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Hint: H.C.F full form is Highest Common Factor whereas L.C.M full form is Lowest Common Multiple. So factors are what we can multiply to get the number and multiples are what we get after multiplying the number by an integer.

Complete step-by-step answer:

Let’s take two fractions \[\dfrac{a}{b}\] and \[\dfrac{c}{d}\] .

To find H.C.F of \[\dfrac{a}{b}\] and \[\dfrac{c}{d}\] the generalized formula will be:

\[\text{H}\text{.C}\text{.F}=\dfrac{\text{H}\text{.C}\text{.F of

numerators}}{\text{L}\text{.C}\text{.M of denominators}}......(1)\]

To find L.C.M of \[\dfrac{a}{b}\] and \[\dfrac{c}{d}\] the generalized formula will be:

\[\text{L}\text{.C}\text{.M}=\dfrac{\text{L}\text{.C}\text{.M of

numerators}}{\text{H}\text{.C}\text{.F of denominators}}......(2)\]

Now L.C.M of two numbers is the smallest number (not zero) that is a multiple of both.

Now, to find out the LCM and HCF of fractions.

Let’s take the example of \[\dfrac{2}{5}\] and \[\dfrac{3}{7}\] .

As the numerators are (2,3) and denominators are (5,7) so now according to equation (1) we get,

\[\text{H}\text{.C}\text{.F}=\dfrac{\text{H}\text{.C}\text{.F of (2,3)}}{\text{L}\text{.C}\text{.M

of (5,7)}}........(3)\]

So H.C.F of 2,3:

The factors of 2 are: 1, 2

The factors of 3 are: 1, 3

1 is the only common factor as it is the only number which is common to both 2 and 3.

Therefore, \[\text{H}\text{.C}\text{.F of (2,3)}=1........(4)\]

Now L.C.M of 5, 7:

The multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, ....

The multiples of 7 are: 7, 14, 21, 28, 35, 42, ....

35 is the lowest common multiple as it is a multiple common to both 5 and 7.

Therefore, \[\text{L}\text{.C}\text{.M of (5,7)}=35........(5)\]

Putting value of H.C.F from equation (4) and value of L.C.M from equation (5) in equation (3)

we get, \[\text{H}\text{.C}\text{.F=}\dfrac{1}{35}\] .

Now finding L.C.M for the same example,

\[\text{L}\text{.C}\text{.M}=\dfrac{\text{L}\text{.C}\text{.M of (2,3)}}{\text{H}\text{.C}\text{.F

of (5,7)}}........(6)\]

So H.C.F of 5, 7:

The factors of 5 are: 1,5

The factors of 7 are: 1,7

1 is the only common factor as it is the only number which is common to both 5 and 7.

Therefore, \[\text{H}\text{.C}\text{.F of (5,7)}=1........(7)\]

Now L.C.M of 2, 3:

The multiples of 2 are: 2, 4, 6, 8, ....

The multiples of 3 are: 3, 6, 9, 12, ....

6 is the lowest common multiple as it is a multiple common to both 2 and 3.

Therefore, \[\text{L}\text{.C}\text{.M of (2,3)}=6........(8)\]

Putting value of H.C.F from equation (7) and value of L.C.M from equation (8) in equation (6)

we get, \[\text{L}\text{.C}\text{.M=}\dfrac{6}{1}=6\] .

Note: Understanding the concept of H.C.F and L.C.M is the key here and chances of mistakes are when we in a hurry substitute L.C.M in place of H.C.F in equation (3) and equation (6) or we do vice-versa.

Complete step-by-step answer:

Let’s take two fractions \[\dfrac{a}{b}\] and \[\dfrac{c}{d}\] .

To find H.C.F of \[\dfrac{a}{b}\] and \[\dfrac{c}{d}\] the generalized formula will be:

\[\text{H}\text{.C}\text{.F}=\dfrac{\text{H}\text{.C}\text{.F of

numerators}}{\text{L}\text{.C}\text{.M of denominators}}......(1)\]

To find L.C.M of \[\dfrac{a}{b}\] and \[\dfrac{c}{d}\] the generalized formula will be:

\[\text{L}\text{.C}\text{.M}=\dfrac{\text{L}\text{.C}\text{.M of

numerators}}{\text{H}\text{.C}\text{.F of denominators}}......(2)\]

Now L.C.M of two numbers is the smallest number (not zero) that is a multiple of both.

Now, to find out the LCM and HCF of fractions.

Let’s take the example of \[\dfrac{2}{5}\] and \[\dfrac{3}{7}\] .

As the numerators are (2,3) and denominators are (5,7) so now according to equation (1) we get,

\[\text{H}\text{.C}\text{.F}=\dfrac{\text{H}\text{.C}\text{.F of (2,3)}}{\text{L}\text{.C}\text{.M

of (5,7)}}........(3)\]

So H.C.F of 2,3:

The factors of 2 are: 1, 2

The factors of 3 are: 1, 3

1 is the only common factor as it is the only number which is common to both 2 and 3.

Therefore, \[\text{H}\text{.C}\text{.F of (2,3)}=1........(4)\]

Now L.C.M of 5, 7:

The multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, ....

The multiples of 7 are: 7, 14, 21, 28, 35, 42, ....

35 is the lowest common multiple as it is a multiple common to both 5 and 7.

Therefore, \[\text{L}\text{.C}\text{.M of (5,7)}=35........(5)\]

Putting value of H.C.F from equation (4) and value of L.C.M from equation (5) in equation (3)

we get, \[\text{H}\text{.C}\text{.F=}\dfrac{1}{35}\] .

Now finding L.C.M for the same example,

\[\text{L}\text{.C}\text{.M}=\dfrac{\text{L}\text{.C}\text{.M of (2,3)}}{\text{H}\text{.C}\text{.F

of (5,7)}}........(6)\]

So H.C.F of 5, 7:

The factors of 5 are: 1,5

The factors of 7 are: 1,7

1 is the only common factor as it is the only number which is common to both 5 and 7.

Therefore, \[\text{H}\text{.C}\text{.F of (5,7)}=1........(7)\]

Now L.C.M of 2, 3:

The multiples of 2 are: 2, 4, 6, 8, ....

The multiples of 3 are: 3, 6, 9, 12, ....

6 is the lowest common multiple as it is a multiple common to both 2 and 3.

Therefore, \[\text{L}\text{.C}\text{.M of (2,3)}=6........(8)\]

Putting value of H.C.F from equation (7) and value of L.C.M from equation (8) in equation (6)

we get, \[\text{L}\text{.C}\text{.M=}\dfrac{6}{1}=6\] .

Note: Understanding the concept of H.C.F and L.C.M is the key here and chances of mistakes are when we in a hurry substitute L.C.M in place of H.C.F in equation (3) and equation (6) or we do vice-versa.

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