Question

1.Liz buys three apples, a dozen bananas, and one cantaloupe for $2.36. Bob buys a dozen apples and two cantaloupe for $5.26. Carol buys two bananas and three cantaloupe for $2.77. How much do single pieces of each fruit cost?

Answer 1

• Apple = $0.29

,

• Banana = $0.05

,

• Cantaloupe = $0.89

Step-by-step explanation

First, we have to name the variables:

• x = cost of one apple

,

• y = cost of one banana

,

• z = cost of one cantaloupe

Next, we have to write equations for the purchases each person did:

• Liz,: 3 apples (3x), 12 bananas (12y), and 1 cantaloupe (z) for $2.36

,

• Bob,: 12 apples (12x), no bananas (0y), and 2 cantaloupe (2z) for $5.26

,

• Carol,: no apples (0x), 2 bananas (2y), and 3 cantaloupe (3z) for $2.77

We have the following system of equations,

`\begin{cases}3x+12y+z=2.36 \\ 12x+2z=5.26 \\ 2y+3z=2.77\end{cases}`

We can solve this system using the method of substitution.

Solve the second equation for x,

`\begin{gathered} 12x+2z=5.26 \\ \\ 12x=5.26-2z \\ \\ x=(5.26)/(12)-(2z)/(12) \end{gathered}`

Solve the third equation for y,

`\begin{gathered} 2y+3z=2.77 \\ \\ 2y=2.77-3z \\ \\ y=(2.77)/(2)-(3z)/(2) \end{gathered}`

Next, replace x and y with these two expressions as functions of z in the first equation,

`3\mleft((5.26)/(12)-(2z)/(12)\mright)+12\mleft((2.77)/(2)-(3z)/(2)\mright)+z=2.36`

We have an equation for z. Apply the distributive property to eliminate the parenthesis,

`3\cdot(5.26)/(12)-3\cdot(2z)/(12)+12\cdot(2.77)/(2)-12\cdot(3z)/(2)+z=2.36`

Solve the products and quotients - with the help of a calculator,

`1.315-0.5z+16.62-18z+z=2.36`

Add like terms,

`\begin{gathered} (1.315+16.62)+(-0.5z-18z+z)=2.36 \\ \\ 17.935-17.5z=2.36 \end{gathered}`

Subtract 17.935 from both sides,

`\begin{gathered} 17.935-17.935-17.5z=2.36-17.935 \\ \\ -17.5z=-15.575 \end{gathered}`

And divide both sides by -17.5,

`\begin{gathered} (-17.5z)/(-17.5)=(-15.575)/(-17.5) \\ \\ z=0.89 \end{gathered}`

Now, knowing that z = 0.89, we can replace this value into the second and third equations that we solved for x and y before,

`x=(5.26)/(12)-(2z)/(12)=(5.26)/(12)-(2\cdot0.89)/(12)`

Solving this with the help of a calculator, we get x = 0.29.

For y,

`y=(2.77)/(2)-(3z)/(2)=y=(2.77)/(2)-(3\cdot0.89)/(2)`

Again, using a calculator, we have that y = 0.05.

Hence, the cost of each fruit is:

• Apple = $0.29

,

• Banana = $0.05

,

• Cantaloupe = $0.89