Question : Which smallest positive number should be subtracted from each of 9 and 13 so that 18 is the third proportion to them?
Option 1: 2
Option 2: 4
Option 3: 3
Option 4: 1
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Correct Answer: 1
Solution :
Let the required number be $x$.
For third proportional of $9-x$ and $13-x$, $(9-x):(13-x)::(13-x):18$
So, $\frac{(9-x)}{(13-x)}=\frac{(13-x)}{18}$
⇒ $18\times (9-x)=(13-x)^2$
⇒ $162-18x=169-26x+x^2$
⇒ $x^2-8x+7=0$
⇒ $(x-1)(x-7)=0$
⇒ $x=1,2$
Since we need the smallest number, $x=1$
Hence, the correct answer is 1.
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