Bunuel wrote:
Three table runners have a combined area of 200 square inches. By overlapping the runners to cover 80% of a table of area 175 square inches, the area that is covered by exactly two layers of runner is 24 square inches. What is the area of the table that is covered with three layers of runner?
(A) 18 square inches
(B) 20 square inches
(C) 24 square inches
(D) 28 square inches
(E) 30 square inches
Kudos for a correct solution.
VERITAS PREP OFFICIAL SOLUTION:Let’s first try to understand what exactly is given to us. The area of all the runners is equal to 200 square inches.
Runner 1 + Runner 2 + Runner 3 = 200
In our diagram, this area is represented by
(a + d + g + e) + (b + d + g + f) + (c + e + g + f) = 200
(We need to find the value of g i.e. the area of the table that is covered with three layers of runner.)
Area of table covered is only 80% of 175 i.e. only 140 square inches. This means that if each section is counted only once, the total area covered is 140 square inches.
a + b + c + d + e + f + g = 140
So the overlapping regions are obtained by subtracting second equation from the first. We get d + e + f + 2g = 60
But d + e + f (area with exactly two layers of runner) = 24
So 2g = 60 – 24 = 36
g = 18 square inches.
Note that you don’t need to make all these equations and can directly jump to d + e + f + 2g = 60. We wrote these equations down only for clarity. It is a matter of thinking vs solving. If we think more, we have to solve less. Let’s see how.
Combined area of runners is 200 square inches while area of table they cover is only 140 square inches. So what does the extra 60 square inches of runner do? It covers another runner!
Wherever there are two runners overlapping, one runner is not covering the table but just another runner. Wherever there are three runners overlapping, two runners are not covering the table but just the third runner at the bottom.
So can we say that (d + e + f) represents the area where one runner is covering another runner and g is the area where two runners are covering another runner?
Put another way, can we say d + e + f + 2g = 60?
We know that d + e + f = 24 giving us g = 18 square inches
This entire ‘thinking process’ takes ten seconds once you are comfortable with it and your answer would be out in about 30 sec!
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