A future rectangular swimming pool is to have an area

Question

A future swimming pool is to be a rectangle with an area of 480 square meters. Both its length and width must be whole numbers of meters, and its length must not be more than three times its width. How many possibilities are there for the length of the pool?

A. 2
B. 3
C. 4
D. 5
E. 6

A. 2
B. 3
C. 4
D. 5
E. 6

paragw · Answer

Hi, I noticed that the question  does not state that the length must be greater than the width . Without that condition, all 12 pairs of factors in which L<W will also satisfy for the given restriction, so could you please confirm if  L>W  was inadvertently omitted?

kevincan · Answer

I debated that when I wrote the question , but decided to leave it out. There’s certainly no harm in including that clarification

findingmyself · Answer

First Lets factorize 480, which is 2*2*2*2*2*5*3

Now we have following conditions Length (L) * Breadth (B) = L*B=480 and L>B and L<3B

Start with a mid point which is 24*20 (Satisfied)

Now we are minimizing breadth from 20 to 19- not possible as 19 is not divisible by 480
18- Not divisible as there is only one 3 in 480
17- prime number
16- Possible when breadth is 16, length will be 30 (Satisfied)
15- possible when breadth is 15, length is 32 (Satisfied)
14- Not possible
13-Not possible
12- Possible but then L should be less than 3B, as L will be come 40 and 3B will become 36. Now we can stop here as going further will only increase L

Now going in other direction when B is above 20
when B is 21, not possible (no 7 in factorization of 480)
22- NP, 11 is not there
23 Prime number
24- Possible but then L becomes 20 which is not satifying L>B
So we can stop further increasing B as it will lead to a smaller L

So we have only three solutions:
16- Possible when breadth is 16, length will be 30 (Satisfied)
15- possible when breadth is 15, length is 32 (Satisfied)
and 24*20

sanjanaagrawal09 · Answer

how long did it take you to solve this question with this approach?

findingmyself · Answer

It took close to 4 minutes, but I could have saved more time if I knew the mid point to start with, that was where i struggled, I did by starting from 1*480 2*240 etc. Then after completing the question i figured this way.

sanjanaagrawal09

how long did it take you to solve this question with this approach?

kevincan · Answer

Decompose 480 with a tree

480 = 2^5 * 3 * 5

480 can be written as a product of factor pairs, but don’t write them all out. Start with the pair whose ratio is closest to one to one .

Look for the smallest factor greater than sqrt (480) : this number 24 is the smallest possible length . Then look for the next higher factor until the ratio of the length to the width is greater than 3 to 1

Length .......width 
24......... 20 good 
30............16 good 
32 ............15 good 
40.............12 not good since 40/12 > 3

A future rectangular swimming pool is to have an area

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A future rectangular swimming pool is to have an area

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615 to 715 in 15 Days (Ria's Strategies)

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