If 78 is divided into three parts which are proportional to 1, 13, 16, the middle part is:

1959 AMC 12 Problem 4

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78

is divided into three parts which are proportional to

1,

\dfrac13,

\dfrac16,

the middle part is:

9\dfrac13

13

17\dfrac13

18\dfrac13

26

Write the three parts as

x,\frac{x}{3},\frac{x}{6}

Their sum is

78

Let the parts be

x,\frac{x}{3},

and

\frac{x}{6}.

Then

x+\frac x3+\frac x6=\frac32x=78,

x=52.

The middle part is

\frac{x}{3}=\frac{52}{3}=17\frac13.

Thus, the correct answer is C.