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Don Jewett · Answer 1 · 2023-05-23T05:34:22+0000

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,

$x\text{ + y =91 --- equation 1}$

The price of soft-serve is $2.25 each. Thus, for x number of soft-serve, we have

$2.25\text{ }*\text{ x =\$2.25x}$

Similarly, the price of hard-pack is $0.75 each. Thus, for y number of hard-pack, we have

$0.75\text{ }*\text{ y= \$0.75y}$

The total bill was $158.25. This implies that

price of x soft-serve + price of y hard-pack = total bill

$2.25x\text{ + 0.75y = 158.25 ----- equation 2}$

Solve for x and y simultaneously in equations 1 and 2.

$\begin{gathered} x\text{ + y = 91 ---- equation 1} \\ 2.25x\text{ + 0.75y = 158.25 ---- equation 2} \end{gathered}$

Solve by the method of substitution.

From equation 1,

$\begin{gathered} x+y=91 \\ \Rightarrow y=91-x\text{ ---- equation 3} \end{gathered}$

Substitute equation 3 into equation 2.

$\begin{gathered} 2.25x\text{ + 0.75y = 158.25} \\ 2.25x\text{ + 0.75(91-x) = 158.25} \\ \text{open brackets} \\ 2.25x\text{ + 68.25-0.75x = 158.25} \\ \text{collect like terms} \\ 2.25x-0.75x\text{ = 158.25-68.25} \\ 1.5x=90 \\ \text{divide both sides by the coefficient of x, which is }1.5 \\ (1.5x)/(1.5)=(90)/(1.5) \\ \Rightarrow x=60 \\ \end{gathered}$

Substitute the value of 60 for x in either equation 1, 2 or 3.

Substitution into equation 3, we have

$\begin{gathered} y=91-x \\ \Rightarrow y=91-60 \\ y=31 \end{gathered}$

Hence,

the number of soft-serve ordered is 60,

the number of hard-pack is 31.