Problem Solving

Pls. help me solve this question: A number of girls bought red, green and black ball pens. Three bought one of each colour. Of the girls who bought two colours, 3 did not buy red, 5 not green and 2 not black. The same number of girls bought red only as bought red with other colours. The same number bought black only as bought green only. More girls bought red and black but not green than bought black only, but more girls bought green only than bought green and black but not red. How many girls were there? Thanks.

The way this question is worded, one can see that one will likely be able to calculate the number who bought pens of all three colors, the number who bought pens of only two colors and the number who bought pens of only one color.

The total number of girls will equal the following.

The Number Who Bought Only One Color + The Number Who Bought Only Two Colors + The Number Who Bought All Three Colors

Let's break down what the question says to get some subtotals.

Three bought one of each colour.

-- 3 Bought All Three

Of the girls who bought two colours, 3 did not buy red, 5 not green and 2 not black.

-- 3 Bought Only Black And Green

-- 5 Bought Only Black And Red

-- 2 Who Bought Only Red And Green

The same number of girls bought red only as bought red with other colours.

I guess this means the following.

-- 3 Bought All Three

-- 5 Bought Red And Black

-- 2 Bought Red And Green

-- 3 + 5 + 2 = 10 Bought Only Red

More girls bought red and black but not green than bought black only,

-- < 5 Bought Only Black

more girls bought green only than bought green and black but not red.

-- > 3 Bought Only Green

The same number bought black only as bought green only.

-- 3 < Only Green = Only Black < 5

-- 4 Bought Only Green

-- 4 Bought Only Black

Total:

3 All Three

3 Only Black And Green

5 Only Black And Red

2 Only Red And Green

10 Only Red

4 Only Green

4 Only Black
_____________________

31 Girls

Johnny 08 wrote:Pls. help me solve this question: A number of girls bought red, green and black ball pens. Three bought one of each colour. Of the girls who bought two colours, 3 did not buy red, 5 not green and 2 not black. The same number of girls bought red only as bought red with other colours. The same number bought black only as bought green only. More girls bought red and black but not green than bought black only, but more girls bought green only than bought green and black but not red. How many girls were there? Thanks.

Draw a VENN DIAGRAM and work from the CENTER OUT.

Three bought one of each color.


Of the girls who bought two colors, 3 did not buy red, 5 not green and 2 not black.


The same number of girls bought red only as bought red with other colors.


The same number bought black only as bought green only.


More girls bought red and black but not green than bought black only.
Since red and black but not green = 5, we get:


More girls bought green only than bought green and black but not red.
Since green and black but not red = 3, we get:


Since x>3 and x<5, x=4.
The result is the following:


Sum of the values in the resulting Venn Diagram:
10+2+4+5+3+3+4 = 31.

I'd just write it out:

All three = 3 girls
Not red = 3
Not green = 5
Not black = 2
Only red = (Not Green) + (Not Black) + (All Three) = 10
Only green = Only black

For simplicity's sake, let's say that only green and only black = x.

Now for the tricky ones!

Not green > x, or 5 > x

x > not red, or x > 3

So x must = 4.

From there, we're done! Only green = 4, only black = 4, and we can add everything up.