Add the parts to find the whole
A bag holds red and blue counters in the ratio 2 : 3: for every 2 red counters there are 3 blue ones. The ratio compares one part of the bag with the other part.
Add the two numbers and you have the whole. Every 2 red counters come with 3 blue ones, so the bag is made of groups of 2 + 3 = 5 counters, and 2 counters in every group are red.
The whole bar is 2 + 3 = 5 equal parts. 2 of them are colored for red and 3 are plain for blue.
Each share as a fraction of the whole
The red counters are 2 of the 5 parts, so they are of all the counters. The blue counters are 3 of the 5 parts, so they are of all the counters. Together, , which is the whole bag.
The denominator of each fraction is the total number of parts, 2 + 3 = 5. It is never the number in the other share.
in a : b the share is a / (a + b), which is unchanged when both parts are multiplied
Make the colored share 2/5
Drag the handles to set the colored parts and the plain parts until the colored share is of the bar. 2 : 3 does it, and so does 4 : 6, because .
Sharing a real amount
Suppose the bag holds 20 counters. The red counters are of 20. Divide by the denominator: 20 ÷ 5 = 4. Multiply by the numerator: 2 × 4 = 8. So 8 counters are red.
The blue counters are of 20, which is 3 × 4 = 12. Check by adding the shares: 8 + 12 = 20, the whole bag. And 8 : 12 is 2 : 3 with both numbers multiplied by 4, so the counters really are in the ratio 2 : 3.
20 counters in 5 equal parts is 4 in each part. The 2 red parts hold 8 counters and the 3 blue parts hold 12.
The usual mistakes
"Two to three" is not "two thirds". The ratio 2 : 3 compares the red counters with the blue counters, not with the whole bag. The red counters are of the bag, not of it. To avoid the mix-up, whenever you meet a ratio, write down the total number of parts first.
Naming the other share. is the fraction of the bag that is blue. If the question asks about the red counters, the answer is .