Math, asked by yadavsuryakant519, 9 months ago



1. A two digit number becomes five-sixth of itself
when its digits are reversed. The two digits differ
by 1. The number is

2.The ratio of the present ages of two brothers is
1: 2 and 5 years back the ratio was 1: 3. What
will be the ratio of their ages after 5 year​

Answers

Answered by Anonymous
13

» Question (1).

A two digit number becomes five-sixth of itself

when its digits are reversed. The two digits differ

by 1. Then Find the two-digit number.

» Solution :

→ Taken :

Let the digits of the two-digit number be x and y.

  • Original number = (10a + b)

  • No. obtained on reversing the digits = 10(b + a)

→ Concept :

According to the question , it says that the Original number became ⅚th of itself when the digits are interchanged.

So the equation formed is :

\purple{\sf{\dfrac{5}{6}\left(10a + b\right) = 10b + a}}

→ Given :

  • Difference of the digits of the number is 1.i.e,

\purple{\sf{a - b = 1}} (Equation.2)

To Find the Equation 1 :

Given Equation :

\purple{\sf{\dfrac{5}{6}\left(10a + b)\right) = 10b + a}}

By solving it , we get :

\sf{\Rightarrow \dfrac{50a}{6} + \dfrac{5b}{6} = 10b + a}

\sf{\Rightarrow \dfrac{50a + 5b}{6} = 10b + a}

\sf{\Rightarrow 50a + 5b = 6(10b + a)}

\sf{\Rightarrow 50a + 5b = 60b + 6a}

\sf{\Rightarrow 50a + 5b - 60b - 6a = 0}

\sf{\Rightarrow 44a - 55b = 0}

Taking the common 11, we get :

\sf{\Rightarrow 11(4a - 5b) = 0}

\sf{\Rightarrow 4a - 5b = \dfrac{0}{11}}

[Anything divided by 0 is 0]

\sf{\Rightarrow 4a - 5b = 0}

\purple{\sf{\therefore 4a - 5b = 0}} (Equation.1)

→ Calculation :

Putting the Two Equations .i.e,(i) and (ii), we get :

\:\:\:\:\:\:\:\:\sf{4a - 5b = 0} \times 1

\:\:\:\:\:\:\:\:\sf{a - b = 1}\times 5

\:\:\:\:\:__________________

\:\:\:\:\:\:\:\:\sf{4a - \cancel{5b} = 0}

\:\:\:\:\:\:\:\:\sf{5a - \cancel{5b} = 5}

\:\:\:\:\:__________________[By Subtracting]

\:\:\:\:\:\:\:\:\sf{\not{-}a = \not{-}5}

Hence ,the value of a is 5.

Putting the value of a in Equation 1 , we get :

\sf{4a - 5b = 0}

\sf{\Rightarrow 4 \times 5 - 5b = 0}

\sf{\Rightarrow 20 - 5b = 0}

\sf{\Rightarrow - 5b = -20}

\sf{\Rightarrow b = \dfrac{-20}{-5}}

\sf{\Rightarrow b = \dfrac{\cancel{-20}}{\cancel{-5}}}

\purple{\sf{\Rightarrow b = 4}}

Hence ,the value of b is 4.

The original number :

  • a = 5
  • b = 4

Putting the value of a and b in the Equation , (10a + b) , we Get :

10a + b

\sf{\Rightarrow 10 \times 5 + 4}

\sf{\Rightarrow 50 + 4}

\purple{\sf{\Rightarrow 54}}

So, the two-digit number is 54.

 \\

» Question (2) :

The ratio of the present ages of two brothers is

1: 2 and 5 years back the ratio was 1: 3. What

will be the ratio of their ages after 5 years .

» Solution :

→ We Know :

Let the present ages of the two brothers be x and 2x.

Ratio before 5 years is 1 : 3.

So the ages of the two brothers before 5 years with respect to present age will be (x - 5) and (2x - 5)

→ Concept :

According to the question , The Ratio of ages of two brothers before 5 years will be Equal to Ratio before 5 years with respect to the present ages of two brothers.

So the equation formed is :

\purple{\sf{\dfrac{x - 5}{2x - 5} = \dfrac{1}{3}}}

By solving this Equation , we will get the Present age of two brothers.

→ Calculation :

Given Equation :

\purple{\sf{\dfrac{x - 5}{2x - 5} = \dfrac{1}{3}}}

By solving cross multiplication , we get :

\sf{\Rightarrow (x - 5)(3) = (2x - 5)}

\sf{\Rightarrow 3x - 15 = 2x - 5}

\sf{\Rightarrow 3x - 2x = 15 - 5}

\sf{\Rightarrow x = 10}

\purple{\sf{\therefore x = 10}}

Hence ,the present age of one brother is 10 and the present age of another brother is 2x. So his present age is 20 years.

Ratio of Age after 5 years.

  • Age of first brother after 5 years =

➝ Present age + 5 years

➝ 10 + 5 = 15 years.

  • Age of second brother after 5 years =

➝ Present age + 5 years

➝ 20 + 5 = 25 years

Ratio :

Age of first brother after 5 years : Age of second brother after 5 years.

\sf{\Rightarrow 15 : 25}

\sf{\Rightarrow 3 : 5}

\purple{\sf{3 : 5}}

Hence , the ratio of their ages after 5 years is 3 : 5.

» Additional information :

  • Profit Percentage =

\sf{P\% = \left(\dfrac{P}{CP} \times 100\right)\%}

  • Loss Percentage =

\sf{P\% = \left(\dfrac{L}{CP} \times 100\right)\%}

  • Selling Price =

\sf{SP = \left(1 \pm \dfrac{P/L}{100}\right) \times CP}

Answered by Anonymous
5

_________________________

 \tt 1)let \: the \: digits \: of \: double \: digit  \\  \tt be \:  x \: and \: y

 \tt let \: no. \: be  = 10a+b=

A.T.Q

 \tt 10b + a =  \frac{5}{6} (10a + b)

 \tt  =  > 10b + a =  \frac{50} {6}  + \frac{5b}{6}

 \tt  =  > 10b + a =  \frac{50a + 5b}  {6}

 \tt  =  > 6(10b + a )=  50a + 5b

 \tt  =  > 60b + 6a =  50a + 5b

 \tt  =  > 60b  - 5b =  50a  - 6a

 \tt  =  > 55b =  44a

 \tt  =  > 55b  -   44a

 \tt  =  > 11(55b  -   44a)

 \tt  =  > 5b  -   4a = 0...(i)

now,

 \tt 10b + a =  \frac{5}{6} (10a + b) ,nd \: (a - 1) = b

 \tt 5(10a+ a  - 1)=  6 (10a  - 10  + a)

 \tt 50a+ 5a  - 5=  6 0a  - 60  + 6a

 \tt55a - 5 = 66a - 60

 \tt  - 11a=  - 55

 \tt a =  \frac{  \cancel-  \cancel{ 55}}{  \cancel  -  \cancel{ 11} }

 \tt a =  5

 \tt substitute \: value \: of \: a \: in eq(i)

 \tt 5b - 4a = 0

 \tt 5b - 4(5) = 0

 \tt 5b - 20 = 0

 \tt 5b  = 20

 \tt b  = \frac{ \cancel{ 20}}{ \cancel {5} } = 4

_________________________

2) 35:25 or 7:5

EXPLANATION:-

(see in attachment)

_____________________

Attachments:
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