__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 1/12**^{th} is lesser
than 1/8^{th}. The value of any fraction can be converted into
decimal.

**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**** fractions, since 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.**

**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 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__ = __9 __

**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 **__2 __

** 5 **

**(b) 8 **__3 __

** 5 **

**(c) 8**__ 2 __

** 5**

**(d) 7 **__3 __

** 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 2/3**^{rd} 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**

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