Question

12. The sum of the squares of three numbers which are in the ratio 2 : 3:4 is 725. Find the numbers.
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Answer 1

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Given

• Sum of Squares is 725

• Ratio of numbers is 2 : 3 : 4

To Find

• Numbers whose sum of squares is given above

Solution

Let the numbers be 2x, 3x and 4x.

According to question,

$\bf \implies {(2x)}^{2} + {(3x)}^{2} + {(4x)}^{2} = 725$

$\bf \implies {4x}^{2} + {9x}^{2} + {16x}^{2} = 725$

$\bf \implies {29x}^{2} = 725$

$\bf \implies {x}^{2} = 725 \div 29$

$\bf \implies {x}^{2} = 25$

$\bf \implies x = \sqrt{25}$

$\bf \implies x = 5$

Therefore, Numbers are :-

• 2x = 2 × 5 = 10

• 3x = 3 × 5 = 15

• 4x = 4 × 5 = 20

So, the numbers are 10, 15 and 20.

Verification

$\bf \implies {10}^{2} + {15}^{2} + {20}^{2} = 725$

$\bf \implies 100 + 225 + 400 = 725$

$\bf \implies 725 = 725$

$\bf \implies LHS = RHS$

Hence Verified ✔

Answer 2

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$The  sum  of  the  squares  of  three\\numbers  which  are  in  the  ratio  2:3:4\\is  725.  Find  the  numbers.$

$\huge\bold{{\pink{G}}{\blue{I}}{\green{V}}{\red{E}}{\purple{N}}{\green{❥}}}$

• $\large\pink{\boxed{\blue{The  sum  =  725}}}$

• $\large\blue{\boxed{\pink{The  ratio  =  2:3:4}}}$

$\huge\bold{{\pink{T}}{\blue{O}}{\green{  }}{\red{F}}{\purple{I}}{\orange{N}}{\pink{D}}{\green{❥}}}$

$The  numbers.$

$\huge\bold{{\pink{S}}{\blue{O}}{\green{L}}{\red{U}}{\purple{T}}{\orange{I}}{\pink{O}}{\blue{N}}{\green{❥}}}$

$Let  the  numbers  are  2x,  3x  and  4x.$

$According  to  condition,$

${2x}^{2}+{3x}^{2}+{4x}^{2}=725</p><p>$

$⇒{4{x}^{2}}+{9{x}^{2}}+{16{x}^{2}}=725$

$⇒{29{x}^{2}=725}$

$⇒{x}^{2}={\frac{725}{29}}$

$⇒{x}^{2}=25$

$⇒x=√25$

$⇒x=5$

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$x=5$

$2x=(2×5)=10$

$3x=(3×5)=15$

$4x=(4×5)=20$

$\huge\bold{{\pink{T}}{\blue{H}}{\green{E}}{\red{R}}{\purple{E}}{\orange{F}}{\pink{O}}{\blue{R}}{\red{E}}{\green{❥}}}$

$The  numbers  are  10,  15  and  20.$

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${10}^{2}+{15}^{2}+{20}^{2}=725</p><p>$

$⇒{{100}^{2}}+{{225}^{2}}+{{400}^{2}}=725$

$⇒725=725$

$So,  L.H.S=R.H.S.\\Hence,  verified.$

$\huge\bold{{\pink{D}}{\blue{O}}{\green{N}}{\red{E}}{\purple{࿐}}}$

$\bold\red{\boxed{{\blue{✿}}{\pink{H}}{\blue{O}}{\green{P}}{\red{E}}{\purple{  }}{\orange{T}}{\pink{H}}{\blue{I}}{\green{S}}{\red{  }}{\purple{H}}{\orange{E}}{\pink{L}}{\blue{P}}{\green{S}}{\red{  }}{\purple{Y}}{\orange{O}}{\pink{U}}{\blue{✿}}}}$