Ratio, proportion & rates of change · Ratio
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Ratio problems where the amounts change
How to handle ratios where quantities are added, removed or transferred by writing the parts as multiples of a common number and solving for it.
Ratio, proportion & rates of change · Ratio
Ratio problems where the amounts change
How to handle ratios where quantities are added, removed or transferred by writing the parts as multiples of a common number and solving for it.
Why it works
The hardest ratio questions give you a ratio before a change and a ratio after it — "the ratio of red to blue is ; more red are added and it becomes ." You can't just work with and as if they were the real amounts, because there could be and , or and , or and — all in the ratio . The trick is to write the unknown amounts as multiples of a single number:for some number you don't yet know. This is the whole idea: and are automatically in the ratio whatever is, so they capture every possible starting amount in one expression. The change and the "after" ratio then give you an equation to pin down .
Set it up, apply the change, solve. After adding red, the amounts are and , and these are in the ratio . Two ratios being equal is an equation — cross-multiply:
Now un-pack: blue , red . (Check: after adding red, . ✓) The change is applied to the actual amounts (), never to the ratio numbers themselves — you can't "add to the ".
Transfer problems work the same way. If two jars hold marbles in the ratio and you move from the first to the second, the amounts become and — one goes down by , the other up by , because the marbles are moved, not created. A common slip is to add to one jar without taking the same amount off the other.
The three-line habit:
- Write the starting amounts as multiples of (e.g. and ).
- Apply the change to those expressions, and set the result equal to the new
- Solve for , then substitute back to get the actual amounts.