Math, asked by adityareddy752, 7 months ago

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​

Answers

Answered by ButterFliee
41

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

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Answered by Anonymous
17

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}

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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

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