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Thursday, May 21, 2020

FRACTIONS - BASIC AND ITS TYPES, REDUCING TO LOWEST TERM AND CHALLENGING QUESTIONS

FRACTIONS 

A fraction is a part of a whole or total. Suppose Kishan has Rs 100 and he spends Rs 25 out of it, so part of money that Kishan spends is Rs 25 out of total money he had Rs 100. Hence part or fraction of money that he spends is – 25/100. Or for instance Gyanesh has these pieces of chocolate.



NUMERATOR AND DENOMINATOR 

There are two parts of a fraction, numerator and denominator. In 14/28 numerator is 14 and denominator is 28. Numerator is the upper part of fraction, that is 14. This 14 is the number of pieces eaten, so numerator denotes how much and denominator is total parts which is 28, in this case. When reduced to lowest term 14/28 is 1/2, that we would discuss further. In other examples fractions denoted by shaded parts -


IMPORTANCE OF FRACTIONS


Fractions are quite important in real life as it represents a
part

A person makes a budget, how much part of the income to be  allocated to an expenditure, we need fractions .  

Four pieces of candy is to be divided equally among eight friends. So each friend would eat 1/2 ( half ) candy, the use of fractions. 

A person has property whose value is in $ 60,00,000 and that property is to be distributed equally in his three sons and two daughters. So total there are five persons, hence $ 60,00,000 would be divided equally in five parts, and each one would get 1/5th part .

GREATER AND SMALLER FRACTION 

Which is greater 1/4 or 1/8 ?

To understand this, draw two circles of same area, divide the area of first circle into four equal parts, and the area of second circle into eight equal parts as shown -


More the number of denominator, lesser would be its value, provided that the numerator is 1. Thus 1/12th is lesser than 1/8th. The value of any fraction can be converted into decimal.


UNLIKE FRACTIONS

The fraction which have different denominators are known as unlike fractions ,no matter what the numerator is.Therefore fractions 1/4, 2/5, 1/8, 3/4 are unlike fractionssince denominator is different.


It may also happen that numerator are of different  numbers, like 2, 5, 7, 8 etc with the same denominator.  So how would you compare these fractions. Method of comparing those fractions is -


LIKE FRACTIONS


The fraction whose denominator is same is known as like fractions, it is immaterial what the numerator may be. 2/8, 5/8 and 6/8 are like fractions, since denominator is same.

Like fractions can be compared by just comparing their numerators, more the numerator, more its value as shown by four rectangles example. But for comparing unlike fractions we have to convert them into decimal, except in the case the numerator is 1. The fraction 1/4, 1/8 and 1/5 can be compared without converting them into decimal, as numerator is 1 of every fraction, observed by activities.


    EQUIVALENT FRACTIONS

Equivalent fractions are the fractions whose value is same. For instance value of 1/2, 2/4 and 4/8 is equal. How?




The fractions denoted by all hexagons are equivalent fractions, 
as equal portion is shaded. When we divide 4 and 6 by the same number that is 2 it is 2/3.  8 and 12 when divided by 4 is 2/3. 16 and 24 when divided by 8 is 2/3.




PROPER FRACTIONS

The proper fractions are the fractions, when numerator is less than its denominator. For instance 3/8 is a proper fraction.

IMPROPER FRACTIONS


Improper fractions are the fractions whose numerator is more than its denominator. For instance 7/6 is improper fraction, as in this 7 the numerator is more than its denominator. It is not a proper form of fraction, so it is improper fraction. Improper fractions are also known as mixed fractions. 




MIXED FRACTION

Mixed fraction is the fraction which is formed from addition of whole fraction and proper fraction as shown -


CONVERSION OF IMPROPER FRACTION INTO MIXED FRACTION


Improper fraction is also known as mixed fraction. But we have to convert an improper into a mixed fraction to make it as mixed fraction.



The fraction is 7/4. Divide the numerator by its denominator, so 7 is divided by 4, quotient is 1 and 3 is remainder, we write the quotient first of all in the left and then 3 the remainder and denominator below the remainder, 1 3 
                                                               4

Now this 1 is the whole part of mixed fraction as in I rhombus all parts are shaded, so 4/4 is 1, and 3/4 is proper part of fraction as in II rhombus 3/4 parts are shaded.


ADDING TWO OR MORE FRACTIONS

(A) ADDING FRACTIONS WITH SAME  DENOMINATORS

When we have to add fractions with same denominator or like fractions, you can add easily -


 4  +  3 +  2 = 
12   12   12  12

In this just you have to add the numerators and write the denominator which is given so adding numerators 4, 3 and 2 is 9, and fraction is 9/12. You can add as many fractions in this way.

(A) ADDING FRACTIONS WITH DIFFERENT DENOMINATORS


Since the denominators are different, we cannot add without taking something common in the denominators. That’s why we take that multiple that is common in all the denominators. For that purpose we calculate the least common multiple or L.C.M. So in first case 12 is L.C.M as 6, 2 and 4 are all factors of 12, and 12 is least also. So considering first fraction divide 12 by denominator that is 6, result is 2 and multiply 2 by numerator 4, hence 2 X 4 = 8. Proceed in the same manner with other fractions. Then add all the numerators which are 8, 6 and 9, and write the denominator below. 
In second case find L.C.M of 16 and 12 that is 48, and then move further and add.


REDUCING THE FRACTION TO LOWEST TERM


The fraction can be reduced to the lowest term if and only if both the numerator and denominator can be divided by the same number. Suppose the fraction is 12/15. We can clearly observe that 12 that is numerator and 15 that is denominator both can be divided by the same number that is 3. 12 divided by 3 is 4, and 15 divided by 3 is 5. So lowest term of 12/15 is 4/5.

The fraction 9/16 cannot be reduced to lowest term because both 9 and 16 cannot be divided by any same or common number, so we have to write it as 9/16.
Reducing the fraction to lowest term is quite impertinent to ease calculations.
Reduce the fractions  3/9, 16/20, 28/42, 64/80, 90/100 to lowest term.


FINDING OFF OF A FRACTION

How much is 4/5 of 70 ? To find  it – 4/5 X 70. Divide 70 by its denominator that is 5. 70 divided by 5 is 14, and then multiply 14 by its numerator that is 4, which is 56.

Find 3/4 of 100, 5/7 of 42, 4/5 of 30.

QUESTIONS BASED ON FRACTIONS 

Q – 1. How much is 3/4th of these blocks ?



(a) 30  

(b) 45  

(c) 50  

(d) 48  


Q – 2. Convert 42/5 into a mixed fraction

(a) 6 
         5 

(b) 8 
         5 

(c) 8 2 
         5

(d) 7 
         5

Q – 3. Lalit has 40 candies and Rohan has 60 candies.  Lalit eats 2/5 and Rohan eats half of the candies both of them have. How much candies Rohan has now more than Lalit has left after eating ?

(a) 4  

(b) 5  

(c) 6  

(d) 7  


Q - 4. One pizza of 12 pieces and another pizza of 6 pieces is ordered by Ravi from dominos. Ravi eats 2/3rd of total pieces of both the pizza and his mother eats 1/4th of the pieces eaten by  Ravi. How many total pieces of pizza are eaten by Ravi and his mother 

(a) 10  

(b) 15  

(c) 12  

(d) 20  


Q – 5. The option not an equivalent fraction of 4/6 is -

(a) 2/3  

(b) 12/18  

(c) 28/56  

(d) 42/63  


Q – 6. How much is the sum of 3/4th of a dozen and 2/3rd of a gross ?  

(a) 105 

(b) 117 

(c)  96 

(d)  90 

Q - 7. The price of one kilogram sugar one week before was Rs 40. The price at present is 5/4th of the price that was a week ago. How much is the present price of six kilograms sugar?

(a) Rs 200 

(b) Rs 125 

(c)  Rs 220 

(d) Rs 300 


 Q - 8. The side of a square is 30 meters. The length of a rectangle is 6/5th of side of the square, while its breadth is 5/6th of its side. How much is the total boundary of the rectangle ?

(a) 120 meters 

(b) 130 meters 

(c) 124 meters

(d) 122 meters 

 Q - 9. John has Rs 729. He spends 1/9th of the amount everyday he has. How much amount John would have left on third day after spending ?

(a) Rs 510 

(b) Rs 512 

(c)  Rs 612

(d)  Rs 648 


Q - 10. The price of 1/3rd dozen bananas is Rs 20. How much is the price of four kilograms bananas, if in one kilogram there are 8 bananas ?

(a) Rs 160 

(b) Rs 150 

(c)  Rs 240 

(d)  Rs 360 


ANSWERS WITH EXPLANATIONS 

Q - 1. (b) Since total there are 60 blocks, its 3/4 is 3/4 X 60 = 45

Q - 2.(c)Converting 42/5 into a mixed fraction is dividing 42 by 
5, quotient is 8 and remainder is 2, so it is 8 2/5 


Q - 3 (c). Number of candies that Lalitesh has 40. Number of candies he eats is 2/5 X 40 = 16. So candies Lalitesh has now is 40 -16 = 24. Number of candies that Suhesh has is  60, and number of candies he eats is 1/2 X 60 = 30. Hence number of candies that Suhesh has now is 60 – 30 = 30. So candies now that Suhesh has more than Lalitesh is 30 - 24 = 6.

Q - 4.(b) Total pieces of both the pizzas is 12 + 6 = 18. Pieces  eaten by Ravi is 2/3 X 18 = 12. Pieces eaten by Ravi's mother is 1/4 of pieces eaten by Ravi hence it is 1/4 X 12 = 3 pieces. So total pieces eaten by Ravi and his mother is 12 + 3 = 15 pieces. 


Q - 5 (c) 4/6 is equal to 2/3 as 4 and 6 both divided by 2 is 2/3. 12/18 is equal to 4/6, because 12 and 18 both divided by 6 is 2/3 which is equal to 4/6. 28/56 is equal to 1/2. 42/63 is equal to 2/3 as when both 42 and 63 are divided by 21 is 2/3. 

Q - 6. (a) 3/4 of a dozen is 3/4 X 12 = 9. 2/3 of a gross is 2/3 X 144 = 96. Hence sum is 9 + 96 = 105. 

Q - 7. (d) Price of one kg sugar at present is 5/4 X 40 = Rs 50. Hence price of six kilograms sugar is 50 X 6 = Rs 300. 

 Q - 8. (d) Side of the square is 30 meters. So length of the rectangle is 6/5 X 30 = 36 meters. Hence breadth of the rectangle is 5/6 X 30 = 25 meters. Therefore sum of boundary of rectangle is 36 + 25 + 36 + 25 = 122 meters.

Q - 9. (b) Amount that Ramesh spends on first day is Rs 729 X 1/9 = Rs 81. So left on 1st day is Rs 729 - Rs 81 = Rs 648. Amount spend on second day is Rs 648 X 1/9 = Rs 72. Hence amount left on 2nd day is Rs 648 - Rs 72 = Rs 576. Amount spend on third day is Rs 576 X 1/9 = Rs 64. Therefore amount left on 3rd day after spending is Rs 576 - Rs 64 = Rs 512.

    Q - 10. (a) 1/3 dozen is 1/3 X 12 = 4 bananas. So price of 4 bananas is Rs 20. Thus price of one banana is 20/4 = Rs 5. Bananas in one kilogram is 8, so bananas in 4 kilograms is 8 X 4 = Rs 32. Therefore price of 4 kilogram or 32 bananas is 32 X 5 = Rs 160
















1 comment:

  1. Very nicely n easily explained fractions to kids . Will be helpful to children . Keep posting more .

    ReplyDelete

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