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
if we only add 1 to the denominator. What is the fraction?
Solution:
Let numerator
Let denominator
According to given condition, we have
and
And,
(1) And,
(2)
So, we have equations (1) and (2), multiplying equation (1) by 2 we get (3)
(3)
(2)
Subtracting equation (2) from (3), we get
Putting value of y in (1), we get
Therefore, 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?
Solution:
Let present age of Nuri
years
Let present age of Sonu
years
5 years ago, age of Nuri = (x-5) years
5 years ago, age of Sonu = (y-5) years
According to given condition, we have
(1)
10 years later from present, age of Nuri
years
10 years later from present, age of Sonu
years
According to given condition, we have
(2)
Subtracting equation (1) from (2), we get
years
Putting value of y in (1), we get
years
Therefore, present age of Nuri
years
And, present age of Sonu
years
(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.
Solution:
Let digit at ten’s place
Let digit at one’s place
According to given condition, we have
(1)
And,
(2)
Adding (1) and (2), we get
Putting value of x in (1), we get
Therefore, 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.
Solution:
Let number of Rs 100 notes
Let number of Rs 50 notes
According to given conditions, we have
(1)
And,
(2)
Subtracting (2) from (1), we get
Putting value of x in (1), we get
Therefore, number of Rs 100 notes
Number of Rs 50 notes
(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.
Solution:
Let fixed charge for 3 days = Rs
Let additional charge for each day thereafter = Rs
According to given condition, we have
(1)
(2)
Subtracting (2) from (1), we get
Rs 3
Putting value of y in (1), we get
Rs 15
Therefore, fixed charge for 3 days
Rs 15
Additional charge for each day after 3 days
Rs 3
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