Question

At an ice cream shop, three flavors are increasing in demand. Last year, banana, pumpkin, and rocky road ice cream made up 13% of total ice cream sales. This year, the same three ice creams made up 18.1% of ice cream sales. The percentage of rocky road sales doubled, the percentage of banana sales increased by 50%, and the percentage pumpkin sales increased by 20%. If the rocky road ice cream had one less percent of sales than the banana ice cream, find out the percentage of ice cream sales each individual ice cream made last year

Answer 1

Answer:

Banana made=3.22

Pumpkin made=7.86

Rocky road ice cream made=1.89

Step-by-step explanation:

From the question we are told that

1st year sales(banana, pumpkin, and rocky road ice cream) $S_1= 13\%$

2nd year sales:$S_2=18.1\%$

Rocky road sales doubled

Banana sales increased by 50%

Pumpkin sales increased by 20%

Rocky road ice cream had one less percent then banana

Let

Banana =x

Pumpkin=y

Rocky road ice cream=z

Generally the first year equation is given mathematically as

$x+y+z=13......(1)$

Generally the second year equation is given mathematically as

100% increase in Rocky road ice cream sales()z i.e double

$2*z=2z$

50% increase in banana ice cream sales()x

$=>1*0.50=0.5\\=>0.5+1x=1.5x$

20% increase in pumpkin(y)

$=>1*0.20=0.2\\=>0.2+1x=1.2y$

Therefore second year equation is given as

$1.5x+1.27y+2z=18.1.....(2)$

Generally where rocky road ice cream had one less percent of sales than the banana ice cream the equation is given as

$1.5x-2z=1........(3)$

Therefore combining the three equations

x+y+z=13

1.5x+1.27y+2z=18.1

1.5x-2z=1

matrix form

$\begin {bmatrix}1 & 1& 1 & 13\\1.5 & 1.2 & 2 & 18.1\\1.5 & 0 & -2 & 1\end{bmatrix}$

Generally the determinant of the first three columns

$\triangle=1.8$

Solving with Crammers Rule by solving Determinant when;

Column 4 replace column 1

$\triangle_1=5.8$

Column 4 replace column 2

$\triangle_2=14.15$

Column 4 replace column 3

$\triangle_3=3.4$

Therefore x is given my

$x=\frac{\triangle_1}{\triangle}$

$x=\frac{5.8}{1.8}$

$x=3.22$

Therefore y is given my

$y=\frac{\triangle_2}{\triangle}$

$y=\frac{14.15}{1.8}$

$y=7.86$

Therefore z is given my

$z=\frac{\triangle_3}{\triangle}$

$z=\frac{3.4}{1.8}$

$z=1.89$

Therefore

Banana made=3.22

Pumpkin made=7.86

Rocky road ice cream made=1.89