Question

The difference of two numbers is 4 and the difference of their
reciprocals is 4 by 21. Find the numbers.
(CBSE 2008]​

Answer 1

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The difference of two numbers is 4 and the difference of their reciprocals is 4 by 21. Find the numbers...?

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✏One of the numbers is -3 or 7

✏Other number is -7 or 3 respectively...

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✳GIVEN...

• Difference of two no.s is 4
• Difference of their reciprocal = 4/21

✳TO FIND...

• The number...?

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$\large{ \sf{ \star{ \: \: let \: one \: of \: the \: no. \: is \: x}}} \\ \large{ \sf{ \star{ \: \: then \: other \: is \: x - 4}}} \\ \\ \large{ \sf{ \bullet{ \: \: it \: is \: also \: given \: that \: the \: difference \: of}}} \\ \large{ \sf{their \: reciprocal \: is \: \frac{4}{21} }}....... \\ \\ : \large{ \sf{ \implies{\: \frac{1}{x - 4} - \frac{1}{x } = \frac{4}{21} }}}\\ \sf{\dag{\purple{\underline{x-4\:<\:x}}}} \\ \\ : \large{ \sf{ \implies{ \frac{ \cancel{x} - \cancel{ x} + 4}{x(x - 4)} = \frac{4}{21} }}} \\ \\ : \large{ \sf{ \implies{ \frac { \cancel{4}}{ {x}^{2} - 4x } = \frac{ \cancel{4} }{21} }}} \\ \sf{ \dag{ \purple{ \underline{cross \: multiplying..}}}} \\ \\ : \large{ \sf{ \implies{ {x}^{2} - 4x = 21}}} \\ \\ : \large{ \sf{ \implies{ {x}^{2} - 4x - 21 = 0}}} \\ \sf{\dag{ \purple { \underline{by \: middle \: term \: factorisation...}}}}\\ \\ : \large{ \sf{ \implies{ {x}^{2} - 7x + 3x - 21 = 0}}} \\ \\ : \large{ \sf{ \implies{x(x - 7) + 3(x - 7)}}} \\ \\ : \large{ \sf{ \implies{(x + 3)(x - 7) = 0}}} \\ \\ \large{ \sf{ \implies{ \red{ \boxed{x = - 3 \: or \: 7}}}}} \\ \\ \large{ \sf{ \star{ \: \: then \: x - 4 = - 7 \: or \: 3\: respectively...}}}$

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How to solve word problems...

☯️Step I: Denote the unknown quantities by x, y etc.

☯️Step II: use the conditions of the problem to establish in unknown quantities.

☯️Step III: Use the equations to establish one quadratic equation in one unknown.

☯️Step IV: Solve this equation to obtain the value of the unknown in the set to which it belongs.