Chapter 3.7, Problem 55PPS

(a)

To determine

Number of each drink sold using Cramer’s rule.

Answer to Problem 55PPS

Vendor sold 650 small drinks, 325 medium drinks, and 410 large drinks.

Explanation of Solution

Given information:

Small drinks sold for $1.15 each.

Medium drink sold for $1.75 each.

Large drink sold for $2.25 each.

Total sales for 1385 drinks were $2,238.75.

Calculation:

Let

Number of medium drinks be x .

Then

Number of small drinks be 2x .

And

Number of large drinks bey .

Now,

Translate the word problem into equation.

For number of drinks:

  $x+2x+y=1385$

That becomes

  $3x+y=1385$

For price of drinks:

  $1.75x+1.15\left(2x\right)+2.25y=2238.75$

That becomes

  $4.05x+2.25y=2238.75$

Now,

To find the determinant for the given matrix, use Cramer’s rule.

For x :

  $x=\frac{\left|\begin{array}{cc}m& b\\ n& g\end{array}\right|}{\left|C\right|}$

For y :

  $y=\frac{\left|\begin{array}{cc}a& m\\ f& n\end{array}\right|}{\left|C\right|}$

Substitute the values:

For x :

  $x=\frac{\left|\begin{array}{cc}1385& 1\\ 2238.75& 2.25\end{array}\right|}{\left|\begin{array}{cc}3& 1\\ 4.05& 2.25\end{array}\right|}$

For y :

  $y=\frac{\left|\begin{array}{cc}3& 1385\\ 4.05& 2238.75\end{array}\right|}{\left|\begin{array}{cc}3& 1\\ 4.05& 2.25\end{array}\right|}$

Write the diagonal of each element:

For x :

  $x=\frac{1385\left(2.25\right)-1\left(2238.75\right)}{3\left(2.25\right)-1\left(4.05\right)}$

For y :

  $y=\frac{3\left(2238.75\right)-1385\left(4.05\right)}{3\left(2.25\right)-1\left(4.05\right)}$

Then Simplify:

For x :

  $x=\frac{1385\left(2.25\right)-1\left(2238.75\right)}{3\left(2.25\right)-1\left(4.05\right)}=\frac{3116.25-2238.75}{6.75-4.05}=\frac{877.50}{2.7}=325$

For y :

  $x=\frac{3\left(2238.75\right)-1385\left(4.05\right)}{3\left(2.25\right)-1\left(4.05\right)}=\frac{6716.25-5609.25}{6.75-4.05}=\frac{1107}{2.7}=410$

Therefore,

Vendor sold 650 small drinks, 325 medium drinks and 410 large drinks.

(b)

To determine

Vendor’s sales for the second week.

Answer to Problem 55PPS

Vendor’s sale for the second week is $2,426.

Explanation of Solution

Given information:

For the second week,

Small drink sold for $1.25 each instead of $1.15 each.

Vendor sold 140 fewer small drinks, 125 more medium drinks, 35 more large drinks.

Calculation:

We have

Small drink sold for $1.25 each instead of $1.15 each.

In second week,

Vendor sold 140 fewer small drinks.

That means

He sold 510 small drinks for $1.25 each.

Thus,

  $510\times \$1.25=\$637.50$

He sold small drinks for $637.50.

Vendor sold 125 more medium drinks.

That means

He sold 450 medium drinks for $1.75 each.

Thus,

  $450\times \$1.75=\$787.50$

He sold medium drinks for $787.50.

Vendor sold 35 more large drinks.

That means

He sold 445 large drinks for $2.25 each.

Thus,

  $445\times \$2.25=\$1001.25$

He sold medium drinks for $1001.25.

To calculate the sale of second week, sum up the sale of all above three categories:

  $\$637.50+\$787.50+\$1001.25=\$2,426.25$

Therefore,

Vendor’s sale for the second week is $2,426.

(c)

To determine

Whether raising the price of the small drink was a good business move.

Answer to Problem 55PPS

Yes, raising the price of the small drink was a good business move as it brought a nice profit.

Explanation of Solution

Given information:

From Part (a),

Total sales for 1385 drinks were $2,238.75.

From Part (b),

Total sales for 1405 drinks were $2,426.

By increasing the price of the small drinks,

The sale of small drinks went down.

Whereas,

The sale of medium or large drinks went up.

Since the difference between the price of drinks diminished,

Some of the customers preferred to pay a little extra to get a bigger quantity of drink.

Therefore,

The raise of price was a good business move as it brought a nice profit.