# 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?

Jul 26, 2016

$\implies {n}_{2} = 30$

#### Explanation:

Let the first number be ${n}_{1}$

Let the second number be ${n}_{2}$

Let the third number be ${n}_{3}$

Given that $\frac{{n}_{1}}{{n}_{2}} = \frac{2}{3}$

Given that $\frac{{n}_{2}}{{n}_{3}} = \frac{5}{8}$

Given that ${n}_{1} + {n}_{2} + {n}_{3} = 98$ .................................(1)
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Consider $\frac{{n}_{1}}{{n}_{2}} = \frac{2}{3}$

Write this as $3 {n}_{1} = 2 {n}_{2}$

So ${n}_{1} = \frac{2}{3} {n}_{2}$ .................................................(2)

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Consider $\frac{{n}_{2}}{{n}_{3}} = \frac{5}{8}$

Write this as $8 {n}_{2} = 5 {n}_{3}$

So ${n}_{3} = \frac{8}{5} {n}_{2}$......................................................(3)

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

color(brown)(n_1+n_2+n_3=98color(blue)(" "->" "2/3n_2+n_2+8/5n_2=98

$\implies {n}_{2} \left(\frac{10 + 1 + 24}{15}\right) = 98$

$\implies {n}_{2} = 30$