Comparing two amounts
A ratio compares two amounts. Suppose a paint mix uses 2 cans of red paint for every 3 cans of blue paint. We write this as the ratio 2 : 3 and read it as "2 to 3".
The order matters. The ratio of red to blue is 2 : 3, because red is named first. The ratio of blue to red is 3 : 2, which is a different ratio.
A ratio does not say how much paint there is. It says only how the two amounts compare: for every 2 parts of red there are 3 parts of blue, whether a part is a can, a cup or a bucket.
Every part is the same size. Red has 2 parts and blue has 3.
Equivalent ratios
Now make twice as much paint. Two batches of the mix hold 2 + 2 = 4 parts of red and 3 + 3 = 6 parts of blue, so the ratio is 4 : 6. Each batch is the same 2 : 3 mix, so the paint is exactly the same color as before.
Ratios that describe the same mix are called equivalent ratios, and we write 2 : 3 = 4 : 6.
Two batches: red has 4 parts and blue has 6. For every 2 parts of red there are still 3 parts of blue.
Multiply both numbers by the same number
Three batches give 6 : 9, and ten batches give 20 : 30. Each time, both numbers in the ratio were multiplied by the number of batches. Multiplying both numbers in a ratio by the same number gives an equivalent ratio.
This is how to find a missing number. In 2 : 3 = 8 : ?, the 2 became 8 by multiplying by 4. The 3 must be multiplied by 4 as well: 3 × 4 = 12. So 2 : 3 = 8 : 12, which is four batches of the mix.
Three batches: 6 parts of red and 9 parts of blue, so 2 : 3 = 6 : 9.
in a : b the share is a / (a + b), which is unchanged when both parts are multiplied
Make the ratio 1 : 2
Drag the handles to set the number of colored parts and plain parts. Make the ratio 1 : 2, then 2 : 4 and 3 : 6. The colored part of the bar stays the same length, because all three are the same mix.
The usual mistakes
Adding instead of multiplying. In 2 : 3 = 8 : ?, the first number went up by 6, so it is tempting to add 6 to the 3 as well and answer 9. But 8 : 9 is not the same mix: four batches have 4 × 3 = 12 parts of blue, so the answer is 12. Adding the same number to both numbers changes the mix. In the drawing above, 2 : 3 colors more of the bar than 1 : 2, even though each number is just 1 more.
Copying the other number. The missing number in 2 : 3 = 8 : ? is the amount of blue, so it comes from blue's 3, multiplied by 4. It is not the 8, which is the amount of red.