CPhill

e^2x >37      take the ln of both sides

ln e^2x > ln 37     and we can write

2x ln e   > ln 37      and ln e = 1     ...  so we have

2x > ln 37     divide both sides by 2

x > ( ln 37) / 2  

x > 1.805

There are 15 in the Science club.  And of these, 80%,  (12)  are also in the Math club.

Let x be the number in the Math club. And  30% of those in the Math club are also in the Science club.

So

.30x = 12   x = 40    And this is the number of students in the Math club.

Here's a Venn diagram

4/19 x  4/18 =    

4/19 x  2/9  =

8 / 171

Nice graph, Melody....!!!

For simplicity.....let there be 100 students in the school

So.....8 bring both

So....let x be the number who bring raincoats

And let 2x be the number that bring umbrellas

The ones that bring only raincoats = x - 8

The ones that bring only umbrellas = 2x - 8

So....

(Number who bring raincoats only) + (Number who bring both) + (Number who bring umbrellas only) = 100

(x - 8) +  8  +  (2x - 8) = 100

3x - 8 =  100

3x = 108

x  = 36

2x = 72

So

The number (percent) who bring only umbrellas =  2(36) - 8   =   64 (%)

C = 1/4 cup    R = 2/5 cup  and   G has 2(1/4)(6/10) = 12/40  = 3/10 cup

So R and G together have 2/5 + 3/10  =  4/10 + 3/10  = 7/10 cup

We can atually write this as

5 / 32^(1/3)    ......we need to "rationalize" the denominator....we can do this thusly :

Multiply "top and bottom" by  32^(2/3)

[5 * 32^(2/3)] / [ 32^(1/3) * 32^(2/3)]  =

5 ³√[32²] / 32      ...... "rationalizing" the denominator usually makes it easier to add fractions