Question

Customers are asked to stand in the lines. If one customer is extra in a line, then there would be two less lines. If one customer is less in line, there would be three more lines. Find the number of students in the class.

Answer 1

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$\LARGE\textsf{\underline\textcolor{aqua}{⇝ QuEsTiOn :-}}$

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Students are asked to stand in the lines. If one student is extra in a line, then there would be two less lines. If one student is less in line, there would be three more lines. Find the number of students in the class.

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$\LARGE\textsf{\underline\textcolor{aqua}{✯\; AnSwEr :-}}$

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$\boxed{\large\textsf\textcolor{orange}{Total number of students = 60 students}}$

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$\LARGE\textsf{\underline\textcolor{aqua}{✯\; SoLuTiOn :-}}$

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• Let the number of students in a line be ' x '

• Let the number of line's be ' y '

• So , the total number of students will be ' xy '

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↭ According to the first condition

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$\qquad\tt{:}\longrightarrow\large\textsf{( x + 1 )( y - 2 ) = xy}$

$\qquad\tt{:}\longrightarrow\large\textsf{x ( y - 2 ) + 1 ( y - 2 ) = xy}$

$\qquad\tt{:}\longrightarrow\large\textsf{xy - 2x + y - 2 = xy}$

$\qquad\tt{:}\longrightarrow\boxed{\large\textsf\textcolor{purple}{- 2x + y = 2 \; ------ ( i )}}$

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↭ According to the second condition

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$\qquad\tt{:}\longrightarrow\large\textsf{( x - 1 ) ( y + 3 ) = xy}$

$\qquad\tt{:}\longrightarrow\large\textsf{x ( y + 3 ) - 1 ( y + 3 ) = xy}$

$\qquad\tt{:}\longrightarrow\large\textsf{xy + 3x - y - 3 = xy}$

$\qquad\tt{:}\longrightarrow\boxed{\large\textsf\textcolor{purple}{3x - y = 3 \; ------ ( ii )}}$

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↭ Adding eq. ( i ) and ( ii ) :-

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$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\: \: \: \: \: \: \large\textsf{- 2x \: + \: y \: = \: 2} \\ \large\textsf{( \: + \: )} \: \: \: \: \; \; \: \: \: \: \: \: \: \large\textsf{3x \: - \: y \: = \: 3} \\ \: \: \large\textsf{-----------------------------} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{\large\textsf\textcolor{red}{x \: = \: 5 }}$

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↭ Substituting the value of x in ( i )

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$\qquad\tt{:}\longrightarrow\large\textsf{- 2x + y = 2}$

$\qquad\tt{:}\longrightarrow\large\textsf{- 2 × 5 + y = 2}$

$\qquad\tt{:}\longrightarrow\large\textsf{- 10 + y = 2}$

$\qquad\tt{:}\longrightarrow\large\textsf{y = 2 + 10}$

$\qquad\tt{:}\longrightarrow\boxed{\large\textsf\textcolor{red}{y = 12}}$

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$\large\textsf\textcolor{orange}{∴ Total number of students = 5 × 12 = 60}$