Question

on adding 5 and 15 respectively to the two numbers, their ratio becomes 5:4 .if 20 and 15 is respectively subtracted from the two numbers, the ratio becomes 5:2 find the numbers ​

Answer 1

❍ Let's consider the two numbers be x and y respectively. 

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$\bigstar\:{\underline{\boldsymbol{According\:to\:the\:question\::}}}$

• On Adding 5 and 15 respectively to the two numbers, their ratio becomes 5:4.

⠀⠀

$:\implies\sf \dfrac{x + 5}{y + 15} = \dfrac{5}{4}\\\\\\ :\implies\sf 4[x + 5] = 5[y + 15]\\\\\\ :\implies\sf 4x + 20 = 5y + 75\\\\\\ :\implies\sf 4x - 5y = 75 - 20\\\\\\ :\implies\sf 4x - 5y = 55\qquad\qquad\qquad\bigg\lgroup eq^n\:(1)\bigg\rgroup$

• On substracting 20 and 15 respectively from the two numbers, their ratio becomes 5:2.

⠀⠀

$:\implies\sf \dfrac{x - 20}{y - 15} = \dfrac{5}{2}\\\\\\:\implies\sf 2[x - 20] = 5[y - 15]\\\\\\ :\implies\sf 2x - 40 = 5y - 75\\\\\\ :\implies\sf 2x - 5y = - 75 + 40\\\\\\ :\implies\sf 2x - 5y = - 35\qquad\qquad\qquad\bigg\lgroup eq^n\:(2)\bigg\rgroup$

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$\bf{\dag}\: {\underline{\frak{Substracting\:eq^n\:(1)\:from\:eq^n\:(2)\::}}}\\\\\\ \dashrightarrow\sf \bigg(4x - 5y\bigg) - \bigg(2x - 5y\bigg) = 55 - (-35)\\\\\\ \dashrightarrow\sf 4x\: \:\cancel{-\: 5y} - 2x \:\:\cancel{+ \:5y} = 55 + 35\\\\\\ \dashrightarrow\sf 4x - 2x = 90\\\\\\ \dashrightarrow\sf 2x = 90\\\\\\ \dashrightarrow\sf x = \cancel{\dfrac{90}{2}}\\\\\\ \dashrightarrow{\underline{\boxed{\pmb{\frak{\pink{x = 45}}}}}}\:\bigstar\\\\\\ \bf{\dag}\: {\underline{\frak{By\:Putting\:value\:of\:x\:in\:eq^n\:(1)\::}}}\\\\\\ \dashrightarrow\sf 4(45) - 5y = 55\\\\\\ \dashrightarrow\sf 180 - 5y = 55\\\\\\ \dashrightarrow\sf - 5y = 55 - 180\\\\\\ \dashrightarrow\sf - 5y = - 125\\\\\\ \dashrightarrow\sf y = \cancel{\dfrac{-125}{-5}}\\\\\\ \dashrightarrow{\underline{\boxed{\pmb{\frak{\purple{y = 25}}}}}}\:\bigstar$

$\therefore\:{\underline{\sf{Hence\:the\:value\:of\:two\:numbers\:is\:{\pmb{45}}\:\sf and\:{\pmb{25}}\sf \:respectively.}}}$

Answer 2

Answer:

Given :-

• On adding 5 and 15 respectively to the two numbers, their ratio becomes 5 : 4.
• 20 and 15 is respectively subtracted from the two numbers, the ratio becomes 5 : 2.

To Find :-

• What are the numbers.

Solution :-

Let,

$\mapsto$ First number = x

$\mapsto$ Second Number = y

${\small{\bold{\purple{\underline{\dashrightarrow\: In\: the\: {1}^{{st}}\: case\: :-}}}}}$

$\longmapsto$ By adding 5 and 5 respectively to the two two numbers and their ratio becomes 5 : 4.

$\implies \sf (x + 5) : (y + 15) =\: 5 : 4$

$\implies \sf \dfrac{x + 15}{y + 15} =\: \dfrac{5}{4}$

By doing cross multiplication we get,

$\implies \sf 4(x + 5) =\: 5(y + 15)$

$\implies \sf 4x + 4(5) =\: 5y + 5(15)$

$\implies \sf 4x + 20 =\: 5y + 75$

$\implies \sf 4x - 5y =\: 75 - 20$

$\implies \sf 4x - 5y =\: 55$

$\implies \sf\bold{\green{4y - 5y =\: 55\: ------\: (Equation\: No\: 1)}}$

${\small{\bold{\purple{\underline{\dashrightarrow\: In\: the\: {2}^{{nd}}\: case\: :-}}}}}$

$\longmapsto$ By subtracting 20 and 15 respectively from the two numbers and their ratio becomes 5 : 2.

$\implies \sf (x - 20) : (y - 15) =\: 5 : 2$

$\implies \sf \dfrac{x - 20}{y - 15} =\: \dfrac{5}{2}$

By doing cross multiplication we get,

$\implies \sf 2(x - 20) =\: 5(y - 15)$

$\implies \sf 2x - 2(20) =\: 5y - 5(15)$

$\implies \sf 2x - 40 =\: 5y - 75$

$\implies \sf 2x - 5y =\: - 75 + 40$

$\implies \sf 2x - 5y =\: - 35$

$\implies \sf\bold{\green{2x - 5y =\: - 35\: ------\: (Equation\: No\: 2)}}$

Now, by subtracting the equation no 1 from the equation no 2 we get,

$\implies \sf 4x - 5y - (2x - 5y) =\: 55 - (- 35)$

$\implies \sf 4x {\cancel{- 5y}} - 2x {\cancel{+ 5y}} =\: 55 + 35$

$\implies \sf 4x - 2x =\: 55 + 35$

$\implies \sf 2x =\: 90$

$\implies \sf x =\: \dfrac{\cancel{90}}{\cancel{2}}$

$\implies\sf\bold{\red{x =\: 45}}$

Again, by putting the value of x = 45 in the equation no 2 we get,

$\implies \sf 2x - 5y =\: - 35$

$\implies \sf 2(45) - 5y =\: - 35$

$\implies \sf 90 - 5y =\: - 35$

$\implies \sf - 5y =\: - 35 - 90$

$\implies \sf {\cancel{-}} 5y =\: {\cancel{-}} 125$

$\implies \sf 5y =\: 125$

$\implies \sf y =\: \dfrac{\cancel{125}}{\cancel{5}}$

$\implies \sf\bold{\red{y =\: 25}}$

Hence, we get,

$\leadsto \sf\bold{\pink{First\: number\: =\: x =\: 45}}$

$\leadsto\: \sf\bold{\pink{Second\: number\: =\: y =\: 25}}$

$\therefore$ The two numbers are 45 and 25 respectively.