Chapter ISG, Problem 6.4.9P

a.

To determine

The the empty values in the given table.

Answer to Problem 6.4.9P

The complete table is-

	Tropical Breeze	Kona Cooler	Total
Amount of juice (qt)	t	k	10
Amount of pineapple juice (qt)	$\frac{t}{5}$	$\frac{k}{2}$	4

Explanation of Solution

Given: The given table is-

	Tropical Breeze	Kona Cooler	Total
Amount of juice (qt)	t	k	10
Amount of pineapple juice (qt)

Tropical breeze contains 20% pineapple juice whereas Kona cooler contains 50% pineapple juice. A new drink of 10 quarts is made using these two drinks which contains 40% pineapple juice.

Concept Used: Consider a mixture that has only two components, say x and y , where the mixture contains $m\%$ of x and the remaining y . Then, A units of that mixture will have-

  $\frac{m}{100}\times A$ units of x

  $\frac{(100-m)}{100}\times A$ units of y

Calculations: Tropical breeze contains 20% pineapple juice. Thus, t quarts of tropical breeze will have-

  $\frac{20}{100}\times t=\frac{t}{5}$ quarts of pineapple juice.

Kona cooler contains 50% pineapple juice. Thus, k quarts of kona cooler will have-

  $\frac{50}{100}\times k=\frac{k}{2}$ quarts of pineapple juice.

The new drink has 40% of pineapple juice. Therefore 10 quarts of the new drink will have-

  $\frac{40}{100}\times 10=4$ quarts of pineapple juice.

Filling in the table with all these values gives-

	Tropical Breeze	Kona Cooler	Total
Amount of juice (qt)	t	k	10
Amount of pineapple juice (qt)	$\frac{t}{5}$	$\frac{k}{2}$	4

b.

To determine

A system of equations to determine the values of t and k.

Answer to Problem 6.4.9P

The required system of equations is-

  $t+k=10$

  $\frac{t}{5}+\frac{k}{2}=4$

Explanation of Solution

Given: The completed table which shows the amount of Tropical breeze and Kona cooler used to make the new drink and the amount of pineapple juice in all these drinks.

	Tropical Breeze	Kona Cooler	Total
Amount of juice (qt)	t	k	10
Amount of pineapple juice (qt)	$\frac{t}{5}$	$\frac{k}{2}$	4

The above table depicts clearly that t quarts of Tropical breeze is mixed with k quarts of Kona cooler to get 10 quarts of the new drink. Hence, the mathematical equation of this statement is-

  $t+k=10$

Also, the amount of pineapple juice from the given amounts of Tropical breeze and Kona cooler are $\frac{t}{5}$ quarts and $\frac{k}{2}$ quarts respectively which is equal to the total amount of pineapple juice in the new drink, which is 4 quarts. Hence, the mathematical equation of the above statement is-

  $\frac{t}{5}+\frac{k}{2}=4$

Therefore, the required system of equations is-

  $t+k=10$

  $\frac{t}{5}+\frac{k}{2}=4$

c.

To determine

To find: The solution of the equations to find the values of t and k.

Answer to Problem 6.4.9P

The required solution is-

t = 3.33 quarts (approx.) and k = 6.67 quarts (approx.).

Explanation of Solution

Given: The given system of equations is-

  $t+k=10$

  $\frac{t}{5}+\frac{k}{2}=4$

Concept Used: Elimination process is a method of solving simultaneous linear equations where the coefficient of one variable is made same by multiplying the equations with a constant and then that variable is eliminated. What remains is a linear equation of one variable which can be solved easily. That value is then put in one of the equations to get the value of the other variable. Consider a system of equations-

  $a_{1}x+b_{1}y=c_1$ . . . . . ( i )

  $a_{2}x+b_{2}y=c_2$ . . . . . ( ii )

Multiplying equation ( i ) by $a_2$ and equation ( ii ) by $a_1$ and subtracting both equations gives-

  $a_2{b}_{1}y-a_1{b}_{2}y=a_2{c}_1-a_1{c}_2$

The value of y can be determined from this equation and put in either of the given equations to find the value of x.

Calculations: The given system of equations is-

  $t+k=10$ . . . . . .( i )

  $\frac{t}{5}+\frac{k}{2}=4$ . . . . . . ( ii )

Multiplying equation ( i ) by $\frac{1}{2}$ and then subtracting equation ( ii ) from it gives-

  $\frac{t}{2}-\frac{t}{5}=\frac{10}{2}-4$

  $\Rightarrow \frac{5t-2t}{10}=5-4$

  $\Rightarrow \frac{3t}{10}=1$

  $\Rightarrow t=\frac{10}{3}$

  $\Rightarrow t=3.33$ (approx.)

Putting this value in equation ( i ) gives-

  $\frac{10}{3}+k=10$

  $\Rightarrow k=10-\frac{10}{3}$

  $\Rightarrow k=\frac{20}{3}$

  $\Rightarrow k=6.67$ (approx.)

Thus, the solution shows that 3.33 quarts of Tropical breeze is mixed with 6.67 quarts of Kona cooler to get the desired amount of the new drink.

d.

To determine

To show: The solutions obtained are meaningful and correct.

Explanation of Solution

Given: The amount of Tropical breeze and Kona cooler mixed to obtain the new drink is 3.33 quarts (approx.) and 6.67 quarts (approx.) respectively.

The amount of tropical breeze obtained from solving the equations is-

  $3.33=\frac{10}{3}$ quarts

Since, Tropical breeze contains 20% pineapple juice, hence the amount of pineapple juice in $\frac{10}{3}$ quarts of Tropical breeze is-

  $\frac{20}{100}\times \frac{10}{3}=\frac{2}{3}$ quarts

Again, the amount of Kona cooler obtained from solving the equations is-

  $6.67=\frac{20}{3}$ quarts

Since, Kona cooler contains 50% pineapple juice, hence the amount of pineapple juice in $\frac{20}{3}$ quarts of Kona cooler is-

  $\frac{50}{100}\times \frac{20}{3}=\frac{10}{3}$ quarts

Total quantity of mixture obtained by mixing 3.33 quarts and 6.67 quarts of Tropical breeze and Kona cooler

  = 3.33 + 6.67 = 10 quarts, which is the required quantity of the new drink to be made.

Again, total amount of pineapple juice in this mixture will be

  = $\frac{2}{3}+\frac{10}{3}=\frac{12}{3}=4$ quarts

Thus, 10 quarts of the new mixture has 4 quarts of pineapple juice. Hence, percentage of pineapple juice in the new mixture

  = $\frac{4}{10}\times 100=40\%$ , which is the required concentration of pineapple juice.

Hence, the solutions obtained are correct.