Most tests you take in your average math class are straightforward and obvious. As long as you’ve studied the concepts, you know what to do when you encounter a problem. You know what to expect.

On the SAT, however, even great math students may not know where to begin. Even if you know all the concepts, you won’t always know which ones to apply and how. To answer the tough questions, you need a combination of experience and a willingness to try things out.

I always warn new students that the SAT Math is unlike most tests they’ve done in school. But when they ask how it’s different, I usually have to resort to vague terms like “problem-solving, creative thinking, and word questions.”

So here are 12 SAT questions that illustrate frequently tested ideas and ways of thinking that go beyond standard school material. I’ve included questions across all test topics—from algebra to geometry to logic.

All of them are tough! Try them on your own. Hints are available at the bottom and answers will be available in another post.

#### The Questions

- If \(2^x + 2^x + 2^x + 2^x = 2^{16}\), what is the value of \(x\)?
- The lengths of two sides of a triangle are \(4\) and \(9\). What is the greatest possible integer length of the third side?
- A car traveled \(10\) miles at an average speed of \(30\) miles per hour and then traveled the next \(10\) miles at an average speed of \(50\) miles per hour. What was the average speed, in miles per hour, of the car for the \(20\) miles?
- The average (arithmetic mean) age of a group of \(15\) teachers is \(40\) years. If \(6\) additional teachers are added to the group, then the average age of the \(21\) managers is \(44\) years. What is the average age of the \(6\) additional managers?
- Let the function \(f\) be defined by \(f(x) = x^2 + 18\). If \(a\) is a positive number such that \(f(2a) = 3f(a)\), what is the value of \(a\)?
- A tech company bought a number of computers for its employees: \(6\) costing \($1600\) each, \(4\) costing \($1200\) each, and \(y\) costing \($900\) each, where \(y\) is a positive odd integer. If the median price for all the exercise machines purchased by the fitness center was \($1200,\) what is the greatest possible value of \(y\)?
- The coordinates \((x, y)\) of each point on the circle above satisfy the equation \(x^2 + y^2 = 25\). Line \(l\) is tangent to the circle at point \(C\). If the \(x\)-coordinate of point \(C\) is \(3\), what is the slope of \(l?\)
- What is the length of \(CD\) in the figure above?

A) \(4\)

B) \(4\sqrt{2}\)

C) \(6\)

D) \(6\sqrt{2}\) - If \(0\leq a \leq b\) and \((a + b)^2 – (a-b)^2 \geq 9\), what is the least possible value of \(b\)?
- In acute triangle \(ABC\), the measure of angle \(A\) is equal to \(2x^\circ + 20^\circ\) and the measure of angle \(B\) is equal to \(x^\circ + 46^\circ\). If \(\sin\angle A = \cos\angle B\), what is the value of \(x\)?
- \[f(t) = 800(1,000)^{\frac{t}{27}}\] The function \(f\) above represents the current total population of sand sharks off the coast of Greenland \(t\) years after 1980. To the nearest year, how many years did (or will) it take for the population to be 10 times larger than it was in 1980?
- In order to fulfill a prescription for 200 milliliters (mL) of a medication at a 55% solution, a pharmacist must mix \(x\) mL of a 40% solution with \(y\) mL of a 60% solution. What is the value of \(x\)?

#### Hints

- The answer is NOT 4. You can only add exponents when the terms are being multiplied. Think about how many \(2^{x}\)s there are.
- The answer is NOT “as big as you want.” Look up the Triangle Inequality Theorem.
- The answer is NOT 40. This is much harder than that. Remember that

distance \(=\) rate \(\times\) time - Think about the sums of all the ages.
- No hint for this one. If you truly understand functions, you’ll get this.
- No hint for this one either.
- What is the slope of \(\overline{OC}\)?
- Know your special triangles (30-60-90 and 45-45-90).
- First, expand (FOIL) and simply. Now how do the values of \(a\) and \(b\) affect each other when they are increased or decreased?
- You’ll have to use a certain trigonometric identity.
- Use the fact that \(1,000 = 10^3\)
- 200 milliliters of a 55% solution contains how many milliliters of the actual medication?

#### Answers

I’m not going to post answer explanations (at least for now), because I want you to figure these out on your own. What I will do is post the answers, so you can check/review your work.

- \(14\)
- \(12\)
- \(\frac{75}{2}\) or \(37.5\)
- \(54\)
- \(6\)
- \(9\)
- \(-\frac{3}{4}\)
- \(D\)
- \(\frac{3}{2}\) or \(1.5\)
- \(8\)
- \(9\)
- \(50\)

Please can you explain 1,5, 9 to me in brief. For q1 I keep getting 16/3. For q 9 I only got so far in the working out as ab> or =9/4 But did not know what to do after that.

But it is q 5 where i am really stuck and don’t even know where to start. I am sure I am forgetting some really simple rule but please can you guide me. Your help as always will be much appreciated!!

I don’t know where my earlier post has gone about the problems I had with q1), q5), q9).

Q 5 i didn’t even know where to start and i am sure I am making some really obvious mistake but please help me.

For Q1 I keep getting 16/3 and for Q9) I got up to the point when ab> or = to 2.25. But then after that I can’t figure out what to do. Please help me with these. I will remain very grateful.

Hi Maroosh.

For #1, think of it as replacing the \(2^x\) with any variable. To illustrate I’ll use \(y\). That way it’s just \(y+y+y+y\). You can simply rewrite that as \(4(y)\). If you now replace the \(y\) with the original \(2^x\), you’ll have \(4(2^x) = 16\). Divide both sides by \(4\). So \((2^x)=4\). Therefore \(x=2\).

For #5, the key is to take each “a” function one at a time. \(f(2a) = (2a)^2 + 18 = 4a^2 + 18\). Next, \(3f(a)=3 (a^2 + 18) = 3a^2 + 54\). Next step is to set the two “a” functions equal to each other. \(4a^2+54 = 3a^2 + 54\). You should get \(a^2 = 36\). And voila! \(a = 6\).

For #9, I didn’t do this as Phu suggested. The easiest method is to notice that the least possible value of \(b\) occurs when \(b\) is equal to \(a\). With this in mind just write \(b\) in place of \(a\). That is, \((b+b)^2-(b-b)^2\). When simplified, you should get \((2b)^2\) which is \(4b^2\). So \(4b^2\) is greater than or equal to \(9\). Again the least possible value of \(b\) occurs when that inequality is equal to \(9\). So just write the equation: \(4b^2=9\). Jingle bells! You get \(b = \dfrac{3}{2}\).

I hope my explanations were of help Maroosh. Cheers and goodluck!

Jingle Bells! I love it!

Fantastic explanations. I’ve edited your responses with math type.

Thank you for posting such amazing help in the form of these questions. Please make more pages on your website with problems like these. i have finished all your worksheets now i think. Please also respond to my email when i sent to you regarding answers for the 12 tricky sat questions problems for which you said we should email you for answers. Thank you very much

can’t the least possible value for b be 1? Or if a is negative, -1?

Thank you very much. I am very grateful. Yes your explanation of each question’s solution was very clear and I finally understood why I was getting these questions repeatedly wrong. Sorry for not replying sooner but I have just checked my email. The quicker method you suggested for q 9 also works really well. Thank you once again

Ooopsies I made a typo in my explanation for #1. The correct solution is \(4(2^x) = 2^{16}\). \(4\) is just \(2^2\). So really you have \((2^2)(2^x) = 2^{16}\). Divide both sides by \((2^2)\). You get \((2^x) = 2^{14}\). Recall that dividing exponents with the same bases just means keeping the same base and subtracting the bottom exponent from the top. That’s how I got \(2^{(16-2)} = 2^{14}\). So \(x= 14\).

Cheers!

No perfect score for you that time! That was more than a typo since you didn’t check against your own answer key 😉

Hi, I got most of them but I just need help with 1 problem. I can’t understand problem 7.

7) slope = change in x / change in y

change in x – given = 3

change in y ( plug in) = 3^2 + y^2 = 25 —-> y=4

— So slope = -3/4. (Since line is sloping downwards, the gradient is negative so there will a minus sign attached to the slope. )

Actually, the slope is rise/ run or (change of y)/ (change of x). using the equation given, we find that x=3 (given) and y= 4. So the slope of the line is 4/3; but that is not the slope we are looking for. 4/3 is the slope of the line that runs from the origin to point C. We are looking for the line that is tangent to the circle and perpendicular to the line from the origin to point C. So the slope of the tangent line will be the negative reciprocal of the slope 4/3. There for the answer is -3/4. Perpendicular slopes are always negative reciprocals of each other.

For number 2, I’m getting 10. Why is the answer 12?

because the sum of the lengths of any two sides of a triangle must be greater than the third side.

check out this website: regentsprep.org/regents/math/geometry/GP7/LTriIneq.htm

hope it helped! Good luck on your SAT!

How do you do #6?

Shouldn’t the answer for Q.6 be 8?

Nope. First of all, the question stipulates that y is a positive odd integer. If it helps, list all the numbers out.

I am having problems with #1 in one area you have the answer as 14 in an email reply you have answer as 2

Please show me how to do the problem

cant do number 3,5 and 9

can anyone show me how to do 3? i know its probably ez but im just not getting it

It takes 1/3 of an hour to travel the first 10 miles. Then it takes 1/5 of an hour to travel the last 10 miles. That’s 1/3 + 1/5 = 8/15 of an hour. To get the average speed, divide the total distance by the total time: 20/(8/15) = 37.5

can someone please explain number 12?