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Fractions and Divisibility

Comparison of fractions with like denominators
• If two fractions have the like [i.e., same] denominator, then the fraction with greater numerator is greater.
• Thus, as 2 > 1.
Comparison of fractions with like numerators
• If two fractions have the like [i.e., same] numerator, then the fraction with greater denominator is smaller.
• Thus, as 19 > 17.
Comparison of fractions with unlike numerators and unlike denominators
In this comparison, we have two methods: (i) by making denominators equal and (ii) by making numerators equal.
By making denominators equal
• Find the L.C.M. of denominators of given unlike fractions.
• Convert the given unlike fractions into equivalent like fractions with the L.C.M. as common denominator.
• Compare the like fractions so obtained.
By making numerators equal
• Find the L.C.M. of numerators of given unlike fractions.
• Convert the given unlike fractions into equivalent like fractions with the L.C.M. as common numerator.
• Compare the like fractions so obtained.
Addition (or) Subtraction of like fractions
• For addition (or) subtraction of like fractions, numerators are added (or) subtracted while the denominator remains the same.
Eg: .
Addition (or) Subtraction of unlike fractions
• Find the L.C.M. of denominators of given unlike fractions.
• Convert the given unlike fractions into equivalent like fractions with the L.C.M. as common denominator.
• Add (or) Subtract the like fractions so obtained.
Multiplication of fractions
• If there is a common factor in one of the numerators and one of the denominators, it may be removed by canceling before multiplying.
Reciprocal of a non-zero fraction
• Reciprocal of any fraction such that the product of these two fractions is equal to 1, i.e., .
Division of fractions
• In order to divide a fraction by another fraction, we multiply the dividend by the reciprocal of the divisor.
• If are two fractions then the division of these two fractions given as
.
Fractions

Fractions: A fraction is a part of a whole. Mathematically, we define fractions as the numbers of the form , where a and b are whole numbers and b ≠ 0.

Therefore, a fraction consists of a top number and bottom number, separated by a dividing line. The top number is called the numerator and bottom number is called the denominator. For example, consider the fraction . In this fraction, '5' is called the numerator and '13' is called the denominator.

Types of fractions: Following are the different types of fractions:

  • Proper fraction: A fraction whose numerator is less than its denominator, is called a proper fraction. For example: is a proper fraction where numerator is less than its denominator.
  • Improper fraction: A fraction whose numerator is greater than (or) equal to its denominator, is called an improper fraction. For example: is an improper fraction where numerator is greater than its denominator and similarly, is an improper fraction where numerator is equal to its denominator.
    • Eg: Convert into an improper fraction.
      Sol:
  • Simple fraction: A fraction both of whose numerator and denominator [i.e., terms] are whole numbers, is called a simple fraction. Example: .
  • Complex fraction: A fraction whose one or both the terms are fractions, is called a complex fraction. Examples for complex fractions are: , ............, etc.
  • Vulgar fraction: A fraction whose denominator is a whole number other than 10, 100, 1000 . . . etc, is called a vulgar fraction. Example: , .........., etc.
  • Mixed fraction: A fraction which can be expressed as a sum of a natural number and a fraction is called a mixed fraction.Example: .
  • Equivalent fractions: Fractions having the same value are known as equivalent fractions. An equivalent fraction of a given fraction can be obtained by multiplying (or) dividing its numerator and denominator by the same non-zero number. Thus, Clearly, are equivalent fractions.
    • Eg: An equivalent fraction of with numerator 18 is
      Sol:


  • Like fractions: Fractions having the same denominator are called like fractions. For example: are like fractions because their denominators are same.
    Example: Add the given like fractions
    Sol:
  • Unlike fractions: The fractions having different denominators are called unlike fractions. For example: are unlike fractions because denominators of the fractions are not same.
  • Unit fractions: The fractions with 1 as numerator are known as unit fractions. For example: are unit fractions because their numerator is '1'.

Fractions In Lowest Terms (or) In Simplest Form: A fraction is said to be in lowest terms (or) in simplest form if the H.C.F. of its numerator and denominator is 1. For example: the fraction is in simplest form because H.C.F. of 2 and 3 is 1. In order to reduce a fraction to its lowest terms, divide each term by their H.C.F.

Simplification: In order to simplify a given expression involving fractions, we use the following rules: Use of BODMAS rule and Removal of brackets.

  • Use of BODMAS rule: We simplify the expressions by applying the operations strictly in the order (1) Bracket (2) Of (3) Division (4) Multiplication (5) Addition (6) Subtraction.
  • Removal of brackets: We strictly remove the different types of brackets in the following order: (1) Bar or Vinculum (–) (2) Parenthesis () (3) Curly brackets {} (4) Square brackets [].

Comparing decimals
Suppose we have to compare two (or) more decimals. Then, we proceed as follows:
Step 1: Convert the given decimals into like decimals.
Step 2: First compare their whole number parts. The decimal with the greater whole number part is greater.
Step 3: If the whole number parts are equal, compare the tenths digits. The decimal with bigger digit in the tenths place is greater.
Step 4: If the tenths digits are also equal, compare the hundredths digits and so on.
Addition of decimals
Suppose we have to add several decimals. Then, we proceed as follows:
Step 1: Convert the given decimals into like decimals.
Step 2: Write the addends under each other with decimal points in the same vertical column.
Step 3: Add the numbers as whole numbers and in the result, place the decimal point just under all decimal points.
Subtraction of decimals
Suppose we have to subtract a decimal from another decimal. Then, we proceed as follows:
Step 1: Convert the given decimals into like decimals.
Step 2: Write the subtrahend (the number to be subtracted) under the minuend (the number from which subtraction is to be done) such that their decimal points are in the same vertical column.
Step 3: Subtract as in whole numbers and in the result, place the decimal point just under the decimal points in the above numbers.
To convert a decimal into a fraction
Step 1: Write the given decimal without the decimal point as numerator.
Step 2: Take 1 annexed with as many zeros as is the number of decimal places in the given decimal as denominator.
Step 3: Reduce the above fraction in simplest form.
To convert a fraction into a decimal by division method
Step 1: Divide the numerator by the denominator.
Step 2: Complete the division. Let a non-zero remainder be left.
Step 3: Insert a decimal point in the dividend and the quotient.
Step 4: Put a zero on the right of the decimal point in the dividend as well as on the right of the remainder. Divide again just as whole numbers.
Step 5: Repeat step 4 till either the remainder is zero (or) requisite number of decimal places have been obtained.
Decimals

Decimals: A fraction whose denominator is 10 or some positive integral power of 10 is called a decimal fraction. Examples for decimal fractions are: , .., etc. These decimal fractions can be written in the decimal form as 0.5, 0.37, 0.848, ...., etc. When the decimal fractions are expressed in the decimal form, they are known as decimal numbers (or) decimals. Thus, each of the 0.5, 0.37, 0.848 is a decimal.

A decimal has two parts: whole number part and decimal part. These parts are separated by a dot (.) called decimal point. The part which is left side to decimal point is whole number part and the part which is right side to decimal point is decimal part. For example: In the decimal 421.839, the whole number part = 421 and the decimal part = .839.

Decimal places: The number of digits contained in the decimal part of a decimal gives the number of its decimal places. For example: the decimal 3.89 has 2 decimal places.

Like decimals: Decimals having the same number of decimal places are called like decimals. For example: 21.3, 489.2, 987.3, 437.0 all are like decimals because each having one decimal place.

Unlike decimals: Decimals having the different number of decimal places are called unlike decimals. For example: 2.97, 51.3 are unlike decimals because in 2.97 we have two decimal places whereas in 51.3 we have only one decimal place.

  • Eg: If you add 6.75 + 4.3 + 2.913 we get
    Sol:

Conversion of unlike decimals to like decimals: Out of the given unlike decimals find the decimal which has the largest number of decimal places, say 'n'. Convert each of the remaining decimals to the one having 'n' decimal places by annexing the required number of zeros to the extreme right of the decimal part.

Multiplication of a decimal by 10, 100, 1000, ........., etc: When decimal is multiplied by some power of 10 then the decimal point is shifted to the right by as many places as is the number of zeros in the multiplier. For example: 73.41 × 10 = 734.1 and 295.867 × 100 = 29586.7.

Multiplication of a decimal by a whole number: In this multiplication we have two steps: (i) Without taking the decimal point into consideration, multiply the decimal by the given whole number [i.e., just like the multiplication of two whole numbers] (ii) In the product, put the decimal point in such a way that the resultant decimal has as many places as are there in the multiplicand.

Multiplication of two or more decimals: In this multiplication also we have two steps: (i) Multiply the given decimals without considering their decimal points (ii) In the product, the decimal point is fixed in such a way that the product has as many decimal places as is the sum of the decimal places in the given decimals.

Dividing a decimal by 10, 100, ...., etc: To divide a decimal by 10, 100, 1000...etc. shift the decimal point to the left by as many places as is the number of zeros in the divisor. For example: 18.6 ÷ 10 = 1.86 and 219.7 ÷ 100 = 2.197.

Dividing a decimal by a whole number: We make ordinary division and mark the decimal point in the quotient as soon as we cross over the decimal point in the dividend.

Dividing a decimal by a decimal: In this division we have two steps: (i) Convert the divisor into a whole number by multiplying the dividend and divisor by a suitable power of 10 (ii) Divide the new dividend by the whole number obtained above.

  • Eg: If 15064 ÷ 28 = 538 then 15.064 ÷ 0.28 = ?
    Sol:
    =53.8

Simplification using the rule of BODMAS: In order to simplify a given expression, we must follow the order of simplification as (1) Bracket (2) Of (3) Division (4) Multiplication (5) Addition (6) Subtraction.

Rounding off Decimals

The process of obtaining the value of a decimal correct to the required number of decimal places is called rounding off and the value obtained is called the rounded off (or) corrected value of the decimal.

Terminating decimals: In the process of converting a fraction into a decimal by the division, if we obtain a zero remainder after a certain number of steps then the decimal obtained is a terminating decimal. Thus, (7/2) = 3.5; (9/4) = 2.25, ...., etc. all are terminating decimals.

  • Eg: Is a terminating decimal or not ?
    Sol: → 5 ÷ 2 →
    represents terminating decimal.

Repeating (or) Recurring Decimals: If in a decimal, a digit (or) a set of digits in the decimal part is repeated continuously, then such a number is called a recurring decimal (or) a repeating decimal. Thus, , etc. all are repeating decimals.

Pure recurring decimal: A decimal in which all the digits in the decimal part are repeated continuously, is called a pure recurring decimal. Thus, 47.3939........ = 47.39, 87.163163..... = 87.163, ...., etc. all are pure recurring decimals.

Mixed recurring decimal: A decimal in which some of the digits in the decimal part are repeated continuously and the rest are not repeated, is called a mixed recurring decimal. Thus, 0.5733333........ = 0.573, 0.63939..... = 0.639, ...., etc. all are mixed recurring decimals.

Non-terminating decimals: If a rational number in its lowest terms has prime factors other than 2 and 5 (or) in addition to 2 and 5 then the division does not end. During the process of division, we get a digit or a group of digits repeated in the same order. Such decimal fractions are called non-terminating but recurring decimals. For example: 1/3 = 0.33333..... = 0.3. Here 3 is repeating.

Periodicity: The recurring part of the non-terminating recurring decimal is called period and the number of digits in the recurring part is called periodicity. For example: (1/3) = 0.333333.... = 0.3. Here, period = 3 and periodicity = 1.

H.C.F. and L.C.M. of given decimals: In order to find the H.C.F. (or) L.C.M. of given decimals: (i) Convert the given decimals into like decimals (ii) Find the H.C.F. (or) L.C.M. of the numbers without the decimal points (iii) In the result, mark the decimal point to have as many decimal places as there are in each decimal, obtained in (i).

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