Question

Mary wants to grow rose plants in her backyard. So she goes to the local nursery and spends a total of $4000 for 100 rose plants of 3 different colors :
pink : $120 each
white : $50 each
red : $25 each

Whas are the possible counts of each color plant ?

Answer :
(5, 41, 54)
(10, 22, 68)
(15, 3, 82)

Answer 1

I think we can set up a system of equations for this.

Answer 2

mmm
120 r +50 w+ 25 re =4000
r+w+re =100
and anyother condition ?

Answer 3

\[120p+50w+25r = 4000\]
\[p + w + r = 100\]

Answer 4

3 Variables 2 equations!?

Answer 5

yes, that was all the given problem... and I have the answer if it helps..

Answer 6

its not a probelm , its OR problem

u can add other condition which is all r,w,re >0
ans by graph get the integer solution

Answer 7

yes p,w,r have to be whole numbers as they represent the number of plants

Answer 8

I had half a rose before

Answer 9

*natural numbers

Answer 10

wait i have QSB lets see if it could solve it =)

Answer 11

Should we set up another equation using the 3 colors = ?

Answer 12

lol Mary wont buy half/quarter plants, let me add this constraint : she knows only natural numbers :)

Answer 13

another equation ?

Answer 14

I've modified the question slightly..

Answer 15

dividing through by 5 and eliminating a variable we get:

300=19p+5w

since 300 is divisible by 5, and apparently since there's only one answer, she had to have gotten 0 pink flowers.

300=5w

w=60

That means r=40

Check it. =)

Answer 16

how to graph two function in wolfram ?

Answer 17

So apparently "At least 1 flower" is something I didn't consider for my answer to work. Hmm. Now that I see the updated answers, I see I'm wrong.

Answer 18

120p + 50w + 25r = 4000
p+w+r = 100

r = 100 - p - w

Answer 19

24p + 10w + 5(100-p-w) = 800

19p + 5w = 300

Answer 20

yeah if we allow 0, we will get w=60, p = 0, r = 40 as one solution

Answer 21

ok so if u consider the line
(changed variable to feel comfortable =D )

19x+5y=300
can we use number theory xD
i mean we wanna x,y integers

Answer 22

I think in a sense allowing any amount of flowers to be zero, we break it. We could buy all red flowers if we wanted.

Answer 23

true ! but that may not be possible since 120 does not go evenly in 4000

Answer 24

We could easily buy almost any combination of red and white flowers.

Answer 25

ok so since we need integer
16x+5y=300
x should devide 5 right , since we need integer ( remember GCD =) )
so
x= 5 k or something

Answer 26

Yeah I think ikram is on the right track here. I am bad at number theory but this looks like a GCD sort of diophantine euclid's algorithm whatever idk.

Answer 27

ikram yes :) using number theory is fine, but please explain the solution :)

Answer 28

ok do u got the part that x have to be 5k ?

Answer 29

typo in your equation :

19x + 5y = 300

Answer 30

19x = 300-5y
19x = 5(60-y)

since 5 cannot divie 19, 5 has to divide x
so x = 5k

fine till here :)

Answer 31

ok so
x=5k *
19k +y=60
y=60-19k *

so what integer k that give integer y xD ( dumb guess )

Answer 32

k = 0, 1,2 will do ?

Answer 33

yeah from 0 up to 3 gives positive

Answer 34

so we have k=0
y=60

k=1
y=41

k=2
y=22

k=3
y=3

Answer 35

wow this looks real simple now xD
why do we eliminate k=0 case ?

Answer 36

so we have set of solution for x,y

(0,60) (5,41 ) (10,22 ) (15 ,3 )


mm does that work or we need to check something :-\

Answer 37

Gotcha ! k=0 makes x also 0, so we eliminate k=0 case !!

Answer 38

well yeah , but still its solution though

Answer 39

haha nope, 0 is not natural !
it was brilliant work from everyone, thanks a lot =))

Answer 40

why solving like this.
we just have to find the answer
x+y+z=100
120x+50y+25z=4000
try all x ,y and z to satisfy 1st and 2nd equation.
for e.g.,
A:(5,41,54)
so 1:5+41+54=100
2:120*5+50*41+25*54=4000
so 1st is correct.

Answer 41

to me, the solution we have worked here looks better than any other fancy number theoretic solutions using diophantine equations xD

Answer 42

haha well , what we did was diphontine =)
(some how )

Answer 43

speeks of NT ,
im trying to find why a prime number N of all 1's digits are rare :3

Answer 44

I think i came across that topic before, you're referring to prime numbers of form
`111111111...`

right ?

Answer 45

yep

Answer 46

wait ill show u something

Answer 47

@neer2890 that works if we have the options already :)

Answer 48

yes

Answer 49

but you have given all the options at first time, so i thought options are already available.

Answer 50

lol,u gave the options,mee to thought first one,then by solving all options gives 4000