Let the number of soft-serve be x and the number of hard-pack be y.
The total number of cones ordered is 91. Thus,
[tex]x\text{ + y =91 --- equation 1}[/tex]
The price of soft-serve is $2.25 each. Thus, for x number of soft-serve, we have
[tex]2.25\text{ }\times\text{ x =\$2.25x}[/tex]
Similarly, the price of hard-pack is $0.75 each. Thus, for y number of hard-pack, we have
[tex]0.75\text{ }\times\text{ y= \$0.75y}[/tex]
The total bill was $158.25. This implies that
price of x soft-serve + price of y hard-pack = total bill
[tex]2.25x\text{ + 0.75y = 158.25 ----- equation 2}[/tex]
Solve for x and y simultaneously in equations 1 and 2.
[tex]\begin{gathered} x\text{ + y = 91 ---- equation 1} \\ 2.25x\text{ + 0.75y = 158.25 ---- equation 2} \end{gathered}[/tex]
Solve by the method of substitution.
From equation 1,
[tex]\begin{gathered} x+y=91 \\ \Rightarrow y=91-x\text{ ---- equation 3} \end{gathered}[/tex]
Substitute equation 3 into equation 2.
[tex]\begin{gathered} 2.25x\text{ + 0.75y = 158.25} \\ 2.25x\text{ + 0.75(91-x) = 158.25} \\ \text{open brackets} \\ 2.25x\text{ + 68.25-0.75x = 158.25} \\ \text{collect like terms} \\ 2.25x-0.75x\text{ = 158.25-68.25} \\ 1.5x=90 \\ \text{divide both sides by the coefficient of x, which is }1.5 \\ \frac{1.5x}{1.5}=\frac{90}{1.5} \\ \Rightarrow x=60 \\ \end{gathered}[/tex]
Substitute the value of 60 for x in either equation 1, 2 or 3.
Substitution into equation 3, we have
[tex]\begin{gathered} y=91-x \\ \Rightarrow y=91-60 \\ y=31 \end{gathered}[/tex]
Hence,
the number of soft-serve ordered is 60,
the number of hard-pack is 31.
