# 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: $\frac{3}{4}$ cup of white sugar and $\frac{1}{2}$ cup of brown. The chief wishes to change the quantities such that he only uses $\frac{1}{6}$ cup of brown. What proportion of a cup of whit sugar should he use.

Mar 6, 2017

$\frac{1}{4}$ cup of white sugar to $\frac{1}{6}$ cup of brown sugar

#### Explanation:

$\textcolor{b l u e}{\text{First principle method with lots of explanation}}$

Assumption: The other sugar at $\frac{3}{4}$ cup is white sugar

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

A fractions structure is $\left(\text{count")/("size indicator")->("numerator")/("denominator}\right)$

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$\to \textcolor{g r e e n}{\frac{3}{4} \textcolor{red}{\times 1} \text{ " ->" "3/4color(red)(xx3/3)" "=" } \frac{9}{12}}$

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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$\textcolor{b r o w n}{\text{Building the comparison of just the numerators (counts)}}$

Initial condition of $\frac{3}{4} \text{ to } \frac{1}{2}$ is the same as: $\frac{9}{12} \text{ to } \frac{6}{12} = 9 : 6$

Writing this ratio in fraction form gives: $\frac{9}{6}$

$\left(\text{white sugar")/("brown sugar}\right) = \frac{9}{6}$

But we need:

$\left(\text{white sugar")/("brown sugar}\right) = \frac{9}{6} \equiv \frac{x}{2}$

Multiply both sides by 2 giving:

$x = \frac{2 \times 9}{6} = \frac{9}{3} = 3 \textcolor{b r o w n}{\text{ remember that this is in 12ths}}$

So measuring in cups we have $\frac{3}{12} \to \frac{3 \div 3}{12 \div 3} = \frac{1}{4}$
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Thus the ratio of the reduced volumes is:

$\frac{1}{6}$ cup of brown sugar to $\frac{1}{4}$ cup of white.

Changing this to the same order as in the question:

$\frac{1}{4}$ cup of white sugar to $\frac{1}{6}$ cup of brown sugar