Aptitude - Area - Discussion

5. 

A rectangular park 60 m long and 40 m wide has two concrete crossroads running in the middle of the park and rest of the park has been used as a lawn. If the area of the lawn is 2109 sq. m, then what is the width of the road?

[A]. 2.91 m
[B]. 3 m
[C]. 5.82 m
[D]. None of these

Answer: Option B

Explanation:

Area of the park = (60 x 40) m2 = 2400 m2.

Area of the lawn = 2109 m2.

Area of the crossroads = (2400 - 2109) m2 = 291 m2.

Let the width of the road be x metres. Then,

60x + 40x - x2 = 291

x2 - 100x + 291 = 0

(x - 97)(x - 3) = 0

x = 3.

Video Explanation: https://youtu.be/R3CtrAKGxkc


Sai said: (Dec 16, 2010)  
What is the need of 60x + 40X -pow (x, 2) ?

Lekshmi said: (Dec 26, 2010)  
Same doubt as sai said above,

60x + 40x - x2 = 291?

60x+40x?

Prajwal said: (Dec 27, 2010)  
I got it. Its because. 60 and 40 get multiplies to width and. Since there are two cross roads. There will be a common space for two roads. (width of two cross roads is width^2).

Shreya said: (Dec 29, 2010)  
My doubt is

60x+40x-x2=291?.

Tanya said: (Jan 5, 2011)  
Its because the area of 2 roads has come out to be 291m2. Rest of the explaination has been given by prajwal.

Naju said: (Jan 8, 2011)  
Draw the figure you can c two cross road intersect, so we need to subtract it.

Ayoosh said: (Jan 12, 2011)  
For all having probs in the 60x + 40x - x^2!!!!

See road is a rect. and they are intercrossing. let the width be x, then for one road area will be 40x and for the other road will be 60x since 60 and 40 will be the lengths of road.

now, we area of roads is 291 m2, so

ar(road1) + ar(road2) - ar(intercrossing)
60x + 40x - x^2 = 291 and then on you can find the ans

Jhola said: (Jan 13, 2011)  
Very good ayoosh.

Ayoosh said: (Jan 19, 2011)  
If we dont subtract the X^2 THEN the area of intercrossing will be doubled since its accountd in area of both roads !! thats why we subtract x^2

Sujay Ramesh said: (Jan 22, 2011)  
Thanks ayoosh. Your explanation is good :).

Nakul Gowda said: (Jan 26, 2011)  
60x + 40x includes the intersection part twice..so to include the intersection part only once, subtract area of intersection part once, ie, x^2....got it yet..

Nayan said: (Jan 29, 2011)  
Thnks nakul.

Manish said: (Mar 4, 2011)  
Thanks ayoosh.

Golu said: (Mar 19, 2011)  
Why 97 was not taken as answer?

Amrut said: (Apr 6, 2011)  
thnks ayoosh & nakul

Sana said: (Apr 18, 2011)  
hey golu..
we have two options for the value of 'x' :
x=97 or x=3;
97 is not chosen as value of 'x' because the width of the rectangular park is only 40 m.
and only the value 'x=3' is less than 40 m.
so, 'x=97' is not possible at all.

Ankit said: (Jul 23, 2011)  
Thanks prajwal.

Abhi said: (Jul 27, 2011)  
Ayoosh explaination really helped me. Thanks Ayoosh :)

Koushik said: (Sep 12, 2011)  
Excellent Ayoosh. Thank you very much.

Sri said: (Sep 21, 2011)  
@Ayoosh good explanation.

Raji said: (Oct 12, 2011)  
@Aysooh superb yar

Aaru said: (Jan 5, 2012)  
Ayoosh your explaination is nice. I understood well now,

Praveen said: (Feb 7, 2012)  
Thanks ayoosh very nice explanation.

Manogaram said: (Mar 23, 2012)  
Hats off to ayoosh and prajwal for nice explanation.

Shruti said: (Mar 26, 2012)  
Thank you ayoosh.

Rkbm said: (Jun 14, 2012)  
X be the width. (60-x) (40-x) is area of remaining field.

Surendar Sundaram said: (Aug 26, 2012)  
40 X is width of the road.....
_________ We have already find out the area of the road by
| | | | subtracting (total area -lawn area)...
|___| |___| road area == 291 sq units
|___| |___|60 i.e Area of two triangle(roads)-(overlap btwn them)
| | | | over lap is width^2====x^2
|___|_|___| area of two triangle (60*x) and (40*x)
|x| so (60x+40x)-x^2===291
| | by solving tis v can find x===3

Ankanna said: (Oct 8, 2012)  
See road is a rect. and they are intercrossing. let the width be x, then for one road area will be 40x and for the other road will be 60x since 60 and 40 will be the lengths of road.

Now, we area of roads is 291 m2, So

area(road1) + area(road2) - area(intercrossing).

60x + 40x - x^2 = 291 and then on you can find the answer.

Imran Shaikh said: (Nov 26, 2012)  
How is this calculated ?

x2 - 100x + 291 = 0 ----------> (x - 97)(x - 3) = 0

Hashmuddin said: (Nov 29, 2012)  
x2 -100x +291=0
x2 -97x -3x +291=0
x(x -97)-3(x-97)=0
(x-97)(x-3)=0

Mahi said: (Jan 30, 2013)  
40*3=120
60*3=180
TOTAL=300
LESS=291
BALANCE=9 SQCM
THAT IS 3*3
ANS=3.

Ashish Malviya said: (Feb 18, 2013)  
Cosidering Road parallel to 60m side Area :- 60x.
Cosidering Road parallel to 40m side Area :- 40x.

Common area :- x*x.

Hence :- 291=60x+40x-x*x.

Rajan said: (May 17, 2013)  
The road is in the shape of plus mark.

So area of 1st road is 60x.

So area of 2nd road is 40x.

But in the middle the square has been calculated twice above. So subtract that area x^2.

Rupam said: (Jun 19, 2013)  
There is no intercrossing left. When we considered 60x +40x, aren't we considering the intercrossing too? Otherwise the lengths couldn't have been 60 and 40.

Shardul said: (Aug 23, 2013)  
Area of the park = (60 x 40) m2 = 2400 m2.

Area of the lawn = 2109 m2.

Area of the crossroads = (2400 - 2109) m2 = 291 m2.

Let the width of the road be x metres. Then,

60x + 40x - x2 = 291.

x2 - 100x + 291 = 0.

(x - 97)(x - 3) = 0.

x = 3.

Diego said: (Jul 15, 2014)  
So it is a rectangle of 60x40cm. It has 2 crossroads, therefore the area of the roads is the horizontal one: 60 times the with (x). -->60*x plus the vertical road 40*x. This results in 60x+40x but you are adding 2 times the area where the roads intersect (one time for the vertical road and one time for the horizontal road). The double counted area is x*x = x^2. So if you subtract this area from 60x+40x you will eliminare 60x+40x -x^2.
                       60cm
_______________________________
I LAWN I RD I LAWN I
I______________I____I________________I
I ROAD I I ROAD I
I_____________ IX^2I________________I 40cm
I LAWN I I LAWN I
I______________I RD I________________I


Hope this helps you better understand the problem.

Peter said: (May 6, 2015)  
@Rkbm your answer best answer.

Puskar Prasun said: (Sep 6, 2015)  
Guys! it should be 2x^2 instead of simply x^2? don't you all agree?

Saila said: (Jan 4, 2016)  
@Puskar,

You may go through Diego's answer for better understanding.

Priyanka said: (Apr 22, 2016)  
We can also subtract the width of the road from length and breadth of the road and multiply the values to get the area of the lawn i.e.

Let the width be x.

Length of lawn = 60 - x.

Breadth of lawn = 48 - x.

(60 - x) (40 - x) = 2109. As given in question.

X^2 - 100x - 291 = 0.

X = 3 and 97.

The value 3 is there in the given option. So, width of the road is 3m.

Aravind said: (Jun 25, 2016)  
Great explanation @Ayoosh.

Arjun said: (Aug 14, 2016)  
(60x + 40x -x^2 this is the area of cross roads alone. 60 and 4O are lengths of cross roads and x is the breadth.

X^2 is subtracted because the meeting of crossroads, the calculated area overlaps. So - X^2.

Imtiyaj said: (Aug 18, 2016)  
God job @Ayoosh.

Lalitha said: (Sep 6, 2016)  
Thank you @Ayoosh.

Lavanya said: (Sep 10, 2016)  
@Ayoosh,

60 and 40 are not the lengths of the roads. 60 is the length and 40 is the width. Still, have a confusion. Could anyone please explain it clearly?

Ammu said: (Oct 11, 2016)  
Thanks for the explanation.

Savita said: (Nov 13, 2016)  
Thanks @Priyanka for the answer.

T Balaji said: (Dec 7, 2016)  
Thank you @Ayoosh.

Aayushi said: (Dec 9, 2016)  
Where is 291?

How to solve this?

Pratiti Das said: (Dec 31, 2016)  
How come two concentric roads have 60x + 40x?

Noble Phelix said: (Jan 9, 2017)  
Good job @Ayoosh.

Gayatri said: (Feb 19, 2017)  
@All.

Please draw the picture first.

Two roads intersecting each other like " + " sign not like "x" sign. One road has an area of 60x other one has 40x. There is a common area of x^2.

Vijeta said: (May 5, 2017)  
Nice explanation. Thank you all.

Van said: (May 23, 2017)  
So, x is squared because it is assumed that the two x's are of equal value?

Salman said: (May 31, 2017)  
Thanks to all for giving the explanation.

Tamil said: (Jul 23, 2017)  
Well said @Ayoosh.

Bhargavi said: (Sep 11, 2017)  
When two roads intersecting like a +.
So road 1 area is l*b =60*x rectangle road 2 area is l*b =40*x rectangle intersecting place is in a square shape so side =x.; the area is x^2.

In road 1 and road2 we are calculating intersecting point area two times so we have to Subtract that square shape intersecting point.

Bond said: (Oct 26, 2017)  
Thanks @Ayoosh.

Torikul said: (Nov 27, 2017)  
Thanks for your good explanation @Priyanka.

Smita said: (Jan 25, 2018)  
Thanks @Ayoosh.

Theodore said: (Feb 3, 2018)  
What will be the LENGTH of the road? Can Anyone help me to find out the answer?

Pooja said: (Apr 10, 2018)  
Thank you @Ayoosh.

Shivangi Goel said: (Jul 4, 2018)  
How this quadratic equation is formed in x?

Vijay said: (Aug 17, 2018)  
@Ankanna.

Your explanation really helped me. Thanks.

Himanshu said: (Aug 26, 2018)  
Why only square of width is taken and not length?

Sreekanth said: (Sep 11, 2018)  
@All.

Take:
(60-x) * (40-x).
We gets the remaining area,
Already in the questions, it gives that remaining area is 2109 m2,

Verify the equation with 3.
The equation will be satisfied.

Harika said: (Sep 23, 2018)  
The intersection part area should be subtracted it's area = x^2.
Area of one road along length = 40x.
Area of one road along breadth = 60x.
But the area of roads is 2400-2109 = 291.
So 40x+60x-x^2 = 291.
Solve, we get x=3.

Sly said: (Jan 3, 2019)  
Thanks all for explaining the answer in detail.

Sk Rahad Mannan said: (Feb 2, 2019)  
Thanks @Ayoosh, @Tanya.

Sekhar said: (Feb 8, 2019)  
60x+40x-291 = x*x.

Then what about 291?

Can you explain?

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