A specialty baking recipe consists of three ingredients: flour, sugar, and butter. By weig… — GRE General Test

A specialty baking recipe consists of three ingredients: flour, sugar, and butter. By weight, $\frac{1}{2}$ of the recipe is flour and $\frac{1}{3}$ of the recipe is sugar, with the remaining portion consisting entirely of butter. Which of the following statements about the ingredients in the recipe must be true? Select all such statements.

The butter accounts for $\frac{1}{6}$ of the total weight of the recipe.Answer
The ratio of the weight of sugar to the weight of butter is $2 : 1$ .Answer
C
The combined weight of flour and sugar accounts for $\frac{2}{5}$ of the total weight of the recipe.
D
The weight of sugar is $50\%$ less than the weight of flour.
E
The ratio of the weight of flour to the combined weight of the remaining ingredients is $1 : 2$ .

Answer

The statement that butter accounts for 1/6 of the total weight of the recipe and the statement that the ratio of the weight of sugar to the weight of butter is 2:1 are both correct.

The statement asserting that butter accounts for 1/6 of the total weight is correct because subtracting the combined fraction of flour and sugar (5/6) from the total (1) yields 1/6. The statement asserting that the ratio of sugar to butter is 2:1 is correct because dividing 1/3 by 1/6 equals 2.

Step-by-Step Solution

Calculate the fraction of the recipe that is butter.

Butter fraction =

1 - \left(\frac{1}{2} + \frac{1}{3}\right) = 1 - \frac{5}{6} = \frac{1}{6}

The total of all constituent fractions must sum to 1.

Calculate the ratio of sugar to butter.

\text{Ratio} = \frac{1/3}{1/6} = \frac{1}{3} \times \frac{6}{1} = \frac{2}{1} = 2 : 1

Divide the fraction representing sugar by the fraction representing butter.

Evaluate the statement regarding combined flour and sugar weight.

Combined fraction =

\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \neq \frac{2}{5}

Adding fractions requires finding a common denominator (6).

Evaluate the statement regarding percentage difference between sugar and flour.

Percent less =

\frac{1/2 - 1/3}{1/2} \times 100\% = \frac{1/6}{1/2} \times 100\% = \frac{1}{3} \times 100\% = 33.33\% \neq 50\%

Percent decrease must use the original reference quantity (flour) as the denominator.

Evaluate the ratio of flour to the combined remaining ingredients.

Remaining ingredients fraction =

1/3 + 1/6 = 1/2

. Ratio of flour (1/2) to remaining (1/2) =

1 : 1 \neq 1 : 2

Compare the part (flour) to the other part (sugar + butter), not to the whole.

Key Concept

Operations on rational numbers, fraction subtraction/addition using common denominators, part-to-part ratio comparisons, and percentage change base selection.

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