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Difficulty: HardFractions, Decimals, and Percentages

A chef is preparing a salad dressing. The mixture consists of 14\frac{1}{4} cup of olive oil, 15\frac{1}{5} cup of balsamic vinegar, and a certain amount of lemon juice. If the lemon juice accounts for exactly 40%40\% of the total volume of the dressing, what is the total volume, in cups, of the prepared dressing?

  1. A
    1027\frac{10}{27}
  2. B
    59\frac{5}{9}
  3. 34\frac{3}{4}Answer
  4. D
    1720\frac{17}{20}
  5. E
    98\frac{9}{8}

Answer

34\frac{3}{4}
To find the total volume, first calculate the sum of the volumes of olive oil and balsamic vinegar: 14+15=520+420=920\frac{1}{4} + \frac{1}{5} = \frac{5}{20} + \frac{4}{20} = \frac{9}{20} cup. Because the lemon juice accounts for 40%40\% of the total volume of the dressing, the remaining ingredients (olive oil and balsamic vinegar) must account for the other 60%60\% (or 35\frac{3}{5}) of the total volume. Letting TT represent the total volume, we set up the equation 35T=920\frac{3}{5} T = \frac{9}{20}. Solving for TT yields T=920×53=34T = \frac{9}{20} \times \frac{5}{3} = \frac{3}{4} cups. Therefore, the option representing three-fourths of a cup is the correct answer.

Step-by-Step Solution

1
Add the volumes of the two known ingredients, olive oil and balsamic vinegar.
Combined volume is 14+15=520+420=920\frac{1}{4} + \frac{1}{5} = \frac{5}{20} + \frac{4}{20} = \frac{9}{20} cup (or 0.450.45 cup).
To find what fraction of the total volume the known ingredients represent, we first need their combined total.
2
Determine the percentage of the total volume that the olive oil and balsamic vinegar represent.
These two ingredients represent 100%−40%=60%100\% - 40\% = 60\% (or 35\frac{3}{5}) of the total volume.
Since lemon juice accounts for 40%40\% of the total volume, the rest of the ingredients must account for the remaining percentage.
3
Set up a linear equation to solve for the total volume, TT.
35T=920⇒T=920×53=34\frac{3}{5} T = \frac{9}{20} \Rightarrow T = \frac{9}{20} \times \frac{5}{3} = \frac{3}{4} cups (or 0.750.75 cups).
Solving the equation gives the total volume of the entire dressing.

Key Concept

Solving multi-step word problems involving fractions, decimals, and percentages by relating parts to the whole.

Alternative Method

Convert the fractions to decimals first: 14=0.25\frac{1}{4} = 0.25 cups of olive oil and 15=0.20\frac{1}{5} = 0.20 cups of balsamic vinegar. Their combined volume is 0.25+0.20=0.450.25 + 0.20 = 0.45 cups. Since lemon juice is 40%40\% of the total volume, the other ingredients make up 100%−40%=60%100\% - 40\% = 60\% of the total volume. Let the total volume be TT. We set up the equation 0.60T=0.450.60 T = 0.45, which simplifies to T=0.450.60=0.75T = \frac{0.45}{0.60} = 0.75 cups, or 34\frac{3}{4} cups.
Estimated Time:1m 30s
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Topics

Fractions, Decimals, and Percentages

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