Overlapping Set Problem

Question

There are 87 balls in a jar. Each ball is painted with at least one of two colors, red or green. It is observed that 2/7 of the balls that have red color also have green color, while 3/7 of the balls that have green color also have red color. What fraction of the balls in the jar have both red and green colors?

A. 6/14
B. 2/7
C. 6/35
D. 6/29
E. 6/42

Found this problem very confusing. Please help!

A. 6/14
B. 2/7
C. 6/35
D. 6/29
E. 6/42

Found this problem very confusing. Please help!

fluke · Answer

n(R \cup G)=n(R)+n(G)-n(R \cap G)

87=R+G-\frac{2}{7}R --------------1

87=R+G-\frac{3}{7}G --------------2

Solve these equations to find either R or G.

Both=(2/7)R OR (3/7)G

I got 6/29.

Ans: "D"

GyanOne · Answer

Let x be the number of red balls in the jar and y be the number of green balls in the jar.

Then we know that n(red U green) = number of total balls in the jar = 87 (as it is given that each ball has at least one of two colors)

=> 87 = x + y - (3/7)*y   
Also (2/7) * x = (3/7) * y  .

Solving these two equations, we get y = 42 and x = 63.

So there are 63 red balls and 42 green balls in the jar.

Of these, number of only red balls = 63 - (2/7)*63 = 45
Number of only green balls = 42 - (3/7) * 42 = 24
Number of balls that are both green and red = 87 - 45 - 24 = 18

Therefore fraction of balls in the jar that have both red and green colors = 18/87 = 6/29

The correct answer is therefore (D).

Asher · Answer

Thanks fluke and gyanone.

i found this solution in another forum.. is it ok to do it this way as well

is it ok to say that r&g/87 = 6/29. hence ans. is 6/29.

fluke · Answer

Yes, this is good as well. It uses substitution rather than simultaneous solution of equation in 2-variables.

GyanOne · Answer

Asher,

The way you have solved it is fine as well. As fluke has pointed out, your method just uses substitution instead of solving the two equations to get the values of the variables.

The point to note here is that if you can use substitution instead of solving the two equations for the two variables, do that on the exam -> might save you those extra precious 5-10 seconds.

gladvijay · Answer

Nice way of solving.. thanks!

Yalephd · Answer

I took a shortcut to do this one if a few seconds. I said that the denominator had to be a factor of 87. Only answer choice D had a factor of 87--29.

This method wouldn't have worked if there were another answer choice with 29 or 3.

callingsid · Answer

good ways of solving them ..

but a very quick soln (for this prbm )

87 is divisible by 29 and 3

since 29 is one of the deno in all the choices... it has to be tht choice ... !!!

(soln by a trick)

Overlapping Set Problem

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