1961 AMC 12 Problem 23

Attempt Problem 23 of the 1961 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1961 AMC 12 solutions, or check the answer key.

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23.

Points

P

and

Q

are both in the line segment

AB

and on the same side of its midpoint.

P

divides

AB

in the ratio

2:3,

and

Q

divides

AB

in the ratio

3:4.

PQ=2,

then the length of

AB

is:

60

70

75

80

85

Express

\frac{AP}{AB}

and

\frac{AQ}{AB}

from the two division ratios

Their difference is the fraction of

AB

represented by

PQ

The division ratios give

\frac{AP}{AB}=\frac25

and

\frac{AQ}{AB}=\frac37.

Hence

PQ=(\frac{3}{7}-\frac{2}{5})AB=\frac{AB}{35}.

Since

PQ=2,

we obtain

AB=70.

Thus, the correct answer is B.

Problem 23 in other years