- Trending Categories
- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- MS Excel
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP
- Physics
- Chemistry
- Biology
- Mathematics
- English
- Economics
- Psychology
- Social Studies
- Fashion Studies
- Legal Studies

- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who

# 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 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.

To do:

We have to form the pair of linear equations and find the solutions by the elimination method.

Solution:

(i) Let the numerator and denominator of the original fraction be $x$ and $y$ respectively.

The original fraction$=\frac{x}{y}$

The fraction becomes $1$ if 1 is added to the numerator and 1 is subtracted from the denominator.

This implies,

New fraction$=\frac{x+1}{y-1}$

According to the question,

$\frac{x+1}{y-1}=1$

$x+1=1(y-1)$ (On cross multiplication)

$x+1=y-1$

$y=x+1+1$

$y=x+2$.....(i)

When 1 is added to only the denominator, it becomes $\frac{1}{2}$.

This implies,

$\frac{x}{y+1}=\frac{1}{2}$

$2(x)=1(y+1)$ (On cross multiplication)

$2x=y+1$

$2x-y-1=0$

$2x-(x+2)-1=0$ (From (i))

$2x-x-2-1=0$

$x=3$

$\Rightarrow y=x+2$

$y=3+2$

$y=5$

Therefore, the original fraction is $\frac{3}{5}$.

(ii) Let the ages of Nuri and Sonu be $x$ and $y$ respectively.

This implies,

Age of Nuri 5 years ago $= x-5$ years.

Age of Sonu 5 years ago $= y-5$ years.

Age of Nuri after 10 years $= x+10$ years.

Age of Sonu after 10 years $= y+10$ years.

According to the question,

$x+10=2(y+10)$

$x+10=2y+20$

$x=2y+20-10$

$x=2y+10$.....(i)

$x-5=3(y-5)$

$x-5=3y-15$

$3y=(2y+10)+15-5$ (From (i))

$3y=2y+10+10$

$3y-2y=20$

$y=20$

$\Rightarrow x=2(20)+10=40+10=50$

The present ages of Nuri and Sonu are 50 years and 20 years respectively.

(iii) Let the two-digit number be $10x+y$.

$x + y = 9$

$x=9-y$.....(i)

The number formed on reversing the digits is $10y+x$.

Therefore,

$9(10x+y) = 2(10y+x)$

$90x+9y=20y+2x$

$90x-2x+9y-20y=0$

$88x-11y=0$

$11(8x-y) = 0$

$8x-y = 0$

$y=8x$

Substituting $y = 8x$ in equation (i), we get,

$x =9-8x $

$x+8x = 9$

$9x = 9$

$x=1$

This implies,

$y = 8x = 8(1)=8$

The original number is $10(1)+8 = 10+8 = 18$.

The original number is 18.

(iv) Let the number of Rs. 50 notes be $x$ and the number of Rs. 100 notes be $y$.

The total number of notes $= 25$

This implies,

$x + y = 25$

$x = 25-y$....(i)

Total amount only in Rs. 50 notes $= 50x$

Total amount only in Rs. 100 notes $=100y$

Total amount $=Rs.\ 2000$

$50x + 100y = 2000$

$50(25-y) + 100y = 2000$ (From (i))

$50\times25 - 50y + 100y = 2000$

$1250 +50 y = 2000$

$50y = 2000-1250 = 750$

$y = \frac{750}{50} = 15$

$ x = 25-15 = 10$ (From (i))

Therefore, she received $10$ Rs. 50 notes and $15$ Rs. 100 notes.

(v) Let the fixed charge for the first three days and the additional charge for each day thereafter be $x$ and $y$ respectively.

According to the question,

$x + 4y = 27$.....(i)

$x + 2y = 21$.....(ii)

Subtracting equation (ii) from equation (i), we get,

$(x+4y)-(x+2y)=27-21$

$x-x+4y-2y=6$

$2y=6$

$y=\frac{6}{2}$

$y=3$

Substituting $y=3$ in equation (ii), we get,

$x+2(3)=21$

$x+6=21$

$x=21-6$

$x=15$

The fixed charge is Rs. 15 and the charge for each extra day is Rs. 3.

- Related Articles
- 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.
- 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.
- A shopkeeper gives books on rent for reading. She takes a fixed charge for the first two days, and an additional charge for each day thereafter. Latika paid Rs. 22 for a book kept for 6 days, while Anand paid Rs. 16 for the book kept for four days. Find the fixed charges and charge for each extra day.
- 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?
- A lending library has a fixed charges for the first three days and an additional charge for each day thereafter. Aarushi paid Rs. 27 for a book kept for seven days. If fixed charges are Rs. x and per day charges are Rs. y. Write the linear equation representing the above information.
- Twinkle has a total of Rs. 590 as currency notes in the denominations of Rs. 50, Rs. 20 and Rs. 10. The ratio of the number of Rs. 50 notes and Rs. 20 notes is \( 3: 5 \). If she has a total of 25 notes, how many Rs. 20 notes does she have?
- A lady went to a bank with a cheque of Rs. 100000. She asked the cashier to give her Rs. 500 and Rs. 2000 currency notes in return. She got 125 currency notes in all. Find the number of each kind of currency notes.
- Lakshmi is a cashier in a bank. She has currency notes of denominations Rs. \( 100, Rs.50 \) and \(Rs. 10 \), respectively. The ratio of the number of these notes is 2:3:5. The total cash with Lakshmi is \( Rs. 4,00,000 \). How many notes of each denomination does she have?
- Total value of notes is 2000, number of notes is 50. How many Rs. 50 and Rs. 100 notes are received?
- If we add 1 to the numerator and subtract 1 from the denominator, a fraction becomes 1. It also becomes $\frac{1}{2}$ if we only add 1 to the denominator. What is the fraction?
- Taxi service charge Rs. 8 for kilometre and levies a fixed charge of Rs. 50. Write an algebraic expression for the above situation if the taxi is hired from x kilometre.
- The car hire charges in a city comprise of a fixed charges together with the charge for the distance covered. For a journey of 12 km, the charge paid is Rs. 89 and for a journey of 20 km, the charge paid is Rs. 145. What will a person have to pay for travelling a distance of 30 km?
- The sum of numerator and denominator of a fraction is 3 less than twice the denominator. If each of the numerator and denominator is decreased by 1, the fraction becomes $\displaystyle \frac{1}{2}$, find the fraction.
- A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows:Rs. 200 for the first day, Rs. 250 for the second day, Rs. 300 for the third day, etc. the penalty for each succeeding day being Rs. 50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days?
- Ten years later, A will be twice as old as B and five years ago, A was three times as old as B. What are the present ages of A and B?