# Mathematics

## Mathematics: Standard Multiple Choice

Read the following SAT test question and then select your answer.

The SAT instructions are so simplified, they leave out the most important parts of what you need to do. Remember that you need to disregard the answers, carefully read the question, mark the bottom line, and THINK about the most efficient way to solve the problem. Then do the math, check your answer against the bottom line you wrote down, and finally select the correct answer choice.

To make an orange dye, 3 parts of red dye are mixed with 2 parts of yellow dye. To make a green dye, 2 parts of blue dye are mixed with 1 part of yellow dye. If equal amounts of green and orange dye are mixed, what fraction of the new mixture is yellow?

First, note the bottom line at the top of your scratch work. The problem asks for a fraction of the whole mixture, so write:

$\frac{y}{m}=?$

Next, assess your options. Ask "what could I do?" and then "what should I do?" The trap here is to simply add the parts together. Add up all the numbers to give the denominator, then add up the yellow parts to give the numerator. However, what that misses is the word "equal" in the question. The mixture you just added up would not have equal amounts of orange and green dye.

To make the problem more concrete, replace the vague term "parts" with a more specific word like "ounces."

The problem is concerned with relationships between different amounts, which makes it a ratio problem. What ratio will help you find out how much yellow is in the final mixture? Start with each of the dye mixes separately. Each batch of orange dye has 3 ounces of red dye and 2 ounces of yellow, so it is 5 ounces total. The green is made up of 2 ounces of blue dye and 1ounce of yellow, so it is only 3 ounces total.

$\frac{y}{o}=\frac{2}{5}$                                                                 $\frac{y}{g}=\frac{1}{3}$

What is the next step? The problem mentions that equal amounts of green and orange are mixed together. Right now, you have different amounts of green and orange. How do you make them equal? The easiest way is to use the least common multiple (or least common denominator, which is really just the least common multiple on the bottom of a fraction). The least common multiple of 3 and 5 is 15, so put both fractions in terms of that number:

$\frac{y}{o}=\frac{2}{5}=\frac{6}{15}$                                                    $\frac{y}{g}=\frac{1}{3}=\frac{5}{15}$

Now you have everything you need to solve the problem. Remember that you are looking for what fraction of the final mixture is yellow, so find those two values and put them into a fraction.

$5+6=11$              $15+15=30$            $\frac{11}{30}$   doesn't reduce, so it is your answer.

Check that against the answer choices:

$A) \frac{3}{16}$

$B) \frac{1}{4}$

$C) \frac{11}{30}$

$D) \frac{3}{8}$

$E) \frac{7}{12}$

The answer is C.

On sat.collegeboard.org, 32% of responses to this question were correct.

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