A radiator contains 10 quarts of 30% of which is antifreeze. How much fluid should be drained and replaced with pure antifreeze so that the new mixture 40% antitreeze?

2 Answers
Feb 15, 2017

Answer:

# 1 3/7 "quarts"# should be drained off and replaced with pure antifreeze.

#1 3/7 ~~1.4286# to 4 decimal places

Explanation:

Current amount of antifreeze in quarts is #30/100xx10 =3#

Target is #40% -> 4# quarts
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Let the amount drained of and replaced with pure antifreeze be #x#

The amount left after draining off is #10-x#

The amount of antifreeze in this is #30/100(10-x)#

The replacement is pure antifreeze which is the amount #x#

so #30/100(10-x)+x=4# quarts

#3-3/10x+x=4#

#3+x(1-3/10)=4#

#x=(4-3)/(1-3/10)#

#x=1xx10/7 = 1 3/7 "quarts"#

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#color(blue)("Check")#

#10 -1 3/7 = 8 4/7#

#=(30/100xx8 4/7) + 1 3/7#

#=(3/10xx60/7)+10/7#

#=(3xx6)/7+10/7#

#=28/7 =4 larr" 4 litres of pure antifreeze mixed into 10 quarts" #

#color(red)("Correct")#

Feb 16, 2017

Answer:

Different approach

Drain off and replace #1 3/7# quarts with pure antifreeze

Explanation:

#color(blue)("Preamble")#

This is a very effective method

You are blending two liquid with different concentration of antifreeze. One liquid has 30% concentration and the other 100% concentration.

If you have all of the original the antifreeze is 30%. If you have all the other the antifreeze content is 100%.

Connect the two points on a graph and you have a model for those conditions and all the other blend concentrations in between. From this we can and may determine the blend that gives us 40%. Sometimes it takes a little thought about how so set up the x-axis which represents the quantities of the blend.
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#color(blue)("Answering the question")#
Tony B

The slope of part of the graph is the same as the slope of all of it.

#("change in y")/("change in x")->("change blend concentration")/("change in added pure antifreeze")#

#(100-30)/(10) -=(40-30)/x#

As we need #x# turn everything upside down:

#10/(100-30)-=x/(40-30)#

#10/70=x/10#

#x=100/70 ->(100-:10)/(70-:10) = 10/7 = 1 3/7# in quarts