The sum of three numbers is 98. The ratio of the first to the second is 2/3 and the ratio of the second to the third is 5/8. What is the second number?

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The sum of three numbers is 98. The ratio of the first to the second is 2/3 and the ratio of the second to the third is 5/8. What is the second number?

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Answer 1 · 2016-07-26T16:09:36

$\Rightarrow n_{2} = 30$ $=>n_2=30$

Explanation:

Let the first number be $n_{1}$ $n_1$

Let the second number be $n_{2}$ $n_2$

Let the third number be $n_{3}$ $n_3$

Given that $\frac{n_{1}}{n_{2}} = \frac{2}{3}$ $(n_1)/(n_2) = 2/3$

Given that $\frac{n_{2}}{n_{3}} = \frac{5}{8}$ $(n_2)/(n_3) = 5/8$

Given that $n_{1} + n_{2} + n_{3} = 98$ $n_1+n_2+n_3=98$ .................................(1)
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Consider $\frac{n_{1}}{n_{2}} = \frac{2}{3}$ $(n_1)/(n_2) = 2/3$

Write this as $3 n_{1} = 2 n_{2}$ $3n_1=2n_2$

So $n_{1} = \frac{2}{3} n_{2}$ $n_1=2/3n_2$ .................................................(2)

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Consider $\frac{n_{2}}{n_{3}} = \frac{5}{8}$ $(n_2)/(n_3) = 5/8$

Write this as $8 n_{2} = 5 n_{3}$ $8n_2=5n_3$

So $n_{3} = \frac{8}{5} n_{2}$ $n_3=8/5n_2$ ......................................................(3)

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Substitute (3) and (2) into (1) giving:

$n_{1} + n_{2} + n_{3} = 98 \to \frac{2}{3} n_{2} + n_{2} + \frac{8}{5} n_{2} = 98$ $color(brown)(n_1+n_2+n_3=98color(blue)(" "->" "2/3n_2+n_2+8/5n_2=98$

$\Rightarrow n_{2} (\frac{10 + 1 + 24}{15}) = 98$ $=>n_2((10+1+24)/15)=98$

$\Rightarrow n_{2} = 30$ $=>n_2=30$

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The sum of three numbers is 98. The ratio of the first to the second is 2/3 and the ratio of the second to the third is 5/8. What is the second number?

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