Pair of Linear Equations in Two Variables

If 1 is added to both of the numerator and denominator of a fraction, it becomes \frac{4}{5}. If however, 5 is subtracted from both numerator and denominator, the fraction becomes \frac{1}{2}. Find the fraction.

Solution: Suppose the required fraction be $\frac{x}{y}$. Therefore, we have: $\begin{array}{l} \frac{x+1}{y+1}=\frac{4}{5} \\ \Rightarrow 5(x+1)=4(y+1) \\ \Rightarrow 5 x+5=4 y+4 \\ \Rightarrow 5...

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5 years hence, the age of a man shall be 3 times the age of his son while 5 years earlier the age of the man was 7 times the age of his son. The present age of the man is
(a) 45 years
(b) 50 years
(c) 47 years
(d) 40 years

Answer: (d) 40 years Solution: Suppose the present age of the man be $\mathrm{x}$ years. And his son's present age be $y$ years. 5 years later: $\begin{array}{l} (x+5)=3(y+5) \\ \Rightarrow x+5=3...

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A lending library has fixed charge for the first three days and an additional charge for each day thereafter. Mona paid { }^{2} 27 for a book kept for 7 days, while Tanvy paid \gtrless 21 for the book she kept for 5 days find the fixed charge and the charge for each extra day.

Solution: Suppose that the fixed charge be Rs.$x$ and the charge for each extra day be Rs.$y$. In case of Mona, according to the question $x+4 y=27\dots \dots(i)$ In case of Tanvy, according to the...

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On selling a tea-set at 5 \% loss and a lemon-set at 15 \% gain, a shopkeeper gains Rs. 7 . However, if he sells the tea-set at 5 \% gain and the lemon-set at 10 \% gain, he gains Rs. 14 . Find the price of the tea-set and that of the lemon-set paid by the shopkeeper.

Solution: Suppose that the actual price of the tea and lemon set be Rs.$x$ and Rs.$y$ respectively. When gain is Rs.7, then $\begin{array}{l} \frac{y}{100} \times 15-\frac{x}{100} \times 5=7 \\...

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The present age of a woman is 3 years more than three times the age of her daughter. Three years hence, the woman’s age will be 10 years more than twice the age of her daughter. Find their present ages.

Solution: Suppose the woman's present age be $\mathrm{x}$ years. and her daughter's present age be $y$ years. Therefore, we have: $\begin{array}{l} x=3 y+3 \\ \Rightarrow x-3 y=3\dots \dots(i)...

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If twice the son’s age in years is added to the mother’s age, the sum is 70 years. But, if twice the mother’s age is added to the son’s age, the sum is 95 years. Find the age of the mother and that of the son.

Solution: Suppose the mother's present age be $\mathrm{x}$ years. and her son's present age be $y$ years. Therefore, we have: $x+2 y=70\dots \dots(i)$ And, $2 x+y=95\dots \dots(ii)$ On multiplying...

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A railway half ticket costs half the full fare and the reservation charge is the some on half ticket as on full ticket. One reserved first class ticket from Mumbai to Delhi costs โ‚น4150 while one full and one half reserved first class ticket cost โ‚น 6255. What is the basic first class full fare and what is the reservation charge?

Solution: Let us suppose the basic first class full fare be Rs.$x$ and the reservation charge be Rs.$y$. Case 1: One reservation first class full ticket cost Rs.4,150 $x+y=4150\dots \dots(i)$ Case...

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The area of a rectangle gets reduced by 67 square meters, when its length is increased by 3 \mathrm{~m} and the breadth is decreased by 4 \mathrm{~m}. If the length is reduced by 1 \mathrm{~m} and breadth is increased by 4 \mathrm{~m}, the area is increased by 89 square meters, Find the dimension of the rectangle.

Solution: Let us suppose the length and the breadth of the rectangle be $x \mathrm{~m}$ and $\mathrm{y} \mathrm{m}$, respectively. Case 1: When length is increased by $3 \mathrm{~m}$ and the breadth...

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The area of a rectangle gets reduced by 8 \mathrm{~m}^{2}, when its length is rcduccd by 5 \mathrm{~m} and its breadth is increased by 3 \mathrm{~m}. If we increase the length by 3 \mathrm{~m} and breadth by 2 \mathrm{~m}, the area is increased by 74 \mathrm{~m}^{2}. Find the length and the breadth of the rectangle.

Solution: Let us suppose the length and the breadth of the rectangle be $\mathrm{x} \mathrm{m}$ and $\mathrm{y} \mathrm{m}$, respectively. $\therefore$ Area of the rectangle $=(x y)$ sq.m Case 1:...

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The length of a room exceeds its breadth by 3 meters. If the length is increased by 3 meters and the breadth is decreased by 2 meters, the area remains the same. Find the length and the breadth of the room.

Solution: Let us suppose the length of the room be $\mathrm{x}$ meters and he breadth of the room be y meters. Therefore, we have: Area of the room $=x y$ As per the question, we have: $x=y+3$...

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2 men and 5 boys can finish a piece of work in 4 days, while 3 men and 6 boys can finish it in 3 days. Find the time taken by one man alone to finish the work and that taken by one boy alone to finish the work.

Solution: Let one man alone can finish the work in $\mathrm{x}$ days and one boy alone can finish it in $y$ days. $\therefore$ One man's one day's work $=\frac{1}{x}$ And, one boy's one day's work...

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Places A and B are 160 km apart on a highway. A car starts from A and another car starts from B simultaneously. If they travel in the same direction, they meet in 8 hours. But, if they travel towards each other, they meet in 2 hours. Find the speed of each car.

Solution: Let us suppose the speed of the car A and B be x $\mathrm{km} / \mathrm{h}$ and y $\mathrm{km} / \mathrm{h}$ respectively. Let $\mathrm{x}>\mathrm{y}$. Case-1: When they travel in the...

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Abdul travelled 300km by train and 200km by taxi taking 5 hours and 30 minutes. But, if he travels 260 km by train and 240 km by he takes 6 minutes longer. Find the speed A \quad N taxi, of the train and that of taxi.

Solution: Let us suppose that the speed of the train and taxi be $\mathrm{x} \mathrm{km} / \mathrm{h}$ and $\mathrm{y} \mathrm{km} / \mathrm{h}$ respectively. Then according to the question...

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A train covered a certain distance at a uniform speed. If the train had been 5 \mathrm{kmph} faster, it would have taken 3 hours less than the scheduled time. And, if the train were slower by 4 \mathrm{kmph}, it would have taken 3 hours more than the scheduled time. Find the length of the journey.

Solution: Let us suppose that the original speed be $\mathrm{x} \mathrm{kmph}$ and let the time taken to complete the journey be $y$ hours. $\therefore$ Length of the whole journey $=(\mathrm{xy})...

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Points A and B are 70 \mathrm{~km} apart on a highway. A car starts from A and another car starts from B simultaneously. If they travel in the same direction, they meet in 7 hours. But, if they travel towards each other, they meet in 1 hour. Find the speed of each car.

Solution: Let us suppose that $\mathrm{X}$ and $\mathrm{Y}$ be the cars starting from points $\mathrm{A}$ and $\mathrm{B}$, respectively and let their speeds be $\mathrm{x}$ $\mathrm{km} /...

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A man sold a chair and a table together for Rs. 1520 , thereby making a profit of 25 \% on chair and 10 \% on table. By selling them together for Rs. 1535, he would have made a profit of 10 \% on the chair and 25 \% on the table. Find the cost price of each.

Solution: Let us suppose the cost price of the chair and table be Rs.x and Rs.y respectively. Then according to the question The selling price of chair $+$ Selling price of table $=1520$...

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A man invested an amount at 10 \% per annum simple interest and another amount at 10 \% per annum simple interest. He received an annual interest of Rs. 1350 . But, if he had interchanged the amounts invested, he would have received Rs. 45 less. What amounts did he invest at different rates?

Solution: Let us suppose that the amounts invested at $10 \%$ and $8 \%$ be Rs.x and Rs.y respectively. Therefore $\begin{array}{l} \frac{x \times 10 \times 1}{100}=\frac{y \times 8 \times...

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A part of monthly hostel charges in a college are fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 25 days, he has to pay Rs. 4550 as hostel charges whereas a student \mathrm{B}, who takes food for 30 days, pays Rs. 5200 as hostel charges. Find the fixed charges and the cost of the food per day.

Solution: Let us suppose the fixed charges be Rs.x and the cost of food per day be Rs.y. Therefore $\begin{array}{l} x+25 y=4500\dots \dots(i) \\ x+30 y=5200\dots \dots(ii) \end{array}$ Subtracting...

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Taxi charges in a city consist of fixed charges per day and the remaining depending upon the distance travelled in kilometers. If a person travels 80 \mathrm{~km}, he pays Rs. 1330 , and for travelling 90 \mathrm{~km}, he pays Rs. 1490 . Find the fixed charges per day and the rate per km.

Solution: Let us suppose the fixed charges be Rs.x and rate per km be Rs.y. Therefore $\begin{array}{l} x+80 y=1330\dots \dots(i) \\ x+90 y=1490\dots \dots(ii) \end{array}$ Subtracting equation(i)...

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There are two classrooms A and B. If 10 students are sent from A to B, the number of students in each room becomes the same. If 20 students are sent from B to A, the number of students in A becomes double the number of students in B. Find the number of students in each room.

Solution: Let us suppose the no. of students in classroom $\mathrm{A}$ be $\mathrm{x}$ Let's suppose the no. of students in classroom B be y. If 10 students are transferred from A to B, then we...

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Find a fraction which becomes \left(\frac{1}{2}\right) when 1 is subtracted from the numerator and 2 is added to the denominator, and the fraction becomes \left(\frac{1}{3}\right) when 7 is subtracted from the numerator and 2 is subtracted from the denominator.

Solution: Let us suppose the required fraction be $\frac{x}{y}$. Therefore, we have: $\begin{array}{l} \frac{x-1}{y+2}=\frac{1}{2} \\ \Rightarrow 2(x-1)=1(y+2) \\ \Rightarrow 2 x-2=y+2 \\...

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If three times the larger of two numbers is divided by the smaller, we get 4 as the quotient and 8 as the remainder. If five times the smaller is divided by the larger, we get 3 as the quotient and 5 as the remainder. Find the numbers.

Solution: It is known that: Dividend $=$ Divisor $\times$ Quotient $+$ Remainder Let's assume the larger no. be $\mathrm{x}$ and the smaller be y. Then, we have: $\begin{array}{l} 3 x=y \times 4+8...

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