Question

Each of the points $A,B,C,D,E,$ and $F$ in the figure below represents a different digit from $1$ to $6.$ Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is $47.$ What is the digit represented by $B?$

$1$

$2$

$3$

$4$

$5$

Answer: E

Concepts:double counting summation

Difficulty rating: 1270

Video solution:

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Written solution:

Every point is counted in two of the line sums except $B,$ which is counted in three. Thus the total of the five line sums is twice the sum of all six digits, plus one extra $B.$

The digits are exactly $1,2,3,4,5,6,$ whose sum is $21.$ Therefore $2\cdot21+B=47,$ so $B=5.$

Thus, the correct answer is E.

2020 AMC 8 Problem 16

Answer: E

Difficulty rating: 1270

Video solution:

Written solution:

Problem 16 in Other Years