Let the original number of red marbles be x and the number of blue marbles y.
According to the beginning of the card, we have the following number of marbles: x+50 , y+30.
We have a system of equations: x/y = 2/3 >> 3x-2y= 0 (1)
(x+50)/(y+30)=5/6
>> 6x+300 = 5y + 150
>> 6x – 5y = -150 (2)
>From (1) and (2) >> y = 150, x= 100

picos school

For this question, we will parts and units to solve it.

at first, there will be 2 units for red marbles and 3 units for blue marbles. After 50 red and 30 blue marbles were added, there will be 5 parts of red marbles and 6 parts of blue marbles.

Hennce, the equation will be as such.

2u + 50 = 5p

3u + 30 = 6p

making parts the same,

12u + 300 = 30p

15u + 150 = 30p

hence 12u + 300 = 15u + 150

3u = 150

1u = 50

total marbles will be 50 x 5 = 250

There were some marbles at a shop. The ratio of the number of red marbles to the number of blue marbles was 2:3. When 50 more red marbles and 30 more blue marbles were added, the ratio of the number of red marbles to the number of blue marbles became 5:6. How many marbles were there at first?= 

Total = 5 Units (2 Units Red and 3 Units Blue)

6* (2 Units + 50) = 5 * (3Units + 30)

12 Units + 300 = 15 Units + 150

3 Units = 150

5 Units = 250

Before adding:
Red : Blue
= 2: 3
= 4: 6

After adding, (based on the above ratio)
Red : Blue
= 4u+50 : 6u+30
= 4u+50 : 6(u+5)

The given ratio, after adding is :
Red : Blue
= 5:6

Compare blue ratio, 6(u+5) = 6
hence, red ratio = 5(u+5)

Equate this to the red ratio found earlier.

5(u+5) = 4u+50
5u+25 = 4u +50
u = 25

At first ,
red marbles = 4u = 4 x 25 = 100
blue marbles = 6u = 6 x 25 = 150

Ref: http://www.kiasuparents.com/kiasu/forum/viewtopic.php?p=1820#p1820

x: red marbles
y:blue marbles

x/y=2/3
(x+50)/(y+30)=5/6

Solving the equation,
3x=2y
6x+300=5y+150—–*

if 3x=2y,(here you times 2 for both side right?)
6x=4y

sub into *
4y+300=5y+150
150=y
y=150

if y=150,
use this formula u have found jus now
3x=2y
3x=2(150)=300
after that,bring 3 over,hence 300 /3=100.
x=100

hence red marbles=100
blue marbles=150

Ref: http://www.kiasuparents.com/kiasu/forum/viewtopic.php?p=2231#p2231