In the small hamlet of abaze, two base systems are in common use. also, everyone speaks the truth, and no one changes which base they are using in mid-sentence. one resident said, "26 people use my base, base 10, and only 22 people speak base 14." another said, "of the 25 residents, 13 use both bases and 1 can't use either base." how many residents are there? (use base 10, ! )

Suppose that the first resident speaks in base $B$. Converting to base $10$, his statement implies that $2B + 6$ people uses base $B$ (notice that base $10$ is actually base $1 \cdot B + 0 = B$), and that $2B+2$ people speak using base $B + 4$. Clearly, the second resident is not talking in base $B$, so he must speak base $B + 4$. Then we can also write all of his statements in terms of the variable $B$. It follows that there are $2(B+4) + 5 = 2B + 13$ total residents, of which $(B+4)+3 = B+7$ are bilingual and $1$ cannot use either base. By the Principle of Inclusion-Exclusion, there are $$(2B + 6) + (2B+2) + 1 - (B+7) = 3B + 2$$total people, and setting this equal to $2B + 13$, it follows that $3B + 2 = 2B + 13 \Longrightarrow B = 11$. Thus, the answer is $2B + 13 = 2 \cdot 11 + 13 = \boxed{35}$.

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explanation

the graph has a slope of 1/3 because each time you move up 1, you go to the right 3. so, slope = rise/run = 1/3. we can cross choice a off the list since we want a slope of 3, rather than 1/3.

the first table shown has a slope of -3. this is because the rise is -6 and the run is 2, so slope = rise/run = -6/2 = -3. note how x increases by 2 each time (run) and y drops by 6 each time (rise). choice b can be eliminated.

the second table has a slope of 3. each time x goes up by 1, the y values increase by 3. slope = rise/run = 3/1 = 3. alternatively you can use the slope formula to prove that the slope is 3 here. use any two rows of values to generate the two points. choice c is one of the answers

the third table also has a slope of 3. let's use the slope formula to prove as such. i'm going to use the first two rows of the table. so i'll use the two points (2,1) and (5,10)

m = slope

m = (y2-y1)/(x2-x1)

m = (10-1)/(5-2)

m = 9/3

m = 3 so we get a slope of 3

choice d is the other answer