Mensa Puzzle - Can you solve it? Explain your answer?

Question

I recently took advantage of a local nursery's sale featuring "you pay no sales taxes".

I bought rose bulbs at $11.55/bag, and violet bulbs at $13.65/bag, as well as some gardenia at $21.45, camellia at $30.03 and japanese maple at $50.05. The final bill came to $600.92.

How many gardenia did I buy?

Answer 1

First you should factor the prices (in cents):
rose 1155 = 3*5*7*11
violet 1365 = 3*5*7*13
gardenia 2145 = 3*5*11*13
carmellia 3003 = 3*7*11*13
maple 5005 = 5*7*11*13
The important thing to note is that all of these prices have 4 of the 5 factors 3,5,7,11&13. Each is missing a different factor. Considering gardenia for example, the other four prices are divisible by 7. So we can divide the grand total (in cents) by 7 and get a good clue as to how many gardenia there are. 60092/7 = 8584 with a remainder of 4.
We need to find how many gardenia we need to buy to get a remainder of 4. It turns out if you buy 6 gardenia, the cost is $128.70. Dividing 12870 by 7 gives a remainder of 4.
This means that there must be either 6 or 13 gardenia. Similarly there are 10 or 23 bags of rose bulbs, 10 or 21 bags of violet bulbs, 4 or 9 carmellia and 2 or 5 japanese maples.
If we add up the price for the minimum of each we get 115.50+136.50+128.70+120.12+100.10 = 600.92. So we know that you purchased 10 rose bulbs, 10 violet bulbs, 4 carmellia, 2 japanese maples and 6 gardenia :-)

Answer 2

First, I'm assuming all purchases are whole numbers >0

Second. notice that the cents are $0.92. all the rest of the numbers end in $.05 any multiple of 5 ends in a 0 or a 5. so to make a "2" in the last digit, you have to have a multiple of the $30.03 number = 4x or 14x or 24x or 34x. etc... since 4 x 30.03 = 120.12, and 14 x 30.03 = 420.42 and 24 x 30.03 = 720.72 (which is more than the total), then the number of camellias you bought has to be either 4 or 14.

let's start with 14. if you bought 14 camellias, then you bought 600.92 -420.42 = $180.50 of everything else.

if that is the case, then you bought either 1, 2, or 3 japanese maples. let's start with 3. 3 x $50.05 = 150.15. leaving 30.35 for the rest. since 1 x 11.55+ 1 x 13.65 + 1 x 21.45 >30.35, you couldn't have bought 3 japanese maples (and 14 camellias). so the number must be 1 or 2. so let's try 2. 2 x 50.05 = 100.10 leaving 80.40 for the rest. so if again we assume 1 of each minimum, 1 x 11.55 + 1x13.65 x 1 x 21.45 = 46.65 which leaves $33.75 remaining. there are no combinations of 11.55, 13.65, and 21.45 that add to 33.75 so you couldn't have bought 14 camellias and 2 japanese maples. so let's try 1 maple....

600.92 - 14x30.03 - 1x 50.05 -1 x 11.55 - 1 x 13.65 - 1 x 21.45 = $83.80. so are there any multiples of 11.55, 13.65 and 21.45 that add to 83.80?

start with 21.45. 3 x 21.45 = 64.35 leaves 19.45. can't be that. 2 x 21.45 = 40.90. can't be that either. what about 1x 21.45 leaves $62.35. well in that case if we then say 4 x 13.65 = 54.6 leaving 7.75 (can't be) 3 x 13.65 = 40.95 leaving 21.4 (can't be because 21.4/11.55 does not equal integer) 2 x 13.65 = 27.3 leaving 35.05 (can't be. same reason). same for 1. leaves 48.7. 0 leaves 62.35.

Therefore, we could not have purchased 14 camillas. meaning we purchased 4 camillas..... whew.

now we know this....

the total price we spent for roses, violets, gardenias, and japanese maples = 600.92-120.12 = 480.80

We also know the max number of maples = 480.80 / 50.50 = 9 right?

so let's start there.....9 maples leaves 480.80 - 9x 50.50 = 30.35. since 1 x 11.55 + 1x 13.65 + 1 x 21.45> 30.35, it can't be 9......

let's try 8. 480.80 - 8x 50.05 = 80.40. since 1 x 11.55 + 1x 13.65 + 1 x 21.45 = 46.65, we have 33.75 left. since there are no combinations of 11.55, 13.65 and 21.45 that add to 33.75, 8 maples didn't work.

let's try 7. 480.80 - 7x 50.05 = 130.45. gThis is getting complicated.... so let's try this approach. notice that the values of 11.55 13.65 and 21.45 are all divisible by 5? hey they're also divisible by 3. ie they're divisible by 15....

if we let the number of roses = a, violets = b, gardenias = c, then the total cost = 11.55a + 13.65b + 21.45c. factor out 15 to get... 15x (.77a + .91b + 1.43c) so now if we say....

130.45 = 15x (.77a + .91b + 1.43c) and realize that the right side is 15 x ( a number in the form of $de.fg) = a number in form of $hi.jk, this is due to us multiplying by integers, the left side must therefore be divisible by 15. to get into the form $hi.jk. so since 130.45 / 15 = 8.69666666, this doesn't work. ie maples does not = 7

ok. for 6 maples we get 480.80 - 6x 50.05 = 180.50
since 180.50/15 = 12.033333, 6 doesn't work.

try five. same way...480.80 - 5x 50.05 = 230.55

230.55 = 15 x (.77a + .91b + 1.43c)
230.55/15 = 15.37 = .77a + .91b + 1.43c

so far so good. remember also that a =>1, b=>1 and c=>1. So the max can be is 9, if c=9, remainder is .82 and this doesn't fit. neither does 8. and so on. so this value of maples = 5, doesn't work either.

let's try 4. $480.80 - $4x 50.05 = 280.60. 280.60/15 = 18.70667 nope

maples = 3. $480.80 - $3x 50.05 = 330.65. 330.65/15 = 22.0433

maples = 2 $480.80 - $2x 50.05 = 380.70. and 380.70/15 = 25.38

this works so far.. hold that thought.

maples = 1. $480.80 - $1x 50.05 = 430.75. 430.75/15 = 28.7166666. so maples do not equal 1

whew. maples must = 2.....

now we know camellia = 4, maples = 2 remaining plants cost 380.70

and

380.70 / 15 = 25.38 = (.77a + .91b + 1.43c)

well. I know right off the bat that since a =>1, and b=>1, the (25.38 - .77 -.91)/1.43 = 16.6, therefore c<17 that's still a very complicated permutation problem... so if I look at .77 and .91, they are both divisible by 7. ie

25.38 = 7 x (.11a +.13b) + 1.43c. Ah ha... I have to find a value of c such that 25.38 - 1.43c = a number divisible by 7. so.....(keep in mind that c ranges from 1 to 16...)

25.38 - 1.43x1 = 23.95. nope
25.38 - 1.43x2 = 25.52. nope
....
25.38 - 1.43x6 = 16.80 ... 16.80/4 = 2.40 this works!
25.38 - 1.43x13 = 6.79 ... 6.79/4 = .97 this also works!

meaning c = 6 or c=13.....

if c =6, then (.11a+.13b) = 2.40. .11x10 = 110, and .13x 10 = .130. and this works......

a = 10, b =10, c = 6. camellia's = 4 and maples = 2

ie...

10 roses, 10 violets, 6 gardenias, 4 camellias and 2 japanense maples....

Answer 3

Mensa people sure aren't the brightest gardeners. Roses don't have bulbs.

Answer 4

23?