Let's set up and solve an equation system to find the answer.
To do this, we need to introduce variables.
Description
Variable
Number of small cones sold
s
Number of medium cones sold
m
Number of large cones sold
l
Let's use these variables to write equations expressing the information given in the question.
Description
Equation
Income from selling small cones for $0.89 each.
0.89s
Income from selling medium cones for $1.19 each.
1.19m
Income from selling large cones for $1.39 each.
1.39l
The total sales were $58.98.
0.89s+1.19m+1.39l=58.98
Santos sold 52 cones during the day.
s+m+l=52
Santos sold 7 more medium cones than small cones.
m=s+7
We found a system of equations in the three unknowns s, m, and l.
0.89s+1.19m+1.39l=58.98 s+m+l=52 m=s+7
The number of unknowns can be reduced by substituting m=s+7 into the first two equations.
However, if we do this we eliminate variable m, which the question asks us to find.
Let's rearrange m=s+7 to s=m-7, and use this version of the substitution to eliminate s instead of m.
We now have a system of equations in the two unknowns m and l.
Let's write this system as a matrix equation.
2.08m+ 1.39l=65.21 2m+1l=59
⇕
2.08& 1.39 2& 1
m l =
65.21 59
A matrix equation like this can be solved using the inverse matrix.
MX=P ⟹ X=M^(-1)P
We can use the determinant to find the inverse of a 2* 2 matrix.
M= 2.08& 1.39 2& 1
⇓
|M|= 2.08(1)- 1.39( 2)=-0.7
⇓
M^(-1)=1/-0.7 1& - 1.39 - 2& 2.08
Let's use this inverse to find the answer to the question.
We found m=24, so Santos sold 24 medium cones during the day.
The correct answer is C.
Alternative Solution
Alternative way of thinking
Since this is a multiple choice question, instead of setting up and solving an equation system we can check the possible options.
It is still useful to use the variables s, m, and l.
Since in the options the possible values of m are given, let's express s and l in terms of m.
Information
Expression
Santos sold 7 more medium cones than small cones.
m&=s+7 &⇓ s&=m-7
Santos sold 52 cones in total.
s+m+l&=52 &⇓ m-7+m+l&=52 &⇓ l&=59-2m
Let's use these expressions to find the number of small and large cones for the given number of medium cones in the different options.
Option
Medium (m)
Small (s=m-7)
Large (l=59-2m)
A
11
11-7=4
59-2(11)=37
B
17
17-7=10
59-2(17)=25
C
24
24-7=17
59-2(24)=11
D
36
36-7=29
59-2(36)=-13
Since Santos cannot sell a negative number of large cones, we can eliminate option D.
Let's now use the price of each cone to express the income.
Income: 0.89s+1.19m+1.39l
Now we will check the value of this expression for the number of cones in the different options.
Option
Small (s)
Medium (m)
Large (l)
Income (0.89s+1.19m+1.39l)
A
4
11
37
0.89(4)+1.19(11)+1.39(37)=68.08
B
10
17
25
0.89(10)+1.19(17)+1.39(25)=63.88
C
17
24
11
0.89(17)+1.19(24)+1.39(11)=58.98
We can see that option C matches the sale value given in the question.
