At a local farmer's market, a vendor sells two types of fruit baskets: a standard basket a… — SAT

At a local farmer's market, a vendor sells two types of fruit baskets: a standard basket and a deluxe basket. The standard basket contains $3$ apples and $2$ oranges. The deluxe basket contains $5$ apples and $4$ oranges. If a customer bought a combination of these baskets containing a total of $31$ apples and $22$ oranges, what is the total number of baskets the customer bought?

Answer: 9

Answer

The total number of baskets the customer bought is 9.

The correct answer is obtained by setting up a system of equations where

s

is the number of standard baskets and

d

is the number of deluxe baskets. The equations

3s + 5d = 31

and

2s + 4d = 22

represent the totals of apples and oranges respectively. Dividing the second equation by 2 gives

s + 2d = 11

, which can be rearranged to

s = 11 - 2d

. Substituting this expression into the first equation yields

3(11 - 2d) + 5d = 31

, which simplifies to

33 - d = 31

, meaning

d = 2

. Substituting

d = 2

back gives

s = 7

. The total number of baskets is the sum of both types,

7 + 2 = 9

Step-by-Step Solution

Define variables and set up the system of linear equations.

Let

s

be the number of standard baskets and

d

be the number of deluxe baskets. The equations are:

3s + 5d = 31

(for apples) and

2s + 4d = 22

(for oranges).

To represent the relationships between the number of baskets and the total quantities of fruits mathematically.

Solve the system of equations using substitution.

From the second equation, dividing by 2 yields

s + 2d = 11

, so

s = 11 - 2d

. Substituting this into the first equation gives

3(11 - 2d) + 5d = 31

, which simplifies to

33 - d = 31

, leading to

d = 2

To find the number of deluxe baskets purchased.

Calculate the number of standard baskets and sum the two counts to find the total number of baskets.

Substitute

d = 2

into

s = 11 - 2d

to get

s = 7

. The total number of baskets is

s + d = 7 + 2 = 9

To answer the specific question asking for the total number of baskets bought.

Key Concept

Systems of Linear Equations

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic