In problems on ratios, it’s always important to know the fundamental properties and use them in solving questions.
If both the terms of a ratio are multiplied by the same constant, the ratio does not change.
For example, if ratio a:b = 5:6 and we assume that a=5 and b=6, then multiplying both the numbers by 2 will give us 10 and 12, which are also in the ratio of 5:6.
However, we cannot say the same thing about addition or subtraction. Adding the same number to both the terms of the ratio will change the ratio in a certain way, depending on what type of ratio is given. And will subtracting the same number from both the terms of the ratio.
With this information, it is now easy to see that statement I alone is sufficient.
\(\frac{3c}{3d}\)= ¾ only means that \(\frac{c}{d}\) = ¾. This is a unique answer to the question which asked us to find out the ratio of c and d.
Statement I being sufficient, the possible answer options are A or D. Answer options B, C and E can be eliminated.
From statement II, we can say \(\frac{c+3}{d+3}\) = \(\frac{4}{5}\). But this is not sufficient to find unique values for c and d and hence a unique value for c:d.
For example, if c = 1 and d = 2, then, \(\frac{c+3}{d+3}\)= \(\frac{4}{5}\). In this case, c:d = 1:2.
On the other hand, if c = 5 and d = 7, \(\frac{c+3}{d+3}\) = \(\frac{4}{5}\). But, in this case, c:d = 5:7.
Therefore, statement II is insufficient. Answer option D can be eliminated.
The correct answer option is A.
Hope this helps!
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