Question

A railway half ticket costs half the full fare and reservation charge is the same on half ticket as on full ticket . Once reserved first class ticket from bombay to Pune costs rs.226 and one full and one half reserved first ticket costs rs. 354. What is the first class full fare and what is the reservation charge?​

Answer 1

Answer :-

$\:\\\large{\underbrace{\underline{\sf{Question's \;\;Analysis\;:-}}}}$

Here the concept of Linear Equations in Two Variables has been used. We can take the values of cost of fare and reservation cost as variables and find their values through constants.

Let's do it.

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★ Question :-

A railway half ticket costs half the full fare and reservation charge is the same on half ticket as on full ticket . Once reserved first class ticket from bombay to Pune costs rs.226 and one full and one half reserved first ticket costs rs. 354. What is the first class full fare and what is the reservation charge?

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★ Solution :-

Given,

» Cost of first class ticket fron Bombay to Pune = Rs. 226

» Cost of one full and one half reserved ticket of first class = Rs. 354

• Let the cost of first class fare be Rs. 'x'

• Let the charge of reservation be Rs. 'y'

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~ Case I :-

$\\\;\;\;\;\;\large{\sf{:\rightarrow\;\;\:Cost\;of\;first\;class\;fare\;+\;Charge\;of\;Reservation\;=\;\bf{Total\;cost\;first\;class}}}$

$\\\;\;\;\;\;\large{\sf{:\rightarrow\;\;\: x\;+\;y\;=\;\bf{226}}}$

$\\\;\;\;\;\;\large{\sf{:\rightarrow\;\;\: x\;=\;\bf{(226\;-\;y)}}}$

Let this be equation (i),

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~ Case II :-

➡ From above, full ticket cost = x

➡ And, half ticket cost = ½ × x

The reservation charge shall be same for both that is y.

So,

$\\\;\;\;\;\;\large{\sf{:\rightarrow\;\;\:Total\;cost\;of\;full\;ticket\;+\;Total\;Cost\;of\;Half\;Ticket\;=\;\bf{Given\;cost}}}$

$\\\;\;\;\;\;\large{\sf{:\rightarrow\;\;\: x\;+\;y\;+\;\dfrac{x}{2}\;+\;y\;=\;\bf{354}}}$

$\\\;\;\;\;\;\large{\sf{:\rightarrow\;\;\:\dfrac{2x\:+\:x}{2}\;+\;2y\;=\;\bf{354}}}$

$\\\;\;\;\;\;\large{\sf{:\rightarrow\;\;\:\dfrac{3x}{2}\;+\;2y\;=\;\bf{354}}}$

$\\\;\;\;\;\;\large{\sf{:\rightarrow\;\;\:\dfrac{3x\:+\:4y}{2}\;\;=\;\bf{354}}}$

$\\\;\;\;\;\;\large{\sf{:\rightarrow\;\;\:3x\:+\:4y\;\;=\;\bf{2\:\times\:354}}}$

$\\\;\;\;\;\;\large{\sf{:\rightarrow\;\;\:3x\:+\:4y\;\;=\;\bf{708}}}$

Let this be equation (ii),

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From equation (i) and (ii), we get

$\\\;\;\;\;\;\large{\sf{:\longrightarrow\;\;\:3(226\:-\:y)\:+\:4y\;\;=\;\bf{708}}}$

$\\\;\;\;\;\;\large{\sf{:\longrightarrow\;\;\:3(226\:-\:y)\:+\:4y\;\;=\;\bf{708}}}$

$\\\;\;\;\;\;\large{\sf{:\longrightarrow\;\;\:678\:-\:3y\:+\:4y\;\;=\;\bf{708}}}$

$\\\;\;\;\;\;\large{\sf{:\longrightarrow\;\;\:-\:3y\:+\:4y\;\;=\;\bf{708\;-\;678}}}$

$\\\;\;\;\;\;\large{\sf{:\longrightarrow\;\;\:y\;\;=\;\bf{30}}}$

$\\\:\large{\boxed{\tt{Reservation\;\;Charge\;\;of\;\;First\;\;Class\;=\;\bf{Rs.\;\;30}}}}$

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From equation (i) and value of y, we get,

$\\\;\;\;\;\;\large{\sf{:\Longrightarrow\;\;\: x\;=\;\bf{(226\;-\;y)}}}$

$\\\;\;\;\;\;\large{\sf{:\Longrightarrow\;\;\: x\;=\;\bf{(226\;-\;30)}}}$

$\\\;\;\;\;\;\large{\sf{:\Longrightarrow\;\;\: x\;=\;\bf{(196)}}}$

$\\\:\large{\boxed{\tt{Cost\;\;of\;\;Fare\;\;of\;\;First\;\;Class\;=\;\bf{Rs.\;\;196}}}}$

$\:\\\large{\underline{\underline{\rm{Thus,\;reservation\;charge\;is\;\boxed{\bf{Rs.\;30}}\;and\;cost\;of\;fare\;is\;\boxed{\bf{Rs.\;196}}}}}}$

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$\;\\\large{\overline{\underline{\sf{Confused\:?\;Don't\;worry\;let's\;verify\;it\;:-}}}}$

For verification, we need to simply apply the values we got into the equations we formed. Then,

~ Case I :-

$\\\qquad\sf{:\Rightarrow\;\;\: x\;+\;y\;=\;226}$

$\\\qquad\sf{:\Rightarrow\;\;\: 196\;+\;30\;=\;226}$

$\\\qquad\sf{:\Rightarrow\;\;\: Rs. \; 226\;=\;Rs.\;226}$

Clearly, LHS = RHS.

~ Case II :-

$\\\qquad\sf{:\Rightarrow\;\;\: x\;+\;y\;+\;\dfrac{x}{2}\;+\;y\;=\;354}$

$\\\qquad\sf{:\Rightarrow\;\;\: 196\;+\;30\;+\;\dfrac{196}{2}\;+\;30\;=\;354}$

$\\\qquad\sf{:\Rightarrow\;\;\: 196\;+\;30\;+\;98\;+\;30\;=\;354}$

$\\\qquad\sf{:\Rightarrow\;\;\: Rs. \; 354\;=\;Rs.\;354}$

Clearly, LHS = RHS.

Here both the conditions are satisfied, so our answer is correct.

Hence, Verified.

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★ More to know :-

• Linear Equations - are the equations formed using constant and variable term is of single degree.

• Linear Equations in One Variable
• Linear Equations in Two Variables
• Linear Equations in Three Variables

• Polynomials - are the mathematical expressions formed using constant and variable terms where the variable term can be of many degrees.

• Linear Polynomial
• Quadratic Polynomial
• Cubic Polynomial
• Bi - Quadratic Polynomial