# Ratios

The election results are in!  The amount of information after an election day can be overwhelming, but limit yourself to one story from the election, and you will have an excellent current event for your SAT essay.  Many people focus on the presidential election, but there are hundreds of other important issues that were brought before the nation.  One group of United States citizens who currently cannot vote for the president of the United States voted about the possibility of becoming the 51st state.  Read this article about what is happening in Puerto Rico, and think about how this prospective state differs from or is similar to other territories that have become states.  Think of the broad themes raised by this story that could relate this article to SAT essay questions.

## 11/8 Algebra:  Ratios

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

Don’t just read the question; read it carefully.  Make sure you know which labels apply to which numbers.  Identify the bottom line.  Assess your options for solving the problem so that you can choose the most efficient method to attack the problem.  Once you have solved the problem, loop back to make sure that you have solved for the bottom line.

In a class of 80 seniors, there are 3 boys for every 5 girls. In the junior class, there are 3 boys for every 2 girls. If the two classes combined have an equal number of boys and girls, how many students are in the junior class?

Bottom Line:  Number of juniors = ?

Assess your options:  You could work backwards by starting with the answer choices, but it might take you a long time to work through all of the possible answers.  Instead, start turning those ratios into actual numbers of students.

Attack the problem:  You know the most about the seniors, so start with them.  You are given a ratio of 3 boys to 5 girls, and you know that the total number of boys and girls must equal 80.  You know that 3 + 5 = 8, so all you have to do is multiply the 3 and the 5 each by 10 and you will have a total of 80 seniors.  There are 30 senior boys and 50 senior girls.

$\frac{senior\: boys}{senior\: girls}=\frac{3}{5}=\frac{30}{50}$

Now that you know the number of senior boys and senior girls, how does that help you find the number of juniors?  Remember that the two classes combined have an equal number of boys and girls.  That means that the senior boys plus the junior boys must be equal to the senior girls plus the junior girls.

$senior\: boys + junior\: boys = senior\: girls + junior\: girls$

Plug in the numbers that you found for the senior boys and girls.

$30 + junior\: boys = 50 + junior\: girls$

What information do you know about the juniors?  You know that there are 3 boys for every 2 girls.  You do not know the total number of juniors, so use an x to represent this number.  What fraction of the total are the boys?  They are actually three fifths of the total number of juniors because you must add the boys and girls to find the total number of juniors.  That means that the girls are two fifths of the total number of juniors.  Plug this into your formula, remembering that anytime you have “of the total” that means that you must multiply by the unknown total.

$30 + \frac{3}{5}x = 50 + \frac{2}{5}x$

Now solve for x.  Rearrange the equation so that you have like terms on the same sides of the equation, and combine those like terms.  Start by subtracting the two fifths of x from each side.

$30+\frac{1}{5}x = 50$

Get those whole numbers together by subtracting 30 from each side.

$\frac{1}{5}x = 20$

To get rid of the fraction, you will need to multiply both sides by 5.  Your answer is x = 100.

Loop Back:  What does x represent?  It represents the total number of juniors, which matches your bottom line.  You are ready to look down at your answer choices.

(A) 72
(B) 80
(C) 84
(D) 100
(E) 120