Rectangular region QRST is divided into four smaller rectangular regions, each with length l and width w.

Quantity A	Quantity B
\(\frac{QR}{QS}\)	\(\frac{3}{4}\)

A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.

I think the question should have the Quantity 'A' asking for the ratio between QR/RS or QR/QT

In this case the Each rectangle inside the bigger rectangle has a width 'x'( Assume)

Now QR is nothing but the combination of 3 horizontally placed rectangles whose side length is the addition of the widths of those rectangles

QR = x+x+x = 3x ( From this ,the length of each of the rectangle is 3x - we can Conclude this by carefully noticing the 3 Horizontal rectangular widths = one vertically lying rectangular length)

Now coming to the denominator : If we need to Find out RS or QT we can notice the side is the combination of two rectangles which is Length of one Horizontal rect + width of vertical rect

So RS = x + 3x = 4x

Finally we get the ration as 3x/4x and we can cancel 'x' in numerator and denominator which is left with a ration of 3/4 and it is equal to quantity B

Thank you

Correct me in case of any mistakes

Now we can see from the figure that \(QR = 3 \times w\) and RS is \(RS= l+w\)


Now \(QS =\sqrt{QR^2 + RS^2}=\sqrt{(l+w)^2 + (3w)^2}\)

So ratio \(QR:QS=3w:\sqrt{(l+w)^2 + (3w)^2}=3w:\sqrt{(4w)^2 + (3w)^2}= \frac{3}{5}\).


Hence option B is correct.
_________________

Sandy
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