Although students do not usually think so, there is a difference between a ratio and a fraction. It is true that we can simplify a ratio the same way we simplify a fraction. For example, the ratio 6 to 9 or 6/9 is equivalent to the ratio 2 to 3 or 2/3.
However, ratios do not always follow the same rules as fractions.
1. We do not add or subtract ratios although we can add or subtract fractions.
2. A ratio does not always compare things that have the same units although a fraction compare things with the same units. With ratios, the units may or may not be the same.
For example, the fraction 8 cakes/2 sodas does not make sense. Why not? It is because this will mean that we are trying to find out how many sodas can go into 8 cakes. Nonsense!
However, the ratio 8 cakes/2 sodas make perfect sense even though the units are not the same. The ratio 8 cakes/2 sodas just tells us that for every single soda, there are 4 cakes.
Similarly, the ratio 40 miles to 2 gallons or 20 miles to 1 gallon makes perfect sense if you are trying to see how far you can go with 1 gallon of gas.
The ratio 20 cakes / 15 cakes also makes perfect sense if you are comparing the number of cakes you made to the number of cakes your neighbor made in a given year.
3. The denominator in a fraction usually represents the number of parts in the same whole. For example, in the fraction 8 biscuits/4, 4 represents the number parts in the same whole or 8 biscuits.
However, suppose a bag has 80 red balls and 50 blue balls. The ratio 80/50 can be simplified to 8/5. In the ratio 8/5, the denominator 5 represents the number of parts in another whole and that whole is the number of blue balls, not the number of red balls.