Question

Three numbers are in the ratio 5:2:3
The
sum of squares of the three numbers is
608. Find the numbers. Use repeated
subtraction method to find the square
root of a square number:​

Answer 1

Given :-

Three numbers in ratio 5:2:3 and sum of their squares is 608

To Find :-

• We have to find the value of these 3 numbers

Lets assume that the number be n

Numbers = 5n , 2n , 3n

$\underline{\bigstar{\sf\ \ Solution :- }}$

$\longmapsto\sf (5n)^2+(2n)^2+(3n)^2= 608\\ \\ \longmapsto\sf 25n^2+4n^2+9n^2= 608\\ \\ \longmapsto\sf 38n^2= 608\\ \\ \longmapsto\sf n^2= \cancel{\dfrac{608}{38}}\\ \\ \longmapsto\sf n^2= 16\\ \\ \longmapsto\sf n= \sqrt{16}$

• Now find square root of 16 by repeated subtraction method !

$\longmapsto\sf 1) \ \ 16-1= 15\ ( subtracting\ 1st\ odd \ number)\\ \\ \longmapsto\sf 2) \ \ 15-3= 12 ( subtracting\ 2nd \ odd\ number )\\ \\ \longmapsto\sf 3) \ \ 12-5= 7 (\ subtracting\ 3rd\ odd\ number \\ \\ \longmapsto\sf 4)\ \ 7-7= 0 ( subtracting\ 4th\ odd \ number )\\ \\ \sf So ,\ n= 4$

• Now we have the value of n which is 4

Let's find the numbers

$\star\sf 5n = 5\times 4= 20\\ \\ \star\sf 3n= 3\times 4= 12\\ \\ \star\sf 2n= 2\times 4= 8$

$\underline{\textsf{\textbf {\dag\ \ Three \ numbers \ are\ 20 ,\ 12, \ 8}}}$

$\underline{\boxed{\sf VERIFICATION :- }}$

$\longmapsto\sf (5n)^2+(3n)^2+(2n)^2=608\\ \\ \bullet\sf \ put \ the \ values \\ \\ \longmapsto\sf (20)^2+(12)^2+(8)^2=608\\ \\ \longmapsto\sf 400+144+64= 608\\ \\ \longmapsto\sf 608= 608\\ \\ \sf\ \ L.H.S= R.H.S \ \ (hence \ verified\ ! )$

Answer 2

$\sf\red{\underline{\underline{Answer:}}}$

$\sf{The \ numbers \ are \ 20, \ 8 \ and \ 12 \ or}$

$\sf{-20, \ -8 \ and \ -12 \ respectively.}$

$\sf\orange{Given:}$

$\sf{\implies{Three \ numbers \ are \ in \ ratio \ 5:2:3}}$

$\sf{\implies{The \ sum \ of \ squares \ of \ the \ three}}$

$\sf{numbers \ is \ 608.}$

$\sf\pink{To \ find:}$

$\sf{The \ numbers:}$

$\sf\green{\underline{\underline{Solution:}}}$

$\sf{Let \ the \ constant \ be \ x.}$

$\sf{According \ to \ the \ first \ condition.}$

$\sf{Numbers \ are \ 5x, \ 2x \ and \ 3x}$

$\sf{According \ to \ the \ second \ condition.}$

$\sf{(5x)^{2}+(2x)^{2}+(3x)^{2}=608}$

$\sf{25x^{2}+4x^{2}+9x^{2}=608}$

$\sf{38x^{2}=608}$

$\sf{x^{2}=\frac{608}{38}}$

$\sf{x^{2}=16}$

$\sf{On \ taking \ square \ root \ of \ both \ sides}$

$\sf{x=4 \ or \ -4}$

$\sf{If \ x=4, \ then \ numbers \ are}$

$\sf{5(4)=20,}$

$\sf{2(4)=8,}$

$\sf{3(4)=12.}$

$\sf{If \ x=-4, \ then \ numbers \ are}$

$\sf{5(-4)=-20,}$

$\sf{2(-4)=-8,}$

$\sf{3(-4)=-12.}$

$\sf\purple{\tt{\therefore{The \ numbers \ are \ 20, \ 8 \ and \ 12 \ or}}}$

$\sf\purple{\tt{-20, \ -8 \ and \ -12 \ respectively.}}$