The sum of two numbers is 1000 and the difference between their squares is 25600, then find the
numbers.
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The sum of two numbers is 1000 and the difference between their squares is 25600, then find the
numbers.
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Solution :
[tex]\bf{\red{\underline{\bf{Given\::}}}}[/tex]
The sum of two numbers is 1000 and the difference between their squares is 25600.
[tex]\bf{\red{\underline{\bf{To\:find\::}}}}[/tex]
The numbers.
[tex]\bf{\red{\underline{\bf{Explanation\::}}}}[/tex]
Let the first number be r
Let the second number be m
A/q
[tex]\longrightarrow\sf{r+m=1000...............(1)}[/tex]
&
[tex]\longrightarrow\sf{r^{2} -m^{2} =25600}\\\\\longrightarrow\sf{(r+m)(r-m)=25600\:\:[\therefore Using\:fromula\:a^{2} -b^{2} ]}\\\\\longrightarrow\sf{1000(r-m)=25600\:\:[from(1)]}\\\\\longrightarrow\sf{r-m=\cancel{\dfrac{25600}{1000} }}\\\\\longrightarrow\sf{r-m=25.6}\\\\\longrightarrow\sf{r=25.6+m........................(2)}[/tex]
Putting the value of r in equation (1),we get;
[tex]\longrightarrow\sf{25.6+m+m=1000}\\\\\longrightarrow\sf{25.6+2m=1000}\\\\\longrightarrow\sf{2m=1000-25.6}\\\\\longrightarrow\sf{2m=974.4}\\\\\longrightarrow\sf{m=\cancel{\dfrac{974.4}{2} }}\\\\\longrightarrow\sf{\blue{m=487.2}}[/tex]
Putting the value of m in equation (2),we get;
[tex]\longrightarrow\sf{r=25.6+487.2}\\\\\longrightarrow\sf{\blue{r=512.8}}[/tex]
Thus;
[tex]\underbrace{\sf{The\:value\:of\:r={\boxed{\bf{512.8}}}}\:\:\&\:\:m=\boxed{\bf{487.2}}}}[/tex]
[tex] \large \red{ \sf{Answer}} [/tex]
[tex]\large \red{ \sf{Given}}
[/tex]
[tex]\\[/tex]
[tex]\large \red{ \sf{ Concept \: \: used }}
[/tex]
[tex]\\[/tex]
[tex]\large \red{ \sf{Solution}}
[/tex]
Let numbers are x and y
Applying given conditions
[tex]\\[/tex]
[tex]\rightarrow[/tex] x +y = 1000
[tex]\rightarrow[/tex]
[tex]x^2 -y^2[/tex] = (x-y) (x+y)
[tex]\\[/tex]
since
[tex]\rightarrow[/tex]x^2 -y^2 = 25600 ......... (1)
[tex]\rightarrow[/tex]x +y = 1000 ..........(2)
[tex]\\[/tex]
[tex]\rightarrow[/tex]25600 = 1000×(x-y)
[tex]\rightarrow[/tex]x-y = 25.6 .......... (3)
[tex]\\[/tex]
add eq (2) and (3)
[tex]\rightarrow[/tex]2x = 1025.6
[tex]\rightarrow[/tex]x = 512.8
[tex]\\[/tex]
put x in eq 1
[tex]\rightarrow[/tex]y = 1000-512.8
[tex]\rightarrow[/tex]= 487.2
[tex]\large \red{ \sf{Note}}
[/tex]
you can verify also that
[tex]\rightarrow[/tex]512.8+487.2 = 1000
[tex]\rightarrow[/tex][tex](512.8) ^2 -(487.2) ^2[/tex]= 25600