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# In the figure above, rectangle PQRS is a shaded region inside the squa

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In the figure above, rectangle PQRS is a shaded region inside the squa  [#permalink]

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29 May 2020, 02:24
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Difficulty:

45% (medium)

Question Stats:

68% (01:47) correct 32% (01:31) wrong based on 71 sessions

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In the figure above, rectangle PQRS is a shaded region inside the square ABCD. What is the probability that a point chosen at random from the square ABCD will lie inside the shaded rectangle PQRS?

(1) The length of a diagonal of rectangle PQRS is 55% the length of a diagonal of square ABCD
(2) The length of side PQ is 20% greater than the length of side QR

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Re: In the figure above, rectangle PQRS is a shaded region inside the squa  [#permalink]

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29 May 2020, 02:52
2
Suppose PQ=a; QR=b; AB=l

Probability = Area of rectangle/ Area of square $$= \frac{a*b}{l^2} = \frac{a}{l}*\frac{b}{l}$$

Statement 1

Length of diagonal of rectangle $$= \sqrt{a^2+b^2}$$

Length of the diagonal of square = $$\sqrt{2}l$$

$$\frac{\sqrt{a^2+b^2}}{\sqrt{2}l}$$ = $$\frac{11}{20}$$

$$\frac{a^2}{l^2} +\frac{b^2}{l^2} = \frac{121}{200}$$.....(1)

We can't find $$\frac{a}{l}$$ and $$\frac{b}{l}$$, using above equation

Insufficient

Statement 2- We know a/b. But there is no way to find the ratios of the area of rectangle and square.

Insufficient

Combining both statements

$$\frac{b}{l} = \frac{5}{6}(\frac{a}{l})$$ (From statement 2). Also $$\frac{a}{l}, \frac{b}{l} >0$$

Hence, we can find the unique value of $$\frac{a}{l}$$ and $$\frac{b}{l}$$, using equation (1) (From statement 1)

Sufficient

Bunuel wrote:

In the figure above, rectangle PQRS is a shaded region inside the square ABCD. What is the probability that a point chosen at random from the square ABCD will lie inside the shaded rectangle PQRS?

(1) The length of a diagonal of rectangle PQRS is 55% the length of a diagonal of square ABCD
(2) The length of side PQ is 20% greater than the length of side QR

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Re: In the figure above, rectangle PQRS is a shaded region inside the squa  [#permalink]

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29 May 2020, 02:55
1
IMO C

Statement 1: The length of a diagonal of rectangle PQRS is 55% the length of a diagonal of square ABCD (but the diagonal could be anywhere moreover we need value of length and breadth of rectangle to determine area) (not sufficient)

Statement 2: The length of side PQ is 20% greater than the length of side QR
(But the rectangle can of any length and breadth, need some parameter to draw area with respect to square) (not sufficient)

Statement 1&2:
Let diagonal of square be 100
Therefore side S of square will be
100^2 = S^2+S^2
10000 = 2S^2
5000 = S^2
S= 50√2 and area of square is 5000 ------(1)

The length of rectangle diagonal is 55% of that of square= 100*55/100=55
Also L = 120B/100 -------(2)

55^2=L^2+B^2
3025 = (1.2b)^2+b^2
3025 = 1.44b^2+b^2
3025 = 2.44b^2
B^2 = 1239.75
B= 35.21 and L = 35.21*120/100=42.25
Area of rectangle (shaded region) = L*B = 1487.7 ------(3)

Probability of that point being in shaded region is = 1487.7/5000 = 0.2975 or 0.3 (approx)

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Re: In the figure above, rectangle PQRS is a shaded region inside the squa  [#permalink]

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29 May 2020, 10:13
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Re: In the figure above, rectangle PQRS is a shaded region inside the squa  [#permalink]

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30 May 2020, 12:48
IMO Ans A

St 1 is sufficient.

Probability=
Area of Rectangle / Area of Square

If Diagonal of Square is x
Area of Sq= x^2/2

And Diagonal of Rect will be 0.55x

Area of Rect= (0.55x)^2/2

Probability: 0.55^2

In other words,

If side ratio is a/b
Area ratio is (a/b)^2

St 2 is Insufficient because the ratio is between two sides of rectangle.

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Re: In the figure above, rectangle PQRS is a shaded region inside the squa   [#permalink] 30 May 2020, 12:48