Ratios & Proportions: Combining Ratios with Different Multipliers - Equate the Common Member
In a certain aquarium, the number of red fish is three times the number of the green fish, and the number of blue fish is half the number of red fish. If the ratio of green fish to red fish were doubled and the ratio of red fish to blue fish were also doubled, then the ratio of blue fish to green fish would be
Finally, combine the two ratios by equating the common member—red. Expand the first ratio by 4 and the second ratio by 3 to get
| | Red | : | Green | : | Blue |
|--------------|----------------------|---|-----------------------|---|----------------------|
| First Ratio | $$\color{red}{12}$$ | : | $$\color{green}{8}$$ | | |
| Second Ratio | $$\color{red}{12}$$ | | | : | $$\color{blue}{3}$$ |
With the common part equated, the ratio of blue to green fish is $$\color{blue}{3}{:}\color{green}{8}$$.
Incorrect.
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Incorrect.
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Incorrect.
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Incorrect.
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Correct.
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The two ratios are
| | Red | : | Green | : | Blue |
|--------------|-----|---|-------|---|------|
| First Ratio | $$3$$ | : | $$1$$ | | |
| Second Ratio | $$2$$ | | | : | $$1$$ |
Doubling the green$${:}$$red ratio means that there should be twice as many green fish for the same previous number of red fish—basically, double the green portion only to get $$3{:}2$$. Similarly, doubling the red$${:}$$blue ratio means doubling only the red side to get $$4{:}1$$.
So the two ratios are now
| | Red | : | Green | : | Blue |
|--------------|-----|---|-------|---|------|
| First Ratio | $$3$$ | : | $$2$$ | | |
| Second Ratio | $$4$$ | | | : | $$1$$ |
$$3{:}2$$
$$1{:}1$$
$$1{:}2$$
$$3{:}4$$
$$3{:}8$$
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