 and 22.5 = 90.00 (taking 2 decimal places)


2. Find the H.C.F. and the L.C.M. of 0.48, 0.72 and 0.108

Solution:

Given, 0.48, 0.72 and 0.108
Converting each of the following decimals into like decimals we get;
0.480, 0.720 and 0.108
Now, expressing each of the numbers without the decimals as the product of primes we get
480 = 2 × 2 × 2 × 2 × 2 × 3 × 5 = 25 × 3 × 5
720 = 2 × 2 × 2 × 2 × 3 × 3 × 5 = 24 × 32 × 5
108 = 2 × 2 × 3 × 3 × 3 = 22 × 33

Now, H.C.F. of 480, 720 and 108 = 22 × 3 = 12
Therefore, the H.C.F. of 0.48, 0.72 and 0.108 = 0.012 (taking 3 decimal places)

L.C.M. of 480, 720 and 108 = 25 × 33 × 5 = 4320
Therefore, L.C.M. of 0.48, 0.72, 0.108 = 4.32 (taking 3 decimal places)


3. Find the H.C.F. and the L.C.M. of 0.6, 1.5, 0.18 and 3.6

Solution:

Given, 0.6, 1.5, 0.18 and 3.6
Converting each of the following decimals into like decimals we get;
0.60, 1.50, 0.18 and 3.60
Now, expressing each of the numbers without the decimals as the product of primes we get
60 = 2 × 2 × 3 × 5 = 22 × 3 × 5
150 = 2 × 3 × 5 × 5 = 2 × 3 × 52
18 = 2 × 3 × 3 = 2 × 32
360 = 2 × 2 × 2 × 3 × 3 × 5 = 23 × 32 × 5

Now, H.C.F. of 60, 150, 18 and 360 = 2 × 3 = 6
Therefore, the H.C.F. of 0.6, 1.5, 0.18 and 3.6 = 0.06 (taking 2 decimal places)

L.C.M. of 60, 150, 18 and 360 = 23 × 32 × 52 = 1800
Therefore, L.C.M. of 0.6, 1.5, 0.18 and 3.6 = 18.00 (taking 2 decimal places)


Simplification of Decimal

Learn the following identities to apply these in simplifying decimal.

(a) (a + b)2 = a2 + b2 + 2ab
(b) (a - b)2 = a2 + b2 - 2ab
(c) a2 - b2 = (a + b) (a - b)
(d) a3 + b3 = (a + b) (a2 - ab + b2)
(e) a3 - b3 = (a - b) (a2 + ab + b2)


Worked-out examples on simplification of decimal

Let us observe how to simplify decimals using identities with detailed step-by-step explanation.

Simplify the following:

(a) {(0.9 - 0.6)2}/{(0.9)2 - 2(0.9)(0.6) + (0.6)2}

Solution:

Let, a = 0.9 and b = 0.6
So, [(a - b)3]/[a2 - 2(a)(b) + b2]
= (a - b)3/(a - b)2
= (a - b)

Now putting the value of a and b we get,
= 0.9 - 0.6
= 0.3

(b) [(5.8)3 - (2.6)3]/[(5.8)2 + (2.6)2 - 2(5.8) + (2.6)2]

Solution:

Let a = 5.8 and b = 2.6
So, we have
= [a3 - b3]/[a2 - 2ab + b2]
= [(a - b) (a2 + ab + b2)]/[(a - b)2]
= (a2 + ab + b2)/(a - b)

Now putting the value of a and b we get,

= [(5.8)2 + (5.8)(2.6) + (2.6)2]/(5.8 - 2.6)
= 55.48/3.2
= (55.48 × 10)/(3.2 × 10), Multiply both numerator and denominator by 10
= 554.8/32
= 17.3375

(c) [(8.65)2 - (4.35)2]/(8.65 - 4.35)

Solution:

    Let a = 8.65 and b = 4.35
    So, we have
    = [a2 - b2]/(a - b)
    = [(a + b)(a-b)]/(a - b)
    = a + b

    Now putting the value of a and b
    = 8.65 + 4.35
    = 13


Rounding Decimals to the Nearest Whole Number

Rules for rounding decimals to the nearest whole number:

  • To round a decimal to the nearest whole number analyse the digit at the first decimal place i.e., tenths place.
  • If the tenths place value is 5 or greater than 5, then the digit at the ones place increases by 1 and the digits at the tenths place and thereafter becomes 0.

For example;

(i) 9.63
In 9.63 analyse the digit at the tenths place. Here 6 is more than 5. Therefore we have to round the number up to the nearest whole number 10.

(ii) 78.537
In 78.537 analyse the digit at the tenths place. Here 5 is equal to 5. Therefore we have to round the number up to the nearest whole number 79.

If the tenths place value is less than 5, then the digit at the ones place remains the same but the digits at the tenths place and thereafter becomes 0.

For example;

(i) 7.21
In 7.21 analyse the digit at the tenths place. Here 2 is less than 5. Therefore we have to round the number down to the nearest whole number 7.

(ii) 13.48
In 13.48 analyse the digit at the tenths place. Here 4 is less than 5. Therefore we have to round the number down to the nearest whole number 13.


Worked-out examples on rounding decimals to the nearest whole number

To round a decimal to the nearest whole number follow the explanation step-by-step how to round up or round down the decimal to the nearest whole number.

Round off the following to the nearest whole number.

(a) 51.7

Solution:

51.7
The digit at the tenths place is 7 and 7 > 5.
The whole number part of 51.7 increases by 1 and the digit to the right of decimal point means the tenths place becomes zero (rounded up).
Therefore, 51.7 rounded off to the nearest whole number as 52.

(b) 147.28

Solution:

147.28 The digit at the tenths place is 2 and 2 < 5.
The digit at the ones place remains unchanged and the digits to the right of the decimal point means the tenths place and hundredths place becomes 0 (rounded down).
Therefore, 147.28 rounded off to the nearest whole number 147.


Example 1 : Find the decimal expansions of 10/3.

Solution :

numbersystems19

Remainders : 1, 1,1,1
Divisior:3

Example 2 : Show that 0.3333... = 0 3. can be expressed in the form p/ q , where p and q are integers and q ≠ 0.

Solution :

Since we do not know what 0 3. is , let us call it 'x' and so x = 0.3333...
Now here is where the trick comes in.
Look at 10 x = 10 × (0.333...) = 3.333...
Now, 3.3333... = 3 + x, since x = 0.3333...
Therefore, 10 x = 3 + x Solving for x, we get 9x = 3, i.e.,
x = 1/3

Example 3 : Show that 1.272727... = 1 27 .can be expressed in the form p q , where p and q are integers and q ≠ 0.

Solution :

Let x = 1.272727...
Since two digits are repeating,
we multiply x by 100 to get 100 x = 127.2727...
So, 100 x = 126 + 1.272727... = 126 + x
Therefore, 100 x - x = 126, i.e.,
99 x = 126
x = 126/99=14/11= 1.27

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