The ratio of boarders to day scholars at a school is 7 to 16. However, after a few new students join the initial 560 boarders, the ratio changed to 1 to 2, respectively. If no boarders became day scholars and vice versa, and no students left the school, how many boarders joined the school?

Correct.
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The change in the boarders ratio is $$+1$$ (from 7 to 8).
Use this to find the multiplier: the real number of original boarders divided by original boarders ratio is $$\frac{560}{7} = \color{red}{80}$$
Multiply this multiplier by the *change in boarders* ratio to find change in the number of boarders.
>$$+1 \cdot \color{red}{80} = 80$$
Hence, the change of one unit in *ratio* corresponds to a change of $$+80$$ students in *real*. Therefore, this is the correct answer.

Incorrect.
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You may have gotten this answer if you made a calculation error when finding the multiplier.

Incorrect.
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Incorrect.
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Incorrect.
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48

64

70

80

84