Question

12. The numerator of a fraction is 4 less than the denominator. If 1 is added to both its
numerator and denominator, it becomes 1/2
Find the fraction​

Answer 1

Given :-

• The numerator of a fraction is 4 less than the denominator. If 1 is added to both its numerator and denominator, it becomes 1/2

To find :-

• Required fraction

Solution :-

Let the denominator be x

• Numerator of a fraction is 4 less than the denominator

• Numerator = x - 4

According to the given condition

• 1 is added to both its numerator and denominator, it becomes 1/2

→ x - 4 + 1/x + 1 = ½

→ x - 3 /x + 1 = ½

→ 2(x - 3) = x + 1

→ 2x - 6 = x + 1

→ 2x - x = 6 + 1

→ x = 7

Hence,

• x = 7

Therefore,

• Required fraction = Numerator/denominator

• x - 4/x = 7 - 4/7 = 3/7

Answer 2

GIVEN :-

• Tʜᴇ ɴᴜᴍᴇʀᴀᴛᴏʀ ᴏғ ᴀ ғʀᴀᴄᴛɪᴏɴ ɪs 4 ʟᴇss ᴛʜᴀɴ ᴛʜᴇ ᴅᴇɴᴏᴍɪɴᴀᴛᴏʀ .

• Iғ 1 ɪs ᴀᴅᴅᴇᴅ ᴛᴏ ʙᴏᴛʜ ɴᴜᴍᴇʀᴀᴛᴏʀ ᴀɴᴅ ᴅᴇɴᴏᴍɪɴᴀᴛᴏʀ, ᴛʜᴇɴ ᴛʜᴇ ғʀᴀᴄᴛɪᴏɴ ʙᴇᴄᴏᴍᴇs 1/2 .

TO FIND :-

• Tʜᴇ ᴏʀɪɢɪɴᴀʟ ғʀᴀᴄᴛɪᴏɴ .

SOLUTION :-

Lᴇᴛ,

• Nᴜᴍᴇʀᴀᴛᴏʀ ʙᴇ "X" .

• Aɴᴅ Dᴇɴᴏᴍɪɴᴀᴛᴏʀ ʙᴇ "Y" .

Aᴄᴄᴏʀᴅɪɴɢ ᴛᴏ ᴛʜᴇ ǫᴜᴇsᴛɪᴏɴ,

ᴄᴀsᴇ - 1 :-

➪ X = Y - 4 -----(1)

ᴄᴀsᴇ - 2 :-

$\huge\red\checkmark$ $\bf{\dfrac{X\:+\:1}{Y\:+\:1}\:=\:\dfrac{1}{2}\:}$

$\rm{:\implies\:2X\:+\:2\:=\:Y\:+\:1\:}$

Putting the value of X in the above equation .

$\rm{:\implies\:2\times(Y\:-\:4)\:+\:2\:=\:Y\:+\:1\:}$

$\rm{:\implies\:2Y\:-\:8\:+\:2\:=\:Y\:+\:1\:}$

$\rm{:\implies\:2Y\:-\:Y\:-\:6\:-\:1\:=\:0\:}$

$\rm{:\implies\:Y\:-\:7\:=\:0\:}$

$\bf\green{:\implies\:Y\:=\:7\:}$

Now, putting the value of Y in the equation (1) .

➳ X = 7 - 4

➳ X = 3

$\huge\red\therefore$ Tʜᴇ ᴏʀɪɢɪɴᴀʟ ғʀᴀᴄᴛɪᴏɴ is $\bf{\dfrac{X}{Y},\:i.e.\:\pink{\dfrac{3}{7}}\:.}$