Question

A fraction becomes 1/3 when 2 is subtracted from numerator and it becomes 1/2 when 1 is subtracted from denominator. Find the fraction.
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Answer 1

Answer :-

$\: \large{\underline{\underline{\sf{Firstly,\; let's \; understand\; the \; Concept \; Used \; :-}}}}$

Here the concept of Linear Equations in Two Variables has been used. According to this, we will take the numerator and denominator as variables and find them using constants.

Let's do !!

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★ Question :-

A fraction becomes 1/3 when 2 is subtracted from numerator and it becomes 1/2 when 1 is subtracted from denominator. Find the fraction.

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★ Solution :-

Given,

» The fraction is ⅓ = Numerator - 2

» The fraction is ½ = Denominator - 1

• Let the numerator of the fraction be 'x'

• Let the denominator of the fraction be 'y'

Then using variables,

$\: \displaystyle{\sf{The \; Fraction \; is \; = \; \bf{\dfrac{x}{y}}}}$

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~ For the value of x and y :-

=> Case I :-

$\: \qquad \large{\sf{:\Longrightarrow \;\;\;\dfrac{\: (x \; - \; 2)}{\: y} \; \; = \; \; \bf{\dfrac{1}{3}}}}$

By cross multiplication, we get,

✒ 3(x - 2) = y

✒ y = 3x - 6 ... (i)

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=> Case II :-

$\: \qquad \large{\sf{:\Longrightarrow \;\;\;\dfrac{\: x}{\: (y\;-\;1)} \; \; = \; \; \bf{\dfrac{1}{2}}}}$

By cross multiplication, we get,

✒ 2x = y - 1

✒ y = 2x + 1 ... (ii)

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

✒ 3x - 6 = 2x + 1

✒ 3x - 2x = 1 + 6

✒ x = 7

$\: \large{\boxed{\boxed{\tt{Hence,\;\:value\;\:of\:\;numerator,\:\; x \:\; = \:\; \bf{7}}}}}$

By using equation (ii) and value of x, we get

✒ y = 2x + 1

✒ y = 2(7) + 1 = 14 + 1

✒ y = 15

$\: \large{\boxed{\boxed{\tt{Hence,\;\:value\;\:of\:\;denominator,\:\; y \:\; = \:\; \bf{15}}}}}$

$\; \qquad \large{\sf{:\Longrightarrow\;\;\; Thus, \; the \; fraction \; is \; \dfrac{7}{15}}}$

$\: \large{\underline{\underline{\rm{Thus,\; the\; fraction\; is \; \boxed{\bf{\dfrac{7}{15}}}}}}}$

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$\: \large{\underline{\sf{Confused?,\; Don't\; Worry \; Let's \; Start\; It}}}$

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

~ Case I :-

$\: \sf{:\longrightarrow \;\;\dfrac{\: (x \; - \; 2)}{\: y} \; \; = \; \; \bf{\dfrac{1}{3}}}$

$\: \sf{:\longrightarrow \;\;\dfrac{\: (7 \; - \; 2)}{\: 15} \; \; = \; \; \bf{\dfrac{1}{3}}}$

$\: \sf{:\longrightarrow \;\;\dfrac{\: \cancel{5}}{\: \cancel{15}} \; \; = \; \; \bf{\dfrac{1}{3}}}$

$\: \sf{:\longrightarrow \;\;\dfrac{\: 1}{\: 3} \; \; = \; \; \bf{\dfrac{1}{3}}}$

Clearly, LHS = RHS.

~ Case II :-

$\: \sf{:\longrightarrow \;\;\;\dfrac{\: x}{\: (y\;-\;1)} \; \; = \; \; \bf{\dfrac{1}{2}}}$

$\: \sf{:\longrightarrow \;\;\;\dfrac{\: 7}{\: (15\;-\;1)} \; \; = \; \; \bf{\dfrac{1}{2}}}$

$\: \sf{:\longrightarrow \;\;\;\dfrac{\: \cancel{7}}{\: \cancel{14}} \; \; = \; \; \bf{\dfrac{1}{2}}}$

$\: \sf{:\longrightarrow \;\;\;\dfrac{\: 1}{\: 2} \; \; = \; \; \bf{\dfrac{1}{2}}}$

Clearly, LHS = RHS

Here both the conditions satisfy, so our answer is correct.

Hence, Verified.

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★ More to know :-

• Polynomials are the mathematical expressions that are formed using constant and variable terms but variable terms can be of many degrees.

They are :-

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

• Linear Equations are the equations formed using constant and variable terms but the variable terms are of single degrees.

They are :-

• Linear Equation in One Variable
• Linear Equation in Two Variable
• Linear Equation in Three Variable

Answer 2

Given:-

• Fraction is $\sf{\dfrac{1}{3}}$ = -2

• Fraction is $\sf{\dfrac{1}{2}}$ = -1

⠀

Solution:-

$\tt\longmapsto{\dfrac{x - 2}{y} =\dfrac{1}{3}}$

$\tt\longmapsto{3(x - 2) = y}$

$\tt\longmapsto{3x - 6 = y}$

$\tt\longmapsto{3x - y = 6}$⠀⠀.........(i)

⠀

$\tt\longmapsto{\dfrac{x}{y - 1} =\dfrac{1}{2}}$

$\tt\longmapsto{2x = y - 1}$

$\tt\longmapsto{2x - y = -1}$⠀⠀.........(ii)

⠀

From (i) and (ii)

$\tt\longmapsto{3x - y = 6}$

$\tt\longmapsto{2x - y = -1}$

$\tt\longmapsto{x = 7}$

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★ Putting the value of x in the eq. (i)

$\tt\longmapsto{3x - y = 6}$

$\tt\longmapsto{3(7) - y = 6}$

$\tt\longmapsto{21 - y = 6}$

$\tt\longmapsto{-y = 6 -21}$

$\tt\longmapsto{-y = -15}$

$\tt\longmapsto{y = 15}$

∴ Fraction = $\tt\longmapsto{\dfrac{7}{15}}$

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