Comparison of Fraction

Compare which fraction is smaller out of `1/4` and `2/3`

To compare fractions do the following steps.

Step 1: Calculate the LCM (Least Common Multiples or Lowest common multiple) of denominators.

In the given fractions 4 and 3 are denominators.
LCM of 4 and 3 = 12
LCM of denominator is also called Least common denominator.

Step 2: Consider one of the fractions, i.e. ¼

Divide the LCM from its denominator.
12 ÷ 4 = 3
Now, multiply the numerator and denominator of the fraction with the quotient you got.
`1/4=(1xx3)/(4xx3)=(3)/(12)`

Step 3: Consider the second fraction, i.e. 2/3

Divide the LCM from its denominator
12 ÷ 3 = 4
Now multiply the denominator and numerator of the fraction with result (quotient).
`2/3=(2xx4)/(3xx4)=(8)/(12)`

Step 4: Now we have two fractions:
`(3)/(12)` and `(8)/(12)`
Both of the fractions have equal denominator.

Now compare the numerator. The fraction with greater numerator is greater than the fraction with smaller numerator. Here, 8 is greater than 3.

Therefore; `(8)/(12)>(3)/(12)`

Or, `2/3>1/4`

Question 1: Compare the following fractions and put < or > sign between them accordingly:

(a) `1/4` and `3/4`

Solution: Here, denominator of both the fractions is same. Numerator 1 < 3

Hence, `1/4<3/4`

(b) `2/3` and `3/4`

Solution: The LCM of denominators 3 and 4 = 12

Hence, `2/3=(2xx4)/(3xx4)=(8)/(12)`

And, `3/4=(3xx3)/(4xx3)=(9)/(12)`

Since 8 < 9

Hence, `(8)/(12)<(9)/(12)`

Or, `2/3<3/4`

Solution: The LCM of 4 and 6 = 12

So, `3/4=(3xx3)/(4xx3)=(9)/(12)`

And, `5/6=(5xx2)/(6xx2)=(10)/(12)`

Since, 9 < 10

Hence, `(9)/(12)<(10)/(12)`

Or, `3/4<5/6`

(d) `6/7` and `5/6`

Solution: LCM of 7 and 6 = 42

Hence, `6/7=(6xx6)/(7xx6)=(36)/(42)`

And, `5/6=(5xx7)/(6xx7)=(35)/(42)`

Here, 36 > 35

Hence, `(36)/(42)>(35)/(42)`

Or, `6/7>5/6`

(e) `7/8` and `8/7`

Solution: LCM of 8 and 7 = 56

Hence, `7/8=(7xx7)/(8xx7)=(49)/(56)`

And, `8/7=(8xx8)/(7xx8)=(64)/(56)`

Since, 49 < 64

Hence, `(49)/(56)<(64)/(56)`

Or, `7/8<(8)/7`

Note: Numerator is less than denominator in `7/8`while numerator is more than denominator in `8/7`. When numerator is less than denominator, the number is less than 1. When numerator is more than denominator, the number is greater than 1. So, `7/8<8/7`

(f) `3/5` and `5/3`

Solution: LCM of 5 and 3 = 15

Hence, `3/5=(3xx3)/(5xx3)=(9)/(15)`

And, `5/3=(5xx5)/(3xx5)=(25)/(15)`

Since, 9 < 25

Hence, `(9)/(15)<(25)/(15)`

Or, `3/5<5/3`

(g) `(16)/(12)` and `(3)/(12)`

Solution: Here, denominator is same in both the fractions, and 16 > 3

Hence, `(16)/(12)>(3)/(12)`

(h) `3/2` and `4/3`

Solution: LCM of 2 and 3 = 6

Hence, `3/2=(3xx3)/(2xx3)=9/6`

And, `4/3=(4xx2)/(3xx2)=8/6`

Here, 9 > 8

Hence, `9/6>8/6`

Or, `3/2>4/3`

(i) `(12)/(11)` and `7/6`

Solution: LCM of 11 and 6 = 66

Hence, `(12)/(11)=(12xx6)/(11xx6)=(72)/(66)`

And, `7/6=(7xx11)/(6xx11)=(77)/(66)`

Since, 72 < 77

Hence, `(72)/(66)<(77)/(66)`

Or, `(12)/(11)<7/6`

(j) `4/7` and `5/9`

Solution: LCM of 7 and 9 = 63

Hence, `4/7=(4xx9)/(7xx9)=(36)/(63)`

And, `5/9=(5xx7)/(9xx7)=(35)/(63)`

So, `(36)/(63)>(35)/(63)`

Or, `4/7>5/9`

Question 2: Arrange the following in ascending order:

(a) `1/2, 1/4, ¾`

Solution: LCM of 2, 4 and 4 = 4

Hence, `1/2=(1xx2)/(2xx2)=2/4`

Numerator is 4 in remaining fractions. Now, the rational numbers can be arranged in ascending order as follows:

`1/4<2/4<3/4`

Or, `1/4<1/2<3/4`

(b) `3/5, 3/7, (9)/(25)`

Solution: LCM of 5, 7 and 25 = 175

Hence, `3/5=(3xx35)/(5xx35)=(105)/(175)`

`3/7=(3xx25)/(7xx25)=(75)/(175)`

`(9)/(25)=(9xx7)/(25xx7)=(63)/(175)`

It is clear that:

`(63)/(175)<(75)/(175)<(105)/(175)`

Or, `(9)/(25)<3/7<3/5`

Solution: LCM of 5, 7 and 6 = 210

Hence, `2/5=(2xx42)/(5xx42)=(84)/(210)`

`4/7=(4xx30)/(7xx30)=(120)/(210)`

`5/6=(5xx35)/(6xx35)=(175)/(210)`

Or, `2/5<4/7<5/6`

(d) `1/3, 6/9, (9)/(18)`

Solution: LCM of 3, 9 and 18 = 18

Hence, `1/3=(1xx6)/(3xx6)=(6)/(18)`

`6/9=(6xx2)/(9xx2)=(12)/(18)`

It is clear that:

`(6)/(18)<(9)/(18)<(12)/(18)`

Or, `1/3<(9)/(18)<6/9`

(e) `3/9, (9)/(25), (5)/(20)`

Solution: LCM of 4, 20 and 25 = 100

Hence, `3/4=(3xx25)/(4xx25)=(75)/(100)`

`(9)/(25)=(9xx4)/(25xx4)=(36)/(100)`

`(5)/(20)=(5xx5)/(20xx5)=(25)/(100)`

It is clear that:

`(25)/(100)<(36)/(100)<(75)/(100)`

Or, `(5)/(20)&t;(9)/(25)<3/4`

(f) `(2)/(15), (3)/(18), (9)/(10)`

Solution: LCM of 15, 18 and 10 = 90

Hence, `(2)/(15)=(2xx6)/(15xx6)=(12)/(90)`

`(3)/(18)=(3xx5)/(18xx5)=(15)/(90)`

`(9)/(10)=(9xx9)/(10xx9)=(81)/(90)`

Hence, `(2)/(15)<(3)/(18)<(9)/(10)`

(g) `(16)/(15), (15)/(14), (14)/(12)`

Solution: LCM of 15, 14 and 12 = 420

Hence, `(16)/(15)=(16xx28)/(15xx28)=(448)/(420)`

`(15)/(14)=(15xx30)/(14xx30)=(450)/(420)`

`(14)/(12)=(14xx35)/(12xx35)=(490)/(420)`

It is clear that:

`(448)/(420)<(450)/(420)<(490)/(420)`

Or, `(16)/(15)<(15)/(14)<(14)/(12)`

(h) `(11)/(12), (9)/(15), 8/9`

Solution: LCM of 12, 15 and 9 = 180

Hence, `(11)/(12)=(11xx15)/(12xx15)=(165)/(180)`

`(9)/(15)=(9xx12)/(15xx12)=(108)/(180)`

`8/9=(8xx20)/(9xx20)=(160)/(180)`

It is clear that:

`(108)/(180)<(160)/(180)<(165)/(180)`

Or, `(9)/(15)<8/9<(11)/(12)`

(i) `2/3, ¾, 4/5`

Solution: LCM of 3, 4 and 5 = 60

Hence, `2/3=(2xx20)/(3xx20)=(40)/(60)`

`3/4=(3xx15)/(4xx15)=(45)/(60)`

`4/5=(4xx12)/(5xx12)=(48)/(60)`

It is clear that:

`(40)/(60)<(45)/(60)<(48)/(60)`

Or, `2/3<3/4<4/5`

(j) `2/4, 4/6, 6/8`

Solution: LCM of 4, 6 and 8 = 24

Hence, `2/4=(2xx6)/(4xx6)=(12)/(24)`

`4/6=(4xx4)/(6xx4)=(16)/(24)`

`6/8=(6xx3)/(8xx3)=(18)/(24)`

It is clear that:

`(12)/(24)<(16)/(24)<(18)/(24)`

Or, `2/4<4/6<6/8`