Systems In A Park

Question

Part A: A mistake was made in mixing the lemonade for the concession stand, but you can fix it! 

The lemonade comes in 100% juice concentrate, but you only serve it as 70% solution. Unfortunately, one batch got overwatered, so you have 4 quarts of 50% solution.

How much 100% concentrate do you need to add in order to get the 70% solution?

How much of the 70% solution will you have?

Set up a system of equations and then show each step to solve it. (10 points)

Part B: A shooting gallery game awards the player 2 points for hitting a stationary duck-target, and 3 points for hitting a moving duck-target. Escobar hit ten targets, and scored 23 points.

How many stationary targets did he hit? How many moving targets did he hit?

Set up a system of equations by using the following variables and then show each step to solve it. (10 points)

x = number of stationary targets he hit

y = number of moving targets he hit

Answer 1

the concentration level is:

50% * 4 quarts + 100% * x quarts = 70% * (x+4) quarts

quarts cancel out...

50% * 4 + 100* x = 70% * (x+4)

changes to decimal:

0.5*4 + 1.00 * x = 0.7*(x+4)

simplifies:

2 + x = 0.7(x+4)

distributive:

2 + x = 0.7x + 2.8

multiplies by 10 to clear away the decimals:

20 + 10x = 7x + 28

subtracts 20:

10x = 7x + 8

subtracts 7x:

3x = 8

divides by 3:

x = 8/3 = 2 + 2/3

check:

the concentration level is:

0.5*4 + 2+2/3 = 2+2+2/3 = 4+2/3 = 14/3

the total amount is 4+2+2/3 = 6+2/3 = 20/3

14/3 divided by 20/3 = 14/3*3/20 = 14/20 = 7/10 = 70%

final answer: adds 2 + 2/3 = 8/3 quarts of concentrate

to get 6 +2/3 = 20/3 quarts of juice which will have 70% concentration level

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he got 10 hits: x+y=10 --> y = 10-x

he scored 23 points: 2x+3y = 23

substitution 2x + 3(10-x) = 23

2x + 30 -3x = 23

-x + 30 = 23

-x = -7

x = 7

7 stationary targets and 3 moving targets