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# Three table runners have a combined area of 200 square inches. By over

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Math Expert
Joined: 02 Sep 2009
Posts: 59730
Three table runners have a combined area of 200 square inches. By over  [#permalink]

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16 Sep 2015, 07:09
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Difficulty:

85% (hard)

Question Stats:

51% (02:40) correct 49% (02:26) wrong based on 193 sessions

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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.

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Math Expert
Joined: 02 Sep 2009
Posts: 59730
Re: Three table runners have a combined area of 200 square inches. By over  [#permalink]

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20 Sep 2015, 09:04
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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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Joined: 10 Aug 2015
Posts: 101
Re: Three table runners have a combined area of 200 square inches. By over  [#permalink]

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16 Sep 2015, 09:49
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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?[/b]

(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.

Solution : After overlapping, let the area that is covered by single layer of three table runners be x,y and z. Let area covered by first and second, second and third, and third and first be a,b and c respectively. Let the area covered by all three be d.
So, x+y+z+(a+b+c)+d = 0.8(175) = 140. We know that area that is covered by exactly two layers(a+b+c) = 24
x+y+z+d = 116.-->(1)

But, combined area = x+a+c+d+y+a+b+d+z+b+c+d = x+y+z+2(a+b+c)+3d = 200
x+y+z+3d = 152.-->(2)

From (1) and (2), 2d = 36 ==> d = 18.

Option A.
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Joined: 15 May 2014
Posts: 61
Re: Three table runners have a combined area of 200 square inches. By over  [#permalink]

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17 Sep 2015, 08:26
4
Total = a + b + c - (sum of EXACTLY 2-group overlaps) - 2*(all three) + Neither
80%*175 = 200 - 24 - 2*(all three) + 0
2*(all three) = 200 - 24 - 140
all three = 18

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Joined: 29 Jul 2015
Posts: 152
Re: Three table runners have a combined area of 200 square inches. By over  [#permalink]

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17 Sep 2015, 16:52
1
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?[/b]

(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.

Let x be the area that is covered by three layers of runners.
Total area covered by three runners is 200
Total area of table covered is 80% of 175 = 140
So, the overlapping area will be
A+B+C - AuBuC = 200-140 = 60

It is given that sum of exactly 2 overlapping areas is 24
Total overlapping area = sum of exactly 2 groups overlap + 2*(all three)
So,
2x+24 = 60
or 2x= 60- 24 = 36
or x=18

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Joined: 22 May 2015
Posts: 13
Location: India
GMAT 1: 650 Q46 V34
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Re: Three table runners have a combined area of 200 square inches. By over  [#permalink]

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01 May 2016, 00:17
Surplus -> Direct sum of three sets - union of three sets.
-> 200 - .75*175
-> 200 - 140 -> 60
Surplus -> Sum of three combinations of intersection of two sets + 2 * (Intersection of three sets)
i.e. 60 -> 24 + 2 * (Intersection of three sets)
so, Intersection of three sets -> 18
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Re: Three table runners have a combined area of 200 square inches. By over  [#permalink]

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17 Sep 2016, 23:38
How come Neither is 0?.If total table area covered is 80% ,then no area covered is 20%,right.Not able to understand why Neither is 0 and total area is considered 140 instead of 175
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Re: Three table runners have a combined area of 200 square inches. By over  [#permalink]

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16 Dec 2016, 05:29
bhamini1 wrote:
How come Neither is 0?.If total table area covered is 80% ,then no area covered is 20%,right.Not able to understand why Neither is 0 and total area is considered 140 instead of 175

neither here refers to the area of table covers that doesn't cover anything , not that area of the table that isn't covered
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Re: Three table runners have a combined area of 200 square inches. By over  [#permalink]

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10 Jul 2019, 17:21
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.

3 groups only?

Given: Three table runners have a combined area of 200 square inches. => n(A) + n(B) + n(C) = 200

Total = n(A) + n(B) + n(C) + None - (2 groups only) - 2(3 groups only)
.8 * 175 = 200 + 0 - 24 -2(3 groups only)
140 = 200 + 0 - 24 -2x => 2x = 60 - 24 => x = 36/2 = 18 => A
Re: Three table runners have a combined area of 200 square inches. By over   [#permalink] 10 Jul 2019, 17:21
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