Question

the denominator of rational number is greater the its numerator by8.if the numerator is i xreased by17 and the denominator is decreased by 1 the number obtained is 3/2 fjnd rational number​

Answer 1

GIVEN:

• The denominator of rational number is greater the its numerator by 8.
• The numerator is increased by 17 and the denominator is decreased by 1 the number obtained is 3/2

TO FIND:

• What is the fraction ?

SOLUTION:

Let the numerator be 'x' and Denominator be 'y'

• NUMERATOR = x
• DENOMINATOR = y
• FRACTION = x/y

CASE:- 1)

❖ The denominator of rational number is greater the its numerator by 8.

According to question:-

➸ y = x + 8.....❶

CASE:- 2)

❖ The numerator is increased by 17 and the denominator is decreased by 1 the number obtained is 3/2

According to question:-

➸ $\sf{\dfrac{x+17}{y-1} = \dfrac{3}{2}}$

Use cross product

➸ 2(x+17) = 3(y–1)

➸ 2x+34 = 3y –3

➸ 2x –3y = –3 –34

➸ 2x –3y = –37....❷

Put the value of 'y' from equation 1) in equation 2)

➸ 2x –3(x+8) = –37

➸ 2x –3x –24 = –37

➸ –x = –37 +24

➸ $\cancel{-}$x = $\cancel{-}$13

➸ ✫ x = 13 ✫

Put the value of 'x' in equation 1)

➸ y = 13 + 8

➸ ✫ y = 21 ✫

• NUMERATOR = x = 13
• DENOMINATOR = y = 21

$\large{\boxed{\bf{\star \: FRACTION = \dfrac{13}{21} \: \star}}}$

❝ Hence, the fraction formed is 13/21 ❞

______________________

Answer 2

QUESTION:-

✯ᴛʜᴇ ᴅᴇɴᴏᴍɪɴᴀᴛᴏʀ ᴏғ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ ɪs ɢʀᴇᴀᴛᴇʀ ᴛʜᴇ ɪᴛs ɴᴜᴍᴇʀᴀᴛᴏʀ ʙʏ8.ɪғ ᴛʜᴇ ɴᴜᴍᴇʀᴀᴛᴏʀ ɪs ɪ ncʀᴇᴀsᴇᴅ ʙʏ 17 ᴀɴᴅ ᴛʜᴇ ᴅᴇɴᴏᴍɪɴᴀᴛᴏʀ ɪs ᴅᴇᴄʀᴇᴀsᴇᴅ ʙʏ 1 ᴛʜᴇ ɴᴜᴍʙᴇʀ ᴏʙᴛᴀɪɴᴇᴅ ɪs 3/2 ғiɴᴅ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ

ANSWER✔

$\Large\bold\purple{GIVEN,}$

$\sf\dashrightarrow in\:a\:rational\:number\:the\:denominator\:>\:the\:numerator\:by\:8$

$\sf\dashrightarrow numerator\:is\:increased\:by\: 17$

$\sf\dashrightarrow denominator\:is\:decreased\:by\:1$

$\sf\dashrightarrow number\:obtained\:is\:\: \dfrac{3}{2}$

$\Large\underline\bold{TO\:FIND,}$

$\sf\star THE\:FRACTION$

$\Large\underline\bold{SOLUTION,}$

$\sf\therefore assuming\:the\:numerator\:as\:a$

$\sf\therefore assuming\:the\:denominator\:as\:b$

$\sf\therefore numerator:denominator= a:b$

$\sf\therefore \dfrac{numerator}{denominator}= \dfrac{a}{b}$

$\Large\underline\bold{\therefore A.T.Q....i.e.,.... according\:to\:question}$

$\sf\large\therefore taking\:two\:cases\:to\:understand\:better$

$\sf {\fbox {CASE:-1 }}$

$\sf\therefore in\:a\:rational\:number\:the\:denominator\:>\:the\:numerator\:by\:8$

THEREFORE,

$\sf\therefore b=a+8........eq^1$

$\large{\boxed{\sf{b=a+8.....eq^1}}}$

$\sf {\fbox {CASE:-2 }}$

$\sf\dashrightarrow numerator\:is\:increased\:by\: 17$

$\sf\dashrightarrow denominator\:is\:decreased\:by\:1$

$\sf\dashrightarrow number\:obtained\:is\:\: \dfrac{3}{2}$

NOW,

$\sf\implies {\dfrac{a+17}{b-1} = \dfrac{3}{2}}$

$\sf\implies 2 \times (a+17)= 3 \times (b-1)$

$\sf\implies 2a+34=3b-3$

$\sf\implies 2a-3b=(-3)-34$

$\sf\implies (-3b)=(-37)-2a.......eq^2$

$\large{\boxed{\sf{(-3b)=(-37)-2a.......eq^2}}}$

$\sf\large\therefore substituting\:the\:value\:of\:b\:in\:eq^2$

$\sf\implies [-3 \times (a+8)] = (-37)-2a$

$\sf\implies [(-3a)+(-24)] = (-37)-2a$

$\sf\implies (-3a)-24 = (-37)-2a$

$\sf\implies (-3a)+2a = (-37)+(24)$

$\sf\implies (-a)= (-13)$

$\sf\implies a=13$

$\sf {\fbox {a=13}}$

NOW,

$\sf\therefore in\:a\:rational\:number\:the\:denominator\:>\:the\:numerator\:by\:8$

$\sf\therefore substituting\:the\:value\:of\:a\:in\:eq^1$

$\sf\therefore b=a+8$

$\sf\implies b=13+8$

$\sf\therefore b=21$

$\sf {\fbox {b=21}}$

$\sf\therefore a:b=13:21$

$\sf\therefore \dfrac{numerator}{denominator}= \dfrac{13}{21}$

$\large{\boxed{\sf{\therefore fraction= \dfrac{13}{21}}}}$

THEREFORE,

$\sf\large\implies fraction\: required \:is\: \dfrac{13}{21}$

_____________________________

ADDITIONAL INFORMATION,

rational numbers,

• it is in the form of $\sf \frac{p}{q}$ where q≠0
• rational numbers can be expressed in the form of fraction
• it includes positive,negative and zeroes

_____________________________