Question

$\huge \fbox \red{H} \fbox \green{e} \huge \fbox \blue{l} \huge\fbox \orange{l} \fbox \green{o}$


ᴛʜᴇ ᴅᴇɴᴏᴍIɴᴀᴛᴏʀ ᴏғ ᴀ ғʀᴀᴄᴛIᴏɴ Is ɢʀᴇᴀᴛᴇʀ ᴛʜᴀɴ Iᴛs ɴᴜᴍᴇʀᴀᴛᴏʀ ʙʏ 12. Iғ ᴛʜᴇ ɴᴜᴍᴇʀᴀᴛᴏʀ Is ᴅᴇᴄʀᴇᴀsᴇᴅ ʙʏ 2 ᴀɴᴅ ᴛʜᴇ ᴅᴇɴᴏᴍIɴᴀᴛᴏʀ Is Iɴᴄʀᴇᴀsᴇᴅ ʙʏ 7, ᴛʜᴇ ɴᴇᴡ ғʀᴀᴄᴛIᴏɴ Is ᴇǫᴜIᴠᴀʟᴇɴᴛ ᴡIᴛʜ 1/2. ғIɴᴅ ᴛʜᴇ ғʀᴀᴄᴛIᴏɴ.




# ᴀʟʟ ᴛʜᴇ ʙᴇsᴛ
# ᴛʜɴᴋ ᴜ ​

Answer 1

Given:

• The denominator of a fraction is greater than its numerator by 12.
• the numerator is decreased by 2 and the denominator is increased by 7.
• The new fraction is equivalent to ½

To find:

• The original fraction

Solution:

Now,

• Let the numerator be x

➷As the numerator greater that the denominator by 12

• let the denominator be x + 12

hence the fraction is,

$\longrightarrow \tt \: \dfrac{x}{x + 12}$

» Now , as the number is decreased by 2

• the new numerator is x - 2

» And as the denominator is increased by 7

• so, the denominator is x + 12 +7
• x +19

From the given condition:

${ : \implies} \tt \frac{x - 2}{x +19 } = \frac{1}{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ { : \implies} \tt \: 2(x - 2) = 1(x + 19) \\ \\ \\ { : \implies} \tt \: 2x - 4 = x + 19 \: \: \: \: \: \: \: \: \\ \\ \\{ : \implies} \tt \: 2x - x = 19 + 4 \: \: \: \: \: \: \: \: \: \\ \\ \\ { : \implies} \tt \: x = 23 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

❶ hence the numerator of the fraction is 23

• The denominator is 23 + 12

❷ hence , the denominator is 35

Verification:

• Now , let's substitute the values and performe the given condition check weather they equal 1/2

$\longrightarrow \tt \: \frac{23 - 2}{35 + 9} \\ \\ \\ \longrightarrow \tt \: \cancel\frac{22}{44} \: \: \: \: \: \: \: \\ \\ \\ \longrightarrow \tt \: \frac{1}{2} \: \: \: \: \: \: \: \: \:$

${ \blue {\huge{ \sf{\underline{ hence \: verified \dag}}}}}$

Answer 2

Answer:

Given:

The denominator of a fraction is greater than its numerator by 12.

the numerator is decreased by 2 and the denominator is increased by 7.

The new fraction is equivalent to ½

\begin{gathered} \\ \\ \end{gathered}

To find:

The original fraction

\begin{gathered} \\ \\ \end{gathered}

Solution:

Now,

Let the numerator be x

➷As the numerator greater that the denominator by 12

let the denominator be x + 12

hence the fraction is,

\longrightarrow \tt \: \dfrac{x}{x + 12}⟶

x+12

x

» Now , as the number is decreased by 2

the new numerator is x - 2

» And as the denominator is increased by 7

so, the denominator is x + 12 +7

x +19

From the given condition:

\begin{gathered}{ : \implies} \tt \frac{x - 2}{x +19 } = \frac{1}{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ { : \implies} \tt \: 2(x - 2) = 1(x + 19) \\ \\ \\ { : \implies} \tt \: 2x - 4 = x + 19 \: \: \: \: \: \: \: \: \\ \\ \\{ : \implies} \tt \: 2x - x = 19 + 4 \: \: \: \: \: \: \: \: \: \\ \\ \\ { : \implies} \tt \: x = 23 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \end{gathered}

:⟹

x+19

x−2

=

2

1

:⟹2(x−2)=1(x+19)

:⟹2x−4=x+19

:⟹2x−x=19+4

:⟹x=23

❶ hence the numerator of the fraction is 23

\begin{gathered} \\ \end{gathered}

The denominator is 23 + 12

\begin{gathered} \\ \end{gathered}

❷ hence , the denominator is 35

\begin{gathered} \\ \end{gathered}

Verification:

Now , let's substitute the values and performe the given condition check weather they equal 1/2

\begin{gathered} \longrightarrow \tt \: \frac{23 - 2}{35 + 9} \\ \\ \\ \longrightarrow \tt \: \cancel\frac{22}{44} \: \: \: \: \: \: \: \\ \\ \\ \longrightarrow \tt \: \frac{1}{2} \: \: \: \: \: \: \: \: \: \end{gathered}

⟶

35+9

23−2

⟶

44

22

⟶

2

1