The ratio of two quantities is 3 to 4. If each of the quantities is increased by 5, what is that ratio of these two new quantities?
We can't determine this. We could have 3 and 4 items or 3000 and 4000. Adding 5 in the first case drastically changes the ratio while adding 5 in the second case will have a negligble effect on the ratio.
However, if we were told that the second quantity doubled but the first remained the same, then the new ratio would be 3 to 8. Or, if we were told that the first quantity doubled but the second remained constant, then the new ratio would be 6 to 4 or 3 to 2.
Drum x is 1/2 full of oil and Drum Y, which has twice the capacity of Drum X, is 2/3 full of oil. If all of the oil in Drum X is poured into Drum Y, then Drum Y will be filled to what fraction of its capacity?
We can solve algebraically or by picking numbers. Because Drum Y is 2/3 full of oil, we should pick for its capacity a number that is divisible by 3--let's pick 300. Y is 2/3 full, so there is 200 units of oil in it. Drum Y has twice the capacity of that of X. Thus, Drum X's capacity is 1/2 of 300 or 150. Drum X is 1/2 full so it has 75 units of oil in it. So, if we pour those 75 units of oil into Drum Y, then Drum Y will have 200 + 75 = 275 units of oil. So, Drum Y will be filled to 275/300 or 11/12 of its capacity. Although that looks long-winded, if you are organized in your scratchwork, you can answer really quickly by picking numbers:
amount of oil-----75------200
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