MATHEMATICS - Motion

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OPERATION ON FRACTIONS

FRACTIONS

(1) What is a fraction?

A fraction consists of a numerator (part) on

top of a denominator (total) separated by a

horizontal line.

numeratorFractiondenominator

For example, the fraction of the circle

which is shaded is:

2(partsshaded)4(totalparts)

In the square on the right, the fraction

shaded is 38

and the fraction unshaded is 58

.

(2) Equivalent Fractions – Multiplying?

The three circles on the right each have

equal parts shaded, yet are represented by

different but equal fractions. These

fractions, because they are equal, are

called equivalent fractions.

Any fraction can be changed into an

equivalent fraction by multiplying both the

numerator and denominator by the same

number

21 222 4

or

41 442 8

so 1 2 4

2 4 8

Similarly

25 1029 18

or

35 1539 27

so 5 10 15

9 18 27

You can see from the above examples that

each fraction has an infinite number of

fractions that are equivalent to it.

(3) Equivalent Fractions – Dividing

(Reducing)

Equivalent fractions can

also be created if both the

numerator and denominator

can be divided by the same

number (a factor) evenly.

This process is called

“reducing a fraction” by

dividing a common factor

(a number which divides

into both the numerator

and denominator evenly).

4 4 18 4 2

27 9 381 9 9

5 5 130 5 6

6 2 310 2 5

MATHEMATICS OPERATIONS ON FRACTIONS

(NMTC-SUB-JUNIOR) WORKSHEET-3


(4) Simplifying a Fraction (Reducing to its

Lowest Terms)

It is usual to reduce a fraction until it can’t

be reduced any further.

A simplified fraction has no

common factors which will

divide into both numerator

and denominator.

Notice that, since 27 and

81 have a common factor

of 9, we find that 39

is an

equivalent fraction.

But this fraction has a factor of 3 common

to both numerator and denominator.

So, we must reduce this fraction again. It

is difficult to see, but if we had known that

27 was a factor (divides into both parts of

the fraction evenly), we could have arrived

at the answer in one step

e.g. 8 8 124 8 3

, 45 15 360 15 4

TYPES OF FRACTIONS

(1) Proper Fractions

A Proper fraction is one in which the

numerator is less than the denominator (or

a fraction which is less than the number 1).

e.g. 1 3 88 8, , ,2 4 93 15

are all Proper fractions.

(2) Fractions that are Whole Numbers

Some fractions, when reduced, are really

whole numbers (1, 2, 3, 4… etc).

Whole numbers occur if the denominator

divides into the numerator evenly.

e.g. 84

is the same as 8 4 24 4 1 or 2

305

is the same as 30 5 65 5 1 or 6

So, the fraction 305

is really the whole

number 6. Notice that a whole number can

always be written as a fraction with a

denominator of 1. e.g. 10 = 101

(3) Mixed Numbers

A mixed number is a combination of a

whole number and a Proper fraction.

e.g. 325

(two and three-fifths)

2279

(twenty-seven and two-ninths)

396

= 192

(always reduce fractions)

(4) Improper Fractions

An improper fraction is one in which the

numerator is larger than the denominator.

From the circles on the right, we see that

314

(mixed number) is the same as 74

(improper fraction).

An improper fraction, like, can be changed

to a mixed number by dividing the

denominator into the numerator and

expressing the remainder (3) as the

numerator.

e.g. 16 135 5 , 29 53

8 8 , 14 24

3 3

27 9 381 9 9

3 3 19 3 3

27 27 181 27 3


3 714 4

74

= 1

4 7 = 314

–43

A mixed number can be changed to an

improper fraction by changing the whole

number to a fraction with the same

denominator as the common fraction.

325

= 105

and 35

1109

= 909

and 19

A simple way to do this is to multiply the

whole number by the denominator, and

then add the numerator.

e.g. 5 4 9 5 36 5 4149 9 8 9

2 10 7 2 70 2 72107 7 7 7

(5) Like Fractions

The fractions which have the same

denominators are called like fractions. For

example 12

, 32

, 52

, 72

are like fractions.

The simplification of such fractions is

easy, as all the denominators here are the

same. Suppose we need to add all the

above like fractions, then;

12

+ 32

+ 52

+ 72

= 1 3 5 72

= 162

= 8.

(6) Unlike Fractions

The fractions which have unequal

denominators or different denominators

are called, unlike fractions.

For example 12

, 13

, 14

, 15

.

COMPARING FRACTIONS

(1) Proper Fractions

In the diagram on the right, it is easy to see

that 78

is larger than 38

(since 7 is larger

than 3). However, it is not as easy to tell

that 78

is larger than 56

.

78

38

In order to compare fractions, we must

have the same (common) denominators.

This process is called

“Finding the Least Common

Denominator” and is usually abbreviated

as finding the LCM (lowest common

multiple).

Which is larger : 78

or 56

?

In order to compare these fractions, we

must change both fractions to equivalent

fractions with a common denominator.

To do this, take the largest denominator

(8) and examine multiples of it, until the

other denominator (6) divides into it.

Notice that, when we multiply 8 x 3, we

get 24, which 6 divides into.

Now change the fractions to 24th s.

When we change these fractions to equivalent fractions with an LCM of 24,

we can easily see that 78

is larger than 56

since 2124

is greater than.



or 512

?

Examine multiples of the larger

denominator (12) until the smaller

denominator divides into it. This tells us

that the LCM is 36.

Now, we change each fraction to

equivalent fractions with the LCM of 36.

4

4 169 36

4

3

5 1512 36

3

So 49

is larger than 512

.


or 1315

or 1112

?

Find the LCM by examining multiples of

15. Notice that, when we multiply 15 × 4,

we find that 60 is the number that all

denominators divide into.

124 489 60

12

4

13 5212 60

4

5

11 5512 60

5

So 1112

is largest fraction.

Notice that one denominator (9) divides

into the other denominator (18).

This means that the LCM = 18 and we

only have to change one fraction 79

to an

equivalent fraction. 2

7 149 18

2

So, 79

is larger than 1318

.

ADDING FRACTIONS

There are four main operations that we can do

with numbers: addition (+), subtraction (–),

multiplication ( x ), and division ( ÷ ).

In order to add or subtract, fractions must have

common denominators.

This is not required for multiplication or division.

(1) Adding with Common Denominators

To add fractions, if the

denominators are the same,

we simply add the

numerators and keep the same

denominators.

e.g. Add 112

and 512

.

Since the denominators are common,

simply add the numerators. Notice that we

must reduce the answer, if possible.

(2) Adding When One

Denominator is a Multiple

of the Other:

Add 29

and 527

.

12 × 1 = 12

12 × 2 = 24

12 × 3 = 36

(LCM)

15 × 1 = 15

15 × 2 = 30

15 × 3 = 45

15 × 4 = 60 (LCM)

1 2 34 4 4

1 5 6 112 12 12 2

32 69 27

3


Notice that the denominators are not common. Also notice that 27 is a multiple of 9 (since 9 × 3 = 27). This means that the LCM = 27 (see the last example in “Comparing Fractions”).

2 5 6 5 119 27 27 27 27

(3) Adding any Fraction:

Add 712

and 1315

.

We must find a common denominator by

examining multiples of the largest

denominator. We find that the LCM = 60.

Add 516

and 328

.

When adding mixed numbers, add the

whole numbers and the fractions

separately. Find common denominators

and add.

If an improper fraction occurs in the

answer, change it to a common fraction by

doing the following.

(4) The Language of Addition

1 2plus2 3

1 2and2 3

1 2totalof and2 3

1 2sumof and2 3

1 2addition of and2 3

1 2combined with2 3

1 2more than (or greater than)2 3

Note: All of these can be worded with the fractions in reverse order:

e.g. 2 1 1 2plus is thesameas plus3 2 2 3

SUBTRACTING FRACTIONS

(1) Common Fractions

As in addition, we must have common

denominators in order to subtract. Find the

LCM; change the fractions to equivalent

fraction with the LCM as the denominator.

Then subtract the numerators, but keep the

same denominator.

(2) Mixed Numbers

When subtracting whole numbers,

subtract the whole numbers, and then

subtract the fractions separately.

However, if the

common fraction we

are subtracting is

smaller than the other common fraction,

we must borrow the number “1” from

the large whole number.

i.e. 2 7 2 94 3 ,or37 7 7 7

7 13 35 52 87 9112 15 60 60 60 20

total equals

5 201 16 243 92 28 24

29324

29 5 53 3 1 424 24 24

1 5 6 112 12 12 2

5 3 2 1or8 8 8 4

2 3 16 9 73 8 24 24 24

5 3 23 1 29 9 9


To subtract 314

from 263

, first change the

common fractions to equivalent fractions

with the LCM. Since 812

is smaller than

912

, borrow from 6.

8 12 8 206 5 512 12 12 12

(3) The Language of Subtraction

5 2 2min us (NOT min us)6 3 3

2 5subtractedfrom3 6

2 5from3 6

2 5less than3 6

5 2decreased byor lowered by6 3

5 2thedifferenceof and3 3

NOTE: Unlike addition, we cannot

reword the above with the fractions in

reverse order:

i.e. 1 22 3 is NOT the same as 2 1

3 2

MULTIPLYING FRACTIONS

(1) When multiplying fractions, a common

denominator is not needed. Simply

multiply the numerators and multiply the

denominators separately.

Sometimes, we can reduce the fractions

before multiplying.

Any common factor in either numerator

can cancel with the same factor in the

denominator. Multiply after cancelling

(reducing).

Note that any whole number (16) has the

number “1” understood in its denominator.

If more than two fractions are multiplied,

the same principles apply.

2 5 9 5 44 2 3 2 17 7 7 7 7

2 3 8 9 20 9 116 1 6 1 5 1 43 4 12 12 12 12 12

5 2 CAN BE WORDED6 3

2 5 2 5 105 9 3 9 27

3 5 2 55 7

5379

4 3 49 8

39 8

16

2 3 1 2 3 1 19 20 4 9 20 4 120

5 5 16168

810

1


(2) Mixed Numbers

Mixed numbers must be changed to

improper fractions before multiplying.

Remember that a mixed number (like ) can

be changed to an improper fraction by

multiplying the whole number (2) by the

denominator (4) and then adding the

numerator.

(3) The Language of Multiplication

1 2multiplied by2 3

1 2by2 3

1 2of2 3

1 2the product of and2 3

NOTE: When multiplying, it doesn’t

matter which fraction is first.

i.e. 1 22 3 is the same as 2 1

3 2 .

DIVIDING FRACTIONS

To divide fractions, we invert (take the reciprocal

of) the fraction that we are dividing by, then

cancel (reduce), and then multiply.

Taking the reciprocal of a fraction involves

“flipping” the fraction so that the numerator and

denominator switch places.

Note that a whole number is really a fraction (e.g.

4 = 41

).

(1) Common Fractions Simply invert (take the reciprocal of) the

fractions that we are dividing by ( 89

).

Then cancel and multiply.

Note: you can only cancel after the division is changed to a multiplication.

(2) Mixed Numbers As in multiplication, mixed numbers must

be changed to improper fractions. (3) The Language of Division

1 2divided by2 3

2 1into3 2

1 2divide by2 3

3 8 11 522 4 134 11 4 11

4 3 49 17 119 25 2 or 139 7 9 7 9 9


2 3reciprocal3 2

8 19reciprocal19 8

14reciprocal4

5 8 5 9 457 9 7 8 56

9 3 9 32 616 32 16 3

2 1 5 15 5 7 71 23 7 3 7 3 15 9



NOTE: In multiplication, the order of the

fractions was not important.

i.e. 1 22 3 is the same as 2 1

3 2 .

In division, this is not the case. The order

of the fractions is important.

Consider the following:

1 2 1 3 32 3 2 2 4

But 2 1 2 2 4 113 2 3 1 3 3


WORKSHEET

1. Find the value of 2 3×1 5 3 11 + of ÷ 11 6 2 41 – 2

2. Find the value of

1 31 41 3–

4 13– 31 12 1–

1 23 –2

3. If x

1 2 3 4 5 31 1.............4 6 8 10 12 64 2

, then what is the value of x?

4. Production of wheat is 124

times that of

rice, but the cost of rice is 114

times that of

wheat. If a farmer produces wheat in place

of rice, then what is his income in terms of

the previous income?

5. If a man spends 56

th part of money and

then earns 12

part of the remaining money,

what part of the money is with him now?

6. Eight people are planning to share equally

the cost of a rental car. If one person

withdraws from the arrangement and the

others share equally the entire cost of the

car, then by how much is the share of each

of the remaining persons is increased in

terms of original share?

7. Which of the following fractions is the

largest?

(A) 1316

(B) 78

(C) 3140

(D) 6380

8. Which of the following fractions is less

than 78

and greater than 13

?

(A) 14

(B) 2324

(C) 1112

(D) 1724

9. Madan picks up three different digits from

the set {1, 2, 3, 4, 5} and forms a mixed

number by placing the digits in the spaces

. The fractional part of the mixed

number should be less than 1 (for example

243

). What is the difference between the

largest and smallest possible mixed

number that can be formed?

(A) 7420

(B) 3410

(C) 9420

(D) 345

10. What fraction of 47

must be added to itself

to make the sum 1114

?

(A) 12

(B) 47

(C) 78

(D) 1514


11. If a=21

10

, b = 15

and c = 1100

, then

which of the following statements is

correct?

(A) a < b < c (B) a < c < b

(C) b < c < a (D) c < a < b

12. Mohan ate half a pizza on Monday. He ate

half of what was left on Tuesday and so

on. He followed this pattern for one week.

How much of the pizza would he have

eaten during the week?

(A) 99.22% (B) 95%

(C) 98.22% (D) 100%

13. Write five equivalent fractions of 35

.

14. Ramesh solved 27

part of an exercise while

Seema solved 45

of it. Who solved lesser

part?

15. Sameera purchased 13 kg2

apples and

34 kg4

oranges. What is the total weight of

fruits purchased by her?

16. Suman studies for 253

hours daily. She

devotes 425

hours of her time for Science

and Mathematics. How much time does

she devote for other subjects?

17. In a class of 40 students 15

of the total

number of studetns like to study English

25

, of the total number like to study

mathematics and the remaining students

like to study Science.

(i) How many students like to study

English?

(ii) How many students like to study

Mathematics?

(iii) What fraction of the total number of

students like to study Science?

18. Sushant reads 13

part of a book in 1 hour.

How much part of the book will he read in

125

hours?

19. The least fraction that must be added to

1 1 11 1 13 2 9 to make the result an integer

is

(A) 45

(B) 35

(C) 25

(D) 15

20. Find the value of x in the following:

2 2 x 1 2 11 13 7 7 4 3 6

(A) 0.006 (B) 16

(C) 0.6 (D) 6


HINTS & SOLUTIONS

1. Given exp. = 2 3×1 15 51 + ÷1 12 42

2 3= × 15 41 + 2 ×12 5

2 33 1

= 2.

2. Given exp. = 1 31 41 3 –

4 13– 31 125 22

= 1 31 41 3–

4 3 23–225

= 1 31 41 3 –

4 53–12 / 5

= 1 31 11120 53–12

= 1 151 111

16 /12

= 1 1512 11116

= 1 157 114

= 4 157 11

= 44 10577 = 149

77

3. x

1 1 2 3 4 5 30 31.......2 4 6 8 10 12 62 64

=

12 2 2 ....30 times 64

x 30

1 12 2 64

= 30 6 36

1 12 2 2

x = 36

4. Let production of rice = x quintals and cost

of 1 quintal rice = ` y

Then, original income of the farmer = ` (x

+ y) = ` xy

Now as per question, production of wheat

= 124

× production of rice = 94

x quintals

Cost of wheat × 114

= Cost of rice = ` y

Cost of wheat = ` 45

y

Present income of the farmer = 9x 4y4 5

= 95

xy = 415

times of xy

= 415

time the previous income.

5. Let the money with the man at first be ` 1.

Money spent = 56

of ` 1 = ` 56

Remaining money = ` 51–6

= ` 16

Money earned = of ` 16

= ` 12

Total money with the man = ` 1 16 12

= ` 312

= ` 14

The man now has 14

th part of the

money.

6. When there are 8 people, the share of each

person is 18

of the total cost.

When there are 7 people, each person’s

share is 17

of the total cost.


Increase in the share of each person =

1 1–7 8

= 156

, i.e., 17

of 18

, i.e., 17

of the

original share of each person.

7. (B)

13 13 516 16 5

= 65 7,

80 8= 7 10

8 10

= 70 31,80 40

= 31 240 2

= 6280

and last fraction is 6380

Out of these, the largest fraction

= 7080

= 78

8. (D)

13

= 0.333000, 78

= 0.875

14

= 0.25, 2324

= 0.9583000, 1112

= 0.9166000,

1724

= 0.7083000

Since 0.7083000 1724

is greater than

0.333000 13

and less than 0.875

78

Therefore, 1724

lies between 13

and 78

.

9. (A)

The largest number = 354

The smallest number = 215

Required difference = 3 25 –14 5

= 234

– 75

= 115 – 2820

= 8720

= 7420

.

10. (C)

Let the required fraction be x. Then,

x of 4 4 117 7 14

4x 4 157 7 14

4x 157 14

– 47

= 714

= 12

x = 12

× 74

= 78

.

11. (B)

a = 1100

, b = 15

= 20100

, c = 110

= 10100

a < c < b

12. (A)

Pizza he ate on Monday = 12

Pizza left = 12

Pizza he ate on Tuesday = 12

× 12

= 14

Pizza left = 12

– 14

= 14

Pizza he ate on Wednesday = 12

× 14

= 18

Pizza left = 1 1–4 8

= 18

Continuing this pattern, we see that Pizza

he ate on Thursday, Friday, Saturday and

Sunday is 116

, 132

, 164

and 1128

respectively.

Quantity of pizza he ate during the week


= 1 1 1 1 1 1 12 4 8 16 32 64 128

= 64 32 16 8 4 2 1128

= 127128

= 127128

× 100% = 99.22%

13. One of the equivalent fractions of 35

is

. Find the other four.

3 3 2 65 5 2 10

. Find the other four.

14. In order to find who solved lesser part of

the exercise, let us compare 27

and 45

.

Converting them to like fractions we have,

2 10 4 28,7 35 5 35

Since10 < 28 , so 10 2835 35

Thus, 2 47 5

Ramesh solved lesser part than Seema.

15. The total weight of the fruits

1 33 4 kg2 4

=

7 19 14 19kg kg2 4 4 4

=

33 1kg 8 kg4 4

16. Total time of Suman’s study = 2 175 h h3 3

Time devoted by her for Science and

Mathematics = 4 142 h5 5

Thus, time devoted by her for other

subjects =

17 14 h3 5

=

= 43 13h 2 h15 15

17. Total number of students in the class = 40.

(i) Of these 15

of the total number of

students like to study English.

Thus, the number of students who

like to study English = 15

of 40 = 15

×

40 = 8.

(ii) Try yourself.

(iii) The number of students who like

English and Mathematics = 8 + 16 =

24. Thus, the number of students

who like Science = 40 – 24 = 16.

Thus, the required fraction is 1640

.

18. The part of the book read by Sushant in

1 hour = 13

.

so, the part of the book read by him in

hours = 1 125 3

=

11 1 11 1 115 3 5 3 15

Let us now find 1 52 3

. We known that

5 1 53 3

So, 1 5 1 1 1 55 52 3 2 3 6 6

Also,

5 1 56 2 3

. Thus,

1 5 1 5 52 3 2 3 6

.

19. (D)

1 1 11 1 13 2 9 =

4 3 103 2 9

17 5 14 3 85 42h h15 15 15

125


= 4 2 103 3 9 =

8 9 49 10 5

15

should be added to 45

to make it an

integer.

20. (D)

5 2 x 5 2 13 7 7 4 3 6

5 7 x 5 2 63 2 7 4 3 1

5 x 56 x =

5 65

= 6.

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