word problem with inconsistant and dependents

Question

A florist is creating 10 centerpieces for a wedding, each containing 12 flowers. Each piece will contain roses that are $2.50 each lilies that cost $4 each and iris that cost $2 each. The budget is $300. The number of roses will be twice the number of the lilies and iris combined. How many type of each flower will be used?

Answer 1

Hi Madison,

 

Here's the solution - This problem has three variables - namely, the number of roses (R), number of lilies (L) and the number of Iris (I). We need three equations in total to solve for three variables.

 

From the first sentence of the question, because the florist needs 10 centerpieces with 12 flowers each, the total number of flowers is 120, which can be written in mathematical form as,

 

R + L + I = 120 ------------------- A

 

The second sentence in the question states the prices of each of the flowers and the total budget; The total cost of roses is the number of roses (R) times the rate of each rose (2.5). Similarly, the cost of lilies is 4L, and the cost of Iris is 2I. So the total cost of all the flowers is (2.5 R + 4 L + 2 I). Now, the total cost of all the flowers cannot exceed 300 dollars, so we have another equation,

 

(2.5 R + 4 L + 2 I) = 300 --------- B

 

The sentence states that the total number of roses is TWICE the sum of lilies and iris, so our third equation is,

 

R  = 2 (L + I) ----------  C

 

 

Substitute the value of R from equation C equation A, we get,

3L + 3 I =120, or L+I = 40. This implies (if you replace this sum in equation C), R = 80

 

If L + I = 40, then L= 40 -I; Replace this new value of L in equation B to get,

 

2.50 R + 4(40-I) + 2I = 300; we know that R =80, so substitute this value of R in equation to further arrive at,

 

2.50(80) + 4 (40 -I) + 2I = 300

or, 200 + 160 - 4I + 2I = 300

or, 360-300 = 3I => I = 30

 

So L = 10 (because L + I =40)

 

So, the florist will use 80 roses, 10 lilies and 30 iris.

 

 

 

 

Answer 2

Let R = # roses

Let L = # lillies

Let I = # irises

 

We know there will be a total of 10 centerpieces with 12 flowers in each so

the total number of flowers is 10(12)= 120

Therefore R + L + I = 120

 

We know the cost of the flowers should total to $300

total cost of the roses is 2.5R

total cost of Lillies =  4L

total cost of Irises = 2I

 

Total cost equation...   2.5R + 4L + 2I = 300

 

We also know that the number of roses is twice the number of lilies and iris combined

So R = 2(L+I) = 2L + 2I

 

 

So far we have a couple of equations to use to solve this:

 

R + L + I = 120

2.5R + 4L + 2I = 300

 

But since R = 2L + 2I we can substitute that back in:

 

2L + 2I + L + I = 120    ...                 3L + 3I = 120   eqn 1

2.5(2L + 2I) + 4L + 2I = 300  ...        9L + 7I = 300   eqn 2

 

Multiply eqn 1 through by -3 and add it to eqn2 to eliminate the L variable and solve for I

 

-9L - 9I  = -360   eqn1(-3)

9L + 7I =  300

-----------------

        -2I = -60

          I = 30

 

We now know there are a total of 30 irises which will be 3 in each of the 10 centerpieces

Plug that back into one of the original eqn 1 or eqn 2 to solve for L... I will use eqn 1.

3L + 3I = 120

3L + 3(30) = 120

3L = 120 - 90

3L = 30

L = 10

There are a total of 10 Lillies so there will be 1 in each centerpiece

 

To find the number of roses use the fact that R = 2L + 2I

                                                                  R = 2(10) + 2(30)

                                                                 R = 80

There are 80 roses total so each centerpiece will have 8 roses

 

Roses = 80   or  8 per centerpiece

Lillies = 10   or 1 per centerpiece

Irises = 30   or 3 per centerpiece