Ratios & Proportions: Ratio Changes by Addition/Subtraction
The ratio of trees to benches in a park is 4 to 1. If adding 10 more benches in the park changes the ratio to 4 to 3, how many trees are there in the park?
Next, note the change in ratio in the benches column:
| | T | | B | |
|----------------|---|---|----|-------------|
| Original ratio | 4 | : | 1 | |
| Change | | | +2 | +10 benches |
| New Ratio | 4 | : | 3 | |
Use the difference in ratio units and the corresponding change in real values to find the multiplier.
Since a rise of 2 in the benches ratio (from 1 to 3) corresponds to a rise of 10 in real, the multiplier is simply $$\frac{10}{2} = \color{red}{5}$$.
Use the multiplier to find the required quantity. Remember to use the ratios in their expanded/reduced form, rather than the original form.
The question asks for the number of trees, which is simply the original ratio of $$4 \cdot \color{red}{5} = \color{purple}{20}$$.
Incorrect.
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You might have gotten this answer if you calculated the number of benches in the park after 10 are added.
Incorrect.
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You may have gotten this answer if you made a mistake calculated the multiplier.
Incorrect.
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You might have gotten this answer if you calculated the number of benches in the park.
Incorrect.
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You may have gotten this answer if you made a mistake calculated the multiplier.
Correct.
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Compare the two ratios—expand/reduce so that the unchanged quantity is represented by the same number in ratio. Note the change in ratio units.
The common quantity in those ratios is the unchanged number of trees. The ratio units of the trees are already equal, so the ratios may be combined as is—no need to reduce/expand the ratios in order to compare them.
5
10
16
20
24
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