Question

A chief wishes to make a dish that requires a blend of white and brown sugar?

The recipe quantities in the book are to great for the number of people he is preparing the dish for. The proportions in the book are: `3/4` cup of white sugar and `1/2` cup of brown. The chief wishes to change the quantities such that he only uses `1/6` cup of brown. What proportion of a cup of whit sugar should he use.

Answer 1

`1/4` cup of white sugar to `1/6` cup of brown sugar

Explanation:

`color(blue)("First principle method with lots of explanation")`

Assumption: The other sugar at `3/4` cup is white sugar

Let the unknown proportion of white sugar be `x` then we have

A fractions structure is `("count")/("size indicator")->("numerator")/("denominator")`

So we need to change the 'size indicators' such that they are all the same. That way we may directly compare just the 'counts'.

initial white sugar`->color(green)(3/4color(red)(xx1)" " ->" "3/4color(red)(xx3/3)" "=" "9/12)`

initial brown sugar`->color(green)(1/2color(red)(xx1)" "->" "1/2color(red)(xx6/6)" "=" "6/12`

target brown sugar`->color(green)(1/6color(red)(xx1)" "->" "1/6color(red)(xx2/2)" "=" "2/12`
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`color(brown)("Building the comparison of just the numerators (counts)")`

Initial condition of `3/4" to " 1/2` is the same as: `9/12" to "6/12 =9:6`

Writing this ratio in fraction form gives: `9/6`

`("white sugar")/("brown sugar")=9/6`

But we need:

`("white sugar")/("brown sugar")=9/6-=x/2`

Multiply both sides by 2 giving:

`x=(2xx9)/6 = 9/3=3 color(brown)(" remember that this is in 12ths")`

So measuring in cups we have `3/12 ->(3-:3)/(12-:3)=1/4`
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Thus the ratio of the reduced volumes is:

`1/6` cup of brown sugar to `1/4` cup of white.

Changing this to the same order as in the question:

`1/4` cup of white sugar to `1/6` cup of brown sugar