The sum of the squares of three numbers is $116$ $116$ , and the ratio of these numbers is $2 : 3 : 4$ $2:3:4$ . What is the largest number?

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The sum of the squares of three numbers is $116$ $116$ , and the ratio of these numbers is $2 : 3 : 4$ $2:3:4$ . What is the largest number?

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Answer 1 · 2016-08-22T16:57:22

The three numbers are $4$ $4$ , $6$ $6$ and $8$ $8$ or $- 4$ $-4$ , $- 6$ $-6$ and $- 8$ $-8$ .

Hence the largest number is either $8$ $8$ or $- 8$ $-8$

Explanation:

If the smallest is $2 n$ $2n$ then the sum of the squares is:

${(2 n)}^{2} + {(3 n)}^{2} + {(4 n)}^{2} = (2^{2} + 3^{2} + 4^{2}) n^{2} = 29 n^{2}$ $(2n)^2+(3n)^2+(4n)^2 = (2^2+3^2+4^2)n^2 = 29n^2$

Since we are told that this is $116$ $116$ we find:

$n^{2} = \frac{116}{29} = 4$ $n^2 = 116/29 = 4$

Hence $n = \pm 2$ $n = +-2$

So the three numbers are $4$ $4$ , $6$ $6$ and $8$ $8$ or $- 4$ $-4$ , $- 6$ $-6$ and $- 8$ $-8$ .

So the largest number is either $8$ $8$ or $- 8$ $-8$ .

[ Note that the greatest number in the two cases is $8$ $8$ or $- 4$ $-4$ ]

Answer 2 · 2016-08-22T16:57:44

I got $8$ $8$ , assuming positivity. If you wish, you could say that since $- 2 : - 3 : - 4 = 2 : 3 : 4$ $-2:-3:-4 = 2:3:4$ , your highest number is $\pm 8$ $pm8$ , in magnitude.

If you label your unknown variable in lieu of the ratios as $x$ $x$ , then you can have the ratio represented as $2 x : 3 x : 4 x :$ $2x:3x:4x:$ .

So, what you have is:

${(2 x)}^{2} + {(3 x)}^{2} + {(4 x)}^{2} = 116$ $(2x)^2 + (3x)^2 + (4x)^2 = 116$

$4 x^{2} + 9 x^{2} + 16 x^{2} = 116$ $4x^2 + 9x^2 + 16x^2 = 116$

$29 x^{2} = 116$ $29x^2 = 116$

$x^{2} = \frac{116}{29} = 4$ $x^2 = 116/29 = 4$

$x = \pm 2$ $color(blue)(x = pm2)$

Therefore, the largest number is $4x = 4(pm2) = bb(pm8)$ .

To check our work:

$2x = 2(pm2) = pm4$
$3x = 3(pm2) = pm6$
$4x = 4(pm2) = pm8$

Indeed, $pm4:pm6:pm8 = pm2:pm3:pm4$ , and:

$(pm4)^2 + (pm6)^2 + (pm8)^2 = 16 + 36 + 64$

$= 52 + 64 = color(green)(116)$

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The sum of the squares of three numbers is $116$ $116$ , and the ratio of these numbers is $2 : 3 : 4$ $2:3:4$ . What is the largest number?

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