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

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.

Answer 2

Step-by-step explanation:

First number = x

\mapsto↦ Second Number = y

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

⇢Inthe1

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⟹(x+5):(y+15)=5:4

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

y+15

x+15

=

4

5

By doing cross multiplication we get,

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

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

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

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

\implies \sf 4x - 5y =\: 55⟹4x−5y=55

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

⟹4y−5y=55−−−−−−(EquationNo1)

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

⇢Inthe2

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⟹(x−20):(y−15)=5:2

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

y−15

x−20

=

2

5

By doing cross multiplication we get,

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

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

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

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

\implies \sf 2x - 5y =\: - 35⟹2x−5y=−35

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

⟹2x−5y=−35−−−−−−(EquationNo2)

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

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

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

−5y

−2x

+5y

=55+35

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

\implies \sf 2x =\: 90⟹2x=90

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

2

90

\implies\sf\bold{\red{x =\: 45}}⟹x=45

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

\implies \sf 2x - 5y =\: - 35⟹2x−5y=−35

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

\implies \sf 90 - 5y =\: - 35⟹90−5y=−35

\implies \sf - 5y =\: - 35 - 90⟹−5y=−35−90

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

−

5y=

−

125

\implies \sf 5y =\: 125⟹5y=125

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

5

125

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

Hence, we get,

\begin{gathered}\leadsto \sf\bold{\pink{First\: number\: =\: x =\: 45}}\\\end{gathered}

⇝Firstnumber=x=45

\begin{gathered}\leadsto\: \sf\bold{\pink{Second\: number\: =\: y =\: 25}}\\\end{gathered}

⇝Secondnumber=y=25

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