Moonshine Problem: How many gallons of 20% moonshine and how many gallons of 30% moonshine have to be mixed together to make 100 gallons of 25% moonshine? (x = gallons of 20% moonshine; y = gallons of 30% moonshine. 20% moonshine is called 20% because 20% of any amount (like x) is PURE MOONSHINE!)

  50 gallons of each

Step-by-step explanation:

When the mix percentage is halfway between the percentages of the constituents, they are mixed half and half.

  50 gallons of 20% and 50 gallons of 30% have to be mixed to make 100 gallons of 25% moonshine.

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For the variable definitions in the problem statement, you can write the equations ...

  x + y = 100

 0.20x +0.30y = 0.25(100)

I like to use a single variable representing the greatest contributor. Along that line, we have ...

  x = 100 -y

  0.20(100 -y) +0.30y = 25 . . . . . substitute for x

  20 +0.10y = 25 . . . . . . simplify

  0.10y = 5 . . . . . . . . . . . subtract 20

  y =  50 . . . . . . . . . . . . . multiply by 10

  x = 100 -50 = 50

 The mixture will be 50 gallons of each constituent.

stops on an even number.

ratios represent a pattern or relationship that can be scaled.

equivalent ratios are pairs of ratios that represent the same relationship between quantities, and can be generated by multiplying or dividing both quantities in the ratio by the same factor.

in this lesson, students will use double number lines and ratio reasoning to solve ratio problems. by partitioning double number lines into proportionate intervals, students will build an understanding of equivalent ratio relationships as those that are generated by multiplying or dividing two related quantities by the same number. this lesson will build upon students' prior understanding of multiplicative comparisons, number line reasoning and equivalent fractions (while highlighting the difference between equivalent fractions and equivalent ratios); and will build toward future work in this unit on building ratio tables, and work in future grades in proportionality, scaling, and expressing and graphing linear functions.

step-by-step explanation:

ratios represent a pattern or relationship that can be scaled.

equivalent ratios are pairs of ratios that represent the same relationship between quantities, and can be generated by multiplying or dividing both quantities in the ratio by the same factor.

in this lesson, students will use double number lines and ratio reasoning to solve ratio problems. by partitioning double number lines into proportionate intervals, students will build an understanding of equivalent ratio relationships as those that are generated by multiplying or dividing two related quantities by the same number. this lesson will build upon students' prior understanding of multiplicative comparisons, number line reasoning and equivalent fractions (while highlighting the difference between equivalent fractions and equivalent ratios); and will build toward future work in this unit on building ratio tables, and work in future grades in proportionality, scaling, and expressing and graphing linear functions.