Studdy AI Logo
Math

Question Find the cost of 4 potted plants and 5 bags of compost given the costs of 7 plants + 2 compost and 3 plants + 4 compost.
Cost of 7 potted plants and 2 bags of compost: £41£ 41 Cost of 3 potted plants and 4 bags of compost: £27£ 27

Studdy Solution

STEP 1

Assumptions
1. The cost of 7 potted plants and 2 bags of compost is £41.
2. The cost of 3 potted plants and 4 bags of compost is £27.
3. Emily wants to buy 4 potted plants and 5 bags of compost.
4. The cost of potted plants and bags of compost is consistent.

STEP 2

Let's denote the cost of one potted plant as p p and the cost of one bag of compost as c c . We can then create two equations based on the given information.

STEP 3

The first equation is based on the cost of 7 potted plants and 2 bags of compost:
7p+2c=41 7p + 2c = 41

STEP 4

The second equation is based on the cost of 3 potted plants and 4 bags of compost:
3p+4c=27 3p + 4c = 27

STEP 5

We now have a system of two linear equations with two variables:
\begin{align*} 7p + 2c &= 41 \\ 3p + 4c &= 27 \end{align*}

STEP 6

To solve this system, we can use the method of substitution or elimination. In this case, we will use the elimination method to eliminate one of the variables.

STEP 7

First, we will multiply the second equation by 2 so that the coefficients of c c in both equations match:
2(3p+4c)=2(27) 2(3p + 4c) = 2(27)

STEP 8

Simplify the equation:
6p+8c=54 6p + 8c = 54

STEP 9

Now we have the new system of equations:
\begin{align*} 7p + 2c &= 41 \\ 6p + 8c &= 54 \end{align*}

STEP 10

Subtract the second equation from the first equation to eliminate c c :
(7p+2c)−(6p+8c)=41−54 (7p + 2c) - (6p + 8c) = 41 - 54

STEP 11

Simplify the equation:
p−6c=−13 p - 6c = -13

STEP 12

Now we have a single equation with two variables. We can solve for one variable in terms of the other. Let's solve for p p :
p=6c−13 p = 6c - 13

STEP 13

Next, we will substitute p p from the equation above into one of the original equations to find the value of c c . Let's use the first original equation:
7(6c−13)+2c=41 7(6c - 13) + 2c = 41

STEP 14

Expand the equation:
42c−91+2c=41 42c - 91 + 2c = 41

STEP 15

Combine like terms:
44c−91=41 44c - 91 = 41

STEP 16

Add 91 to both sides of the equation:
44c=132 44c = 132

STEP 17

Divide both sides by 44 to find c c :
c=13244 c = \frac{132}{44}

STEP 18

Calculate the value of c c :
c=3 c = 3

STEP 19

Now that we have the value of c c , we can substitute it back into the equation we found for p p to find the value of p p :
p=6c−13 p = 6c - 13

STEP 20

Substitute c=3 c = 3 into the equation:
p=6(3)−13 p = 6(3) - 13

STEP 21

Calculate the value of p p :
p=18−13 p = 18 - 13

STEP 22

p=5 p = 5

STEP 23

Now we have the cost of one potted plant (p=5 p = 5 ) and one bag of compost (c=3 c = 3 ).

STEP 24

Emily wants to buy 4 potted plants and 5 bags of compost. We can calculate the total cost for Emily using the values of p p and c c :
Total cost=4p+5c Total\, cost = 4p + 5c

STEP 25

Substitute the values of p p and c c into the equation:
Total cost=4(5)+5(3) Total\, cost = 4(5) + 5(3)

STEP 26

Calculate the total cost:
Total cost=20+15 Total\, cost = 20 + 15

STEP 27

Total cost=35 Total\, cost = 35
Emily will have to pay £35 for 4 potted plants and 5 bags of compost.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord