# 15 Hardest SAT Math Questions

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## Question 8: Calculator not-permitted, multiple choice

Category: Passport to Advanced Math—rewriting equations in terms of another variable

Here’s how you solve it:

1. Students get intimidated by questions that have only variables. Remember to use what you know about manipulating equations—such as whatever you do to one side you must do to the other—to handle these problems.

2. We want to rewrite the equation in terms of $$F$$. To start, we’ll want to get rid of the $$F$$ in the denominator of the equation above by multiplying both sides by $$N+F$$ like so:

$$R(N+F)=\frac{F(N+F)}{(N+F)}$$

$$R(N+F)=F$$

3. Distribute the $$R$$ on the left-hand side, and then move all the F‘s to the same side.

$$RN+RF=F$$

$$F-RF=RN$$

4. Next, we can factor out the $$F$$ on the left-hand side and divide both sides by the remaining factor to isolate $$F$$ and get our answer.

$$F(1-R)=RN$$

$$F=\frac{RN}{1-R}$$

## Question 9: Calculator permitted, multiple choice

Category: Problem Solving and Data Analysis—standard deviation, range, analyzing graphical data

Here’s how you solve it:

1. You won’t have to calculate standard deviation on the SAT, but you will need to know what it means and what affects it. Standard deviation is a measure of spread in a data set, specifically how far the points in the data set are from the mean value. A larger standard deviation means that the points in the data set are more spread out from the mean value, and a smaller one means that the points in the data set are close to the mean value. Likewise, you’ll need to know what is meant by range. Range refers to the difference between the highest and lowest values in a data set. The SAT may require you to calculate the range for a data set.

2. Let’s first determine the standard deviations of each data set relative to each other. In the first data set, most of the data points are clustered around each other, which makes it likely that the mean is somewhere around $$72$$ (even though there are two outliers, they are equidistant from $$72$$ so they will balance each other out when determining the mean). In contrast, the second data set is more spread out, so we can conclude that the standard deviation of the first set is smaller than the standard deviation of the second. We can eliminate choices A and C.

3. Now let’s look at the range for each set. We can subtract the highest and lowest values for each set to find the range. For the first set, $$88-56 = 32$$. For the second set, $$112-80 = 32$$. The ranges of each set are equivalent. This leaves us with choice D as our answer.

## Question 10: Calculator not-permitted, multiple choice

Here’s how you solve it:

1. We can solve this using substitution. Substitute the second equation into the first and expand the quadratic:

$$x=2[(2x-3)(x+9)]+5$$

$$x=2(2x^2+15x-27)+5$$

$$x=4x^2+30x-49$$

2. Next, move the x from the left-hand side over to the right. The roots of this quadratic will give us the solution to the system of equations above.

$$0=4x^2+29x-49$$

3. Since the roots of the above quadratic will give us the solution to the system of equations above, we can use the discriminant of the quadratic formula to find out how many solutions there are. The discriminant is the part of the quadratic formula under the radical, or $$b^2-4ac$$. If the discriminant is positive, there are 2 solutions; if the discriminant is 0, there is 1 real solution (or a repeated solution); if the discriminant is negative, there are no real solutions. Substituting the numbers from the equation into the discriminant formula gives us $$29^2-4(4)(49)=841-784=57$$. This is a positive number, which means there are 2 solutions to the system of equations.

## Question 11: Calculator permitted, grid-in response

Category: Passport to Advanced Math—determining effect of one variable on another

Here’s how you solve it:

1. This is similar to the earlier problem where there are only variables, but if we proceed step-by-step and manipulate the equation carefully, we can find the solution. In this problem, we have two fluids, and one of them is moving at a velocity of $$1$$ and the other at $$1.5$$. We can substitute the $$1.5$$ into the equation to see how it might affect $$q$$, or the dynamic pressure.

$$q=\frac{1}{2}n(1.5v)^2$$

2. Using exponent rules (from Question 3 above) we can rewrite $$(1.5v)^2=1.5^2v^2$$. Squaring $$1.5$$ gives us $$2.25$$.

3. For our fluid moving at a velocity of $$1.5$$, we thus have: $$q=\frac{1}{2}n(2.25)v^2$$. For our fluid moving at a velocity of $$1$$, we have $$q=\frac{1}{2}n(1)v^2$$.  We can thus see that the dynamic pressure of the faster fluid will be $$2.25$$ times that of the slower fluid.

## Question 12: Calculator not-permitted, grid-in response

Here’s how you solve it:

1. If you’re given a word problem that involves geometry or trigonometry and no diagram, you should draw yourself a diagram and label the information. This often makes it very clear how to find the answer. Here’s our diagram:

2. Many students know about the 30-60-90 and the 45-45-90 right triangles, and information about these triangles is included on the first page of each SAT math portion. However, you should also be on the lookout for the 3-4-5 and 5-12-13 right triangles. College Board likes to use these triangles and their similar counterparts because they have nice, whole-number sides that make calculations easy. Knowing this, we can see that Triangle ABC is similar to the 3-4-5 triangle, each side of ABC 4 times the length of the 3-4-5 triangle.

3. Knowing that triangle ABC is similar to a 3-4-5 triangle, and that triangle DEF is also similar to a 3-4-5 triangle, we can find $$sin\:F$$ by substituting 3-4-5 for the sides of DEF and using the definition of sine: $$sin\:\theta\:=\frac{opposite}{hypotenuse}$$

4. With respect to $$F$$, the opposite side is $$3$$ and the hypotenuse is $$5$$. This means that $$sin\:F=\frac{3}{5}$$, or $$0.6$$.

## Question 13: Calculator permitted, multiple choice

Category: Heart of Algebra—linear equations and inequalities in context

Here’s how you solve it:

1. You’ll need to be able to write equations that reflect the context described in a word problem for several questions on the SAT, both linear context and non-linear contexts. If you have trouble “translating” word problems into math, this is a skill you’ll want to work on! Rewriting word problems to include words like equal to, less than, more than, sum, and so on can help you easily translate the problem into equations and inequalities.

2. We’re told that Roberto wanted to sell $$57$$ insurance policies but he didn’t meet his goal, which means that he sold less than $$57$$ insurance policies. Given that $$x$$ represents the number of $$\50,000$$ policies sold and $$y$$ represents the number of $$\100,000$$ policies sold, the sum of these two numbers is less than $$57$$. This is represented as $$x+y\:\lt\:57$$. We can eliminate choices B and D.

3. Next, we’re told that the value of all the policies sold was more than $$\3,000,000$$. We need to multiply the value of the policy by the number of that type of policy, which gives us $$50,000x+100,000y\:\gt\:3,000,000$$. Choice C is the only answer that reflects Roberto’s insurance policy sales.

## Question 14: Calculator permitted, multiple choice

Category: Problem Solving and Data Analysis—Statistics

Here’s how you solve it:

1. These kinds of problems can surprise students because there isn’t any obvious calculation or “math” to do. Instead, you’re being asked to interpret the results based on given information, and you’ll need to know statistics to correctly interpret these types of problems. In this case, you’ll want to draw upon knowledge of population parameters, random sampling, and random assignment:

• A population parameter is a numerical value describing a characteristic of a population. For example, we’re told that the population targeted by this treatment are “people with poor eyesight.” While not given a number, any conclusion we draw can only be made regarding people who fall under this parameter.
• Random sampling means that the subjects in the sample were selected at random (without bias) from the entire population in question. If random sampling is used, the results can be generalized to the entire population.
• Random assignment means that subjects in the sample were assigned to a treatment at random (without bias). If random assignment is used, it might be appropriate to make conclusions regarding cause and effect.

2. The SAT is very particular when including statistical clues and knowing the definitions of statistics. For example, if random sampling is not mentioned in a problem, then the results cannot be generalized to the entire population. Knowing the above, we can see that random sampling and random assignment are explicitly mentioned in this case. We can reasonably conclude cause and effect to be generalized to the population in question, or that it’s likely that Treatment X will improve the eyesight of people who have poor eyesight. The other choices are either too confident in their conclusion or do not specify the population in question.

## Question 15: Calculator not-permitted, grid-in response

Here’s how you solve it:

1. The point $$A$$ is $$(3,1)$$. We can create a triangle to find the degree measure of $$\angle{AOB}$$. When we do this, we can use the information the SAT provides to see that this is a 30-60-90 triangle and that the hypotenuse (or the radius of the circle) is $$2$$.

2. This means that $$\angle{AOB}$$ is $$30$$ degrees. To convert from degrees to radians, we multiply by $$\frac{\pi}{180}$$. This is what we get: $$30\:\cdot\:\frac{\pi}{180}=\frac{\pi}{6}$$

3. Since $$a$$ is the denominator, the answer is $$6$$.

## Final Tips

We hope that you have a good sense of the type of questions that you might see on the SAT, but just remember that these questions aren’t a complete representation of the topics you could be tested on. Review the questions above and ask yourself: which ones were the most challenging to you? Which topics do you need to brush up on?

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Gianna Cifredo
Blogger at CollegeVine
Short bio
Gianna Cifredo is a graduate of the University of Central Florida, where she majored in Philosophy. She has six years of higher education and test prep experience, and now works as a freelance writer specializing in education. She currently lives in Orlando, Florida and is a proud cat mom.
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