Question

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 \(\frac{1}{2}\) 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 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.

Answer 1

(i) Let the fraction be \(\frac{x}{y}\). Given: \(\frac{ x+1}{y-1 }\)= 1 → x - y = -2 .............(1) \(\frac{x}{ y+1}\) =\(\frac{ 1}{2}\) → 2x - y = 1 ............(2) Subtracting (1) from (2): x = 3 ………….(3) Substituting x = 3 in (1): 3 - y = -2 -y = -5 y = 5 The fraction is \(\frac{3}{5}\).

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(ii) Let Nuri's present age be x years and Sonu's present age be y years. Given: (x - 5) = 3(y - 5)   x - 3y = -10.............(1) (x + 10) = 2(y + 10) x - 2y = 10...............(2) Subtracting (1) from (2): y = 20 .....................(3) Substituting y = 20 in (1): x - 60 = -10 x = 50 Nuri's age = 50 years Sonu's age = 20 years

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(iii) Let the unit digit be x and the tens digit be y. The number is 10y + x. The number after reversing the digits is 10x + y. Given: x + y = 9 …………………....(1) 9(10y + x) = 2(10x + y) 88y − 11x = 0 − x + 8y = 0 ………………….(2) Adding (1) and (2): 9y = 9 y = 1 ..........................(3) Substituting y = 1 in (1): x = 8 The number is 10(1) + 8 = 18.

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(iv) Let the number of Rs 50 notes be x and the number of Rs 100 notes be y. Given: x + y = 25 .............……....(1) 50x + 100y = 1250..…..(2) Multiplying (1) by 50: 50x + 50y = 1250....…...(3) Subtracting (3) from (2): 50y = 750 y = 15 Substituting y = 15 in (1): x = 10 Meena has 10 notes of Rs 50 and 15 notes of Rs 100.

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(v) Let the fixed charge for the first three days be Rs x and the charge for each subsequent day be Rs y. Given: x + 4y = 27 ............(1) x + 2y = 21 .........…(2) Subtracting (2) from (1): 2y = 6 y = 3 .....................(3) Substituting y = 3 in (1): x + 12 = 27 x = 15 Fixed charge = Rs 15 Charge per day = Rs 3