Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method:
(i) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes ½ if we only add 1 to the denominator. What is the fraction?
(ii)Five years ago, Nuri was thrice as old as sonu. Ten years later, Nuri will be twice as old as sonu. How old are Nuri and Sonu?
(iii)The sum of the digits of a two–digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.
(iv) Meena went to a bank to withdraw Rs 2000. She asked the cashier to give her Rs 50 and Rs 100 notes only. Meena got 25 notes in all. Find how many notes of Rs 50 and Rs 100 she received.
(v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs 27 for a book kept for seven days, while Susy paid Rs 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.

Ans.(i)Let numerator =x and let denominator =y

According to given condition,

=>\({{x + 1}\over{y - 1}} = 1\) and \({{x}\over{y + 1}} = {{1}\over{2}}\)
=>\( x + 1 = y – 1\) and \(2x = y + 1\)
=>\( x – y = -2 \)….......(1) and,
\( 2x – y = 1\)…...... (2)

So, we have equations (1) and (2), multiplying equation (1) by 2 we get (3)

\(2x - 2y = -4\)… (3)
\(2x – y = 1\)… (2)

Subtracting equation (2) from (3), we get
\(-y = -5 => y = 5\)

Putting value of y in (1), we get
\(x – 5 = -2 => x = -2 + 5 = 3\)

Therefore, fraction = \({{x}\over{y}} = {{3}\over{5}}\)

(ii) Let present age of Nuri = x years and let present age of Sonu = y years
5 years ago, age of Nuri = \((x – 5)\) years
5 years ago, age of Sonu = \((y – 5)\) years

According to given condition,
\((x - 5) = 3 (y - 5)\)
=> \(x – 5 = 3y – 15\)
=> \(x - 3y = -10\)… (1)

10 years later from present, age of Nuri = \((x + 10)\) years

10 years later from present, age of Sonu = \((y + 10)\) years

According to given condition,
\((x + 10) = 2 (y + 10)\)
=> \(x + 10 = 2y + 20\)
=> \(x - 2y = 10 … (2)\)

Subtracting equation (1) from (2),
\(y = 10 - (-10) = 20 \)years

Putting value of y in (1),
\(x – 3 (20) = -10\)
=> \(x – 60 = -10\)
=> \(x = 50 \)years

Therefore, present age of Nuri = 50 years and present age of Sonu = 20 years

(iii) Let digit at ten’s place = x and Let digit at one’s place = y

According to given condition,
\(x + y = 9 \)… (1)
And 9 (10x + y) = 2 (10y + x)
=>\( 90x + 9y = 20y + 2x\)
=>\( 88x = 11y\)
=>\( 8x = y\)
=>\( 8x – y = 0 \)… (2)

Adding (1) and (2),
\(9x = 9 => x = 1\)

Putting value of x in (1),
\(1 + y = 9\)
=>\( y = 9 – 1 = 8\)

Therefore, number = \(10x + y = 10 (1) + 8 = 10 + 8 = 18\)

(iv)Let number of Rs 100 notes = x and let number of Rs 50 notes = y

According to given conditions,
\(x + y = 25\) … (1)
and \(100x + 50y = 2000\)
=>\( 2x + y = 40\) … (2)

Subtracting (2) from (1),
\(-x = -15 => x = 15\)

Putting value of x in (1),
\(15 + y = 25\)
=>\( y = 25 – 15 = 10\)

Therefore, number of Rs 100 notes = 15 and number of Rs 50 notes = 10

(v) Let fixed charge for 3 days = Rs x

Let additional charge for each day thereafter = Rs y

According to given condition,
\(x + 4y = 27 \)… (1)
\(x + 2y = 21 \)… (2)

Subtracting (2) from (1),
\(2y = 6 => y = 3\)

Putting value of y in (1),
\(x + 4 (3) = 27\)
=>\( x = 27 – 12 = 15\)

Therefore, fixed charge for 3 days = Rs 15 and additional charge for each day after 3 days = Rs 3