Here are the 10 toughest SAT questions from our New SAT Prep, as well as the 10 toughest questions from our current “old” SAT Prep. See how many you get right by checking your answers at the end of the post!

(P.S. If you can get these right, then you should check out Harvard SAT scores and Yale SAT scores…)

## 10 Toughest New (Redesigned) SAT Questions

These are the ten most difficult SAT practice questions you’ll find in our New SAT Prep (which helps students study for the redesigned SAT, which will debut in March 2016). Let us know how you do!

_____

*The common setup is researchers will divide subjects into two groups, one of which is allowed to use the Internet after finishing the task, the other of which must finish the task until completion. Yet another common setup allows subjects unfettered use of the Internet when trying to complete the task. Not surprisingly this last group acted 1 worse on tests of productivity. 2 Not so surprisingly the group that used 10 minutes of web access as an incentive, tended not only to finish the task sooner than the group without any web access but also 3 worked with more vigor when their Internet time was up.*

1.

A) NO CHANGE

B) the worst

C) badly

D) more poorly

*Tip: Carefully read the sentences that come before this question.*

2.

Within the context of the paragraph, the underlined portion should be changed to which of the following?

A) NO CHANGE

B) Unsurprisingly

C) Less surprisingly

D) What is surprising is that

*Tip: The New SAT Writing section will be a lot about context. What I mean by that is to find the answer you’ll have to read the sentences before and after the sentence in which the question appears. Only that way will you have the evidence you need to support your answer.*

3.

A) NO CHANGE

B) to work

C) they worked

D) to be working

*Tip: Whenever you have the construction “not only VERB A but also VERB B”, the two verbs must always be parallel. *

_____

*Humans make errors. We make errors of fact and errors of judgment. We have blind spots in our field of vision and gaps in our stream of attention. Sometimes we can’t even answer the simplest questions. Where was I last week at this time? How long have I had this pain in my knee? How much money do I typically spend in a day? These weaknesses put us at a disadvantage. We make decisions with partial information. We are forced to steer by guesswork. We go with our gut.*

That is, some of us do. Others use data. A timer running on Robin Barooah’s computer tells him that he has been living in the United States for 8 years, 2 months and 10 days. At various times in his life, Barooah — a 38-year-old self-employed software designer — has also made careful records of his work, his sleep and his diet.

A few months ago, Barooah began to wean himself from coffee. His method was precise. He made a large cup of coffee and removed 20 milliliters weekly. This went on for more than four months, until barely a sip remained in the cup. He drank it and called himself cured. Unlike his previous attempts to quit, this time there were no headaches, no extreme cravings. Still, he was tempted, and on Oct. 12 last year, while distracted at his desk, he told himself that he could probably concentrate better if he had a cup Coffee may have been bad for his health, he thought, but perhaps it was good for his concentration.

Barooah wasn’t about to try to answer a question like this with guesswork. He had a good data set that showed how many minutes he spent each day in focused work. With this, he could do an objective analysis. Barooah made a chart with dates on the bottom and his work time along the side. Running down the middle was a big black line labeled “Stopped drinking coffee.” On the left side of the line, low spikes and narrow columns. On the right side, high spikes and thick columns. The data had delivered their verdict, and coffee lost.

He was sad but also thrilled. Instead of a stimulating cup of coffee, he got a bracing dose of truth. “People have such very poor sense of time,” Barooah says, and without good time calibration, it is much harder to see the consequences of your actions. If you want to replace the vagaries of intuition with something more reliable, you first need to gather data. Once you know the facts, you can live by them.

Barooah knows that this behavior is abnormal. He is an outlier. But why does what he does seem so strange? In other contexts, it is normal to seek data. A fetish for numbers is the defining trait of the modern manager. Corporate executives facing down hostile shareholders load their pockets full of numbers. So do politicians on the hustings, doctors counseling patients, and fans abusing their local sports franchise on talk radio. Charles Dickens was already making fun of this obsession in 1854, with his sketch of the fact-mad schoolmaster Gradgrind, who blasted his students with memorized trivia. But Dickens’s great caricature only proved the durability of the type. For another century and a half, it got worse.

Or, by another standard, you could say it got better. We tolerate the pathologies of quantification — a dry, abstract, mechanical type of knowledge — because the results are so powerful. Numbering things allows tests, comparisons, experiments. Numbers make problems less resonant emotionally but more tractable intellectually. In science, in business and in the more reasonable sectors of government, numbers have won fair and square.

For a long time, only one area of human activity appeared to be immune. In the cozy confines of personal life, we rarely used the power of numbers. The techniques of analysis that had proved so effective were left behind at the office at the end of the day and picked up again the next morning. The imposition, on oneself or one’s family, of a regime of objective record keeping seemed ridiculous. A journal was respectable. A spreadsheet was creepy. And yet, almost imperceptibly, numbers are infiltrating the last redoubts of the personal. Sleep, exercise, food, mood, location, alertness, productivity, even spiritual well-being are being tracked and measured, shared and displayed.

4. The use of Gradgrind (sixth paragraph) as a supporting example is most problematic because it

A) conflates the accumulation of academic facts with the process of quantification.

B) undermines the main thesis of the passage by citing a dated example.

C) accepts without reservation that a trend has intensified with the passing of time.

D) provides an example of a process the author ultimately appreciates.

*Tip: Think about the author’s main point. Does it make sense to use Gradgind as an example? Why not? (Your answer to that second question should match the correct answer).*

5. In the last paragraph, the author would likely view the use of numbers to track intimate aspects of people’s lives as

A) intrusive.

B) inevitable.

C) ominous.

D) disruptive.

*Tip: You will need to understand the thrust of the entire passage to be able to answer this. Don’t just jump to the conclusion that the answer must have a negative connotation. *

6. Of the solutions for , which has the greatest value?

A) 0

B)

C)

D)

*Tip: Remember to look at the answers accompanying a question. They will sometimes give you an idea of how to approach this question. In this case, the set up of the answer choices should remind you of a certain formula. *

7. For x * t ≠ 0, which of the following equations is equivalent to ?

A)

B)

C)

D)

*Tip: Be very careful when factoring out each side. And look at the format of the answer choices. They should show you that you are not combining like terms per se but are using FOIL. *

8. The average (arithmetic mean) of 4 different integers is 75. If the largest integer is 90, what is the least possible value of the smallest integer?

A) 1

B) 29

C) 30

D) 33

*Tip: This is a logic question. Setting up an equation for average will only get you so far. Think in terms of what number could be the smallest possible value.*

9. Solution X is 10 percent alcohol by volume, and solution Y is 30 percent alcohol by volume. How many milliliters of solution Y must be added to 200 milliliters of solution X to create a solution that is 25 percent alcohol by volume?

A) 250/3

B) 500/3

C) 480

D) 600

*Tip: You can solve this question by setting up an equation…or you can think of this problem as a weighted average.*

10. If the circle with center O has area 9π, what is area of equilateral triangle ABC?

A) 9√3

B) 18

C) 12√3

D) 24

*Tip: Remember to think of the necessary steps to arrive at the answer. Once you’ve worked those steps at then apply the math. And don’t forget – the fundamental geometry formulas are always in the beginning of each math section.*

*Answers:*

*1) B*

*2) D*

*3) B*

*4) A*

*5) B*

*6) C*

*7) C*

*8) D*

*9) D*

*10) C*

## 10 Toughest (Old) SAT Questions

These are the ten most difficult questions you would have found in our Magoosh SAT Prep if you had taken the old version of the SAT exam (the one given until January 23, 2016). You’re still welcome to try them (they’re tricky), but you probably won’t see anything like this on the redesigned SAT. Maybe be glad you’re missing out? 🙂

**Directions: Choose the words that best fit the blanks:**

1. Cosmologist Martin Rees has cautioned that our present satisfaction with the big bang explanation for the creation of the universe may reflect the ——- of the data rather than the ——- of the theory.

- paucity . . validity
- genius . . accuracy
- relevance . . scope
- destruction . . core
- persuasiveness . . reality

*Tip: Try to come up with your own word(s) for the blank. If you are unable to, it is okay, as a last resort, to plug the answer choices back in the blank. Sometimes meaning emerges this way and the sentence makes sense.*

2. Apparently the groom was very nervous: one moment he would be ——-, rambling on to his best man about silly, meaningless things, and then abruptly he would turn ——- and could not be prompted to say anything

- garrulous . . reticent
- grandiose . . taciturn
- vociferous . . effusive
- melodious . . timorous
- munificent . . utilitarian

*Tip: Match the clues with the blanks and then find a word that matches. Remember you only need to work one blank at a time, eliminating those answer choices that don’t work. Then, when you move on the other blank, you only have a few possible answers to deal with.*

**Directions: Choose the correct version of the sentence:**

3. Regardless of the fact of the ridge-top condominiums’ aesthetics, every investor has enjoyed a high return on their investment.

- Regardless of the fact of the ridge-top condominiums’ aesthetics, every investor has enjoyed a high return.
- Regardless of the ridge-top condominium aesthetic, every investor has had a high return to enjoy.
- Regarding the aesthetics of the ridgetop condominiums, every investor has enjoyed a high return.
- Regardless of the fact of the ridge-top condominiums’ aesthetics, a high return by every investor has been enjoyed.
- Regardless of the aesthetics of the ridge-top condominiums, every investor has enjoyed a high return

*Tip: Remember to retain the original meaning of the sentence – investors are enjoying an investment. If you remove investment than they are enjoying (having a good time) the high return (money). Which, while highly likely, changes the overall meaning of the sentence.*

4. Included in the cost of many services and products sold in Great Britain, American tourists may not realize that they do not necessarily have to pay the value added tax (VAT).

- Included in the cost of many services and products sold in Great Britain, American tourists may not realize that they do not necessarily have to pay the value added tax (VAT).
- Included in the cost of many services and products which are sold in Great Britain, tourists from America may not realize that they do not necessarily have to pay the value added tax (VAT).
- American tourists may not realize that they do not necessarily have to pay the value added tax (VAT) that are included in the cost of many services and products sold in Great Britain.
- In addition to the cost of many services and products sold in Great Britain, American tourists may not realize that they do not necessarily have to pay the value added tax (VAT).
- American tourists may not realize that they do not necessarily have to pay the value added tax (VAT) that is included in the cost of many products and services sold in Great Britain.

*Tip: Remember to make sure that the nouns in the sentence are being modified correctly. American tourists are not included in the cost of many services.*

**Select the answer that best answers the question:**

5. The average (arithmetic mean) of 4 different integers is 75. If the largest integer is 90, what is the least possible value of the smallest integer?

- 1
- 19
- 29
- 30
- 33

*Tip: This is a logic question. Setting up an equation for average will only get you so far. Think in terms of what number could be the smallest possible value.*

6. If square ABCD has area 25, and the area of the larger shaded square is 9 times the area of the smaller shaded square, what is the length of one side of the smaller shaded square?

Note: Figure not drawn to scale

- 3/4
- 5/4
- 6/5
- 4/3
- 5/3

* Tip: If you are not sure how to set up the question algebraically you can also solve using the given information. In this case you can assume the answer is (C). So if the side of the small square is 6/5 do we end up with 25 as the area of the big square? Remember the big square has an area that is twice as big as that of the small square (in this question the algebraic approach is better).*

7. Solution X is 10 percent alcohol by volume, and solution Y is 30 percent alcohol by volume. How many milliliters of solution Y must be added to 200 milliliters of solution X to create a solution that is 25 percent alcohol by volume?

- 250/3
- 500/3
- 400
- 480
- 600

*Tip: You can solve this question by setting up an equation…or you can think of this problem as a weighted average.*

8. On a certain multiple-choice test, 9 points are awarded for each correct answer, and 7 points are deducted for each incorrect or unanswered question. Sally received a total score of 0 points on the test. If the test has fewer than 30 questions, how many questions are on the test?

- Cannot be determined
- 16
- 19
- 21
- 24

*Tip: This is a question based more on logic. Do not try to set up an equation but think in terms of how many 7-point questions you need and how many 9-point questions you need for the two to cancel out.*

9. A computer can perform c calculations in s seconds. How many minutes will it take the computer to perform k calculations?

- 60ks/c
- ks/c
- ks/60c
- 60c/ks
- k/60cs

*Tip: Assign values to k, s, and c if you have difficulty thinking through this question algebraically.*

10. If the circle with center O has area 9π, what is area of equilateral triangle ABC?

- 18
- 24

*Tip: Remember to think of the necessary steps to arrive at the answer. Once you’ve worked those steps at then apply the math. And don’t forget – the fundamental geometry formulas are always in the beginning of each math section.*

**For this last question, try it out in Magoosh SAT to see the answer and video explanation!**

*Answers:*

*1. A*

*2. A*

*3. E*

*4. E*

*5. E*

*6. B*

*7. E*

*8. B*

*9. C*

*10. C*

Which exam do you think is more difficult – the current or new SAT? 🙂

**Looking for answers to the most challenging Official SAT Study Guide math questions? Use our Official SAT Study Guide Question Explanations and watch test prep expert Chris Lele explain the smartest way to solve new SAT math questions.**

Hello! Just want to say thank you for this interesting article! =) Peace, Joy.

Thanks for the kudos!

thanx dude got 9 of dem correct and i hav got 2300 in SAT

Great! That shows that the questions are pretty tough, and that you are very good at the SAT :).

Whoa…I got the exact same stats. THere’s really is a correlation.

Nice Dude It was pretty helpful but got just 6 so sad 🙁

Hi, Julie! 6/10 certainly isn’t bad to start out with– I’m sure that after studying for a few weeks, you could score even higher 🙂

sat exam math is so easy as compared to iits math,its almost like nothing

Hi Ankush,

The general SAT math portion is meant more to test one’s thinking/problem solving skills. It is definitely not too difficult, and sometimes getting a perfect score for us seasoned tutors, is making sure we don’t make a careless error. The SAT math subject tests, on the other hand, are much more difficult, and are targeted towards those who want to study in math/science fields.

Are you Ankush Sharma from FIITJEE Chandigarh?????

Hi there I’m 13 and in 8th grade in this test I got 4/10 ….I’m feeling bad

Dkgladr,

Wow, that’s actually a great score, for someone who isn’t even required to take the PSAT. Some of the questions above are taken from GMAT and GRE tests, which are for those looking to go to graduate school (some of the smartest people around :)). So, I’d say you are doing pretty well :).

I got 9/10 correct and I have a 1960 SAT Score. These questions were helpful though

Hi Ned,

These questions are very tough — so you did well. I’m guessing you do pretty well on the quant, maybe close to 700. Even then, on the SAT, those “easy” and “medium” questions can get you to go :).

Good luck!

Could you please explain how to do number 10? I keep getting the answer 27 times the square root of three. Thank you.

I just answered my question, sorry ^^.

No problem! 🙂 Glad you figured it out!

I’ve been studying for a few weeks and could only get 5 right (barely)…I am so gonna to bomb the SAT…

Hi Ash,

Not at all — these are all difficult questions. Only about 25% of the questions on the SAT are this difficult. So if you are getting half of the difficult ones correct, you are probably do even better on the medium and easy questions.

Good luck–and don’t get discouraged :)!

Well, tell me if I am wrong , but there is something off about question 8. Even I came up with 16(b) at first but then I considered this situation …

Suppose Sally is as bad as a student as myself and managed to get 4/16 questions correct , earning her 4*9=36 marks .

However she skipped or got wrong 12/16 questions , losing a total of 12*7=84 marks .

So, her total score is = ta ta ra ra … negative 48.

You see my point?

Hi Madcap,

It seems that we already have a given piece of information: the student scored zero points on the test. The only way this is possible, given the test has fewer than 30 questions, is if there are 16 questions on the test. The question is not saying, if the test had 16 questions, then how many points must Sally have gotten? She could have gotten a bunch of different points, including, as you mentioned, -48.

Hopefully that makes sense 🙂

Im stuck on number 5 could you please explain? I was thinking that if there are no restrictions on what the other possible integers could be then couldn’t 1 be one of them and then the other two would add up to the rest? That was my thinking but I guess I was wrong. Could you please explain, thank you so much!!

Hi Rose,

So we know that the sum of the average has to be 300 (75×4) = 300. If the largest number is 90, then the next largest numbers could be 89 and 88. Remember, we want to maximize these two numbers so that the remaining number can be as small possible. This gives us 90+89+88+x = 33 (E).

Hope that makes sense 🙂

Hi Chris,

I don’t understand number 5 at all whatsoever. Could you please explain it to me? Also, regarding number 5, why is the answer not 1? The question doesn’t ask for the highest possible smallest integer that could result in the mean? It merely asks for the smallest, so 1 should be applicable, shouldn’t it?

Thank you!

Hi Anzie,

It does seem that way, that ‘1’ should be the answer. But this is a tricky one. If we actually add up the four integers, x + 88 + 89 + 90 = 300 we get 33 for x. x cannot equal ‘1’ because the two integers in the middle have to be less than 90, and the greatest possible numbers they can equal are 88 and 89 respectively.

Hope that helps shed some light :).

I do not get number 9 at all because I tried to plug in 60 for all the values, and only choice B. worked using that method. The only way that I could see choice C. working is if there was a fraction bar between the ks and 60c like this:

ks —–

60c

Is this how it was intended to be written? I tried to follow the choices given to me using order of operations.

Thanks and Happy Holidays!

#9 was the only one I got wrong and I am in 7th grade. 😛

Hi Viraj,

Good job on getting most of the questions correct!

Yes, on question #9, it should be written as ks/60 (that mark is the same as the horizontal line dividing the numerator and the denominator).

Hope that helps!

Hey there,

Thank you very much for these very helpful questions.

I need to understand number 10 and 7, I am stuck in 7 for not knowing how many millimeters contain what percentage.

in 10 it’s because I did 9^2÷π=25.78310078, and that’s close to the correct answer but I need to understand the idea!

Thank you in advance, have a beautiful day 😀

On number 10, the key is to solve for the radius. Another important thing is on the SAT never change π to 3.14. Just keep it as π. If we know the circle has an area of 9π, then its radius must equal 3 (9π = πr^2, r = 3).

Each length of the equi. triangle is therefore 6. Using the area for equi. formula (s^2√3)/4, where s corresponds to the side of the equi. triangle, we get 9√3.

For number 7, think of the problem has a weighted average. If equal parts of each solution were put in a new container, then the resulting mixture would contain 20% alcohol. The resulting mixture, however, contains 25% alcohol. Therefore, more of solution Y has to be in the mix? How much more?

Quick way to figure this is out is find the ratio between solution Y – resulting mixture (30 – 25) = 5, and resulting mixture and solution X (25 – 10) = 15. That gives us 1:3. Therefore there is three times as much solution Y as X: 3 x 200 = 600.

Hope that makes sense!

Hey Chris,

For number 10 the answer is 12 root 3 not 9 root 3. Yes, there is a 6 however that is the diameter of circle O. Once one has the diameter, he/she could simply use the 30-60-90 triangle as a reference and find the base of the triangle, segment CB. CB comes out to be 4 root 3. Using the formula A=(1/2)base times height : [(1/2)(4 root 3)(6)] the area comes out to be 12 root 3.

Hope this helped =)

Oops, it looks like I made a very basic error there. Clearly the diameter is not the same as the length of a leg of the equilateral triangle. Morale of the story: don’t rush :).

So .. trigonometry is in the syllabus of SAT ?

Actually, it’s not. You’ll just have questions that use basic geometry. For instance, you won’t have to know how to derive the length ratio of a 30:60:90 triangle. That’s given to you at the beginning of the section.

The new SAT, the one set to debut in 2016, will have trigonometry, or so it’s rumored :).

I got 7/10 correct (yay), although, my SAT scores suck (except in math) ( ;__; )

Anyway, I don’t understand question 1. The sentence doesn’t make sense to me with those answers.

Good job! Your SAT scores can’t suck too much if you did that well :).

For #1, what the question is saying is that scientists are happy with the Big Bang theory because they simply don’t have enough data–data that could indicate the theory is not valid.

Hope that helps!

wouldn’t choice (E) also work?… I mean what context clues can you draw to get to choice (A) rather than (E)

exactly

got a 2380 on the SAT, these questions were really helpful, thx for posting them

You are welcome 🙂 — I’m happy you enjoyed the questions.

And great score!

Hey man i was wondering about the 5th question

there are no restrictions about the other 3 intgers so couldnt the other 2 numbers also be 90 and 90 as 90 is the largest integer so in retrospect only one of the numbers need to be smaller than 90

90+90+90+x=300

x=30…..?

You have great instincts! The SAT could definitely trap you with something like that. But in this case, the question does specify “four

differentintegers,” so we know there can’t be three 90s. We instead have to use 90, 89, and 88 for the three large numbers, since those are the three largestdifferentintegers allowed. Notice how one word in the question can totally change the answer—this is a perfect example of the SAT being tricky and teaches an important lesson!Do you know of a book that has groups of questions that are ONLY the hard ones from or like the SAT or ACT? Thx.

I didn’t understand how to solve out 6,7, and 9. Also any general tips for the sat because I’m a really hard worker and still did poorly. Thanks!

Problem 6. We know the total area of the square to be 25. Since that’s the case we can also infer that each side is 5. If we call a side of the small square x and of the larger square y then we have x + y = 5 since they span the length of the larger square. The areas of each square are x² and y². We are also given that the area of square y is 9 times the area of square x so y² = 9x². Now we have a system we can solve pretty easily.

x + y = 5

y² = 9x²

Setting each equation equal to y in terms of x we get…

y = 5 – x

y = sqrt(9x²) or y = ±3x

We’re only concerned about y = 3x since the length can only be positive and y = -3x will give a negative answer when setting -3x = 5 – x.

So now we have

y = 5 – x

y = 3x

Setting the right sides equal to one another we have

5 – x = 3x

5 = 3x + x

5 = 4x

5/4 = x

Since x is the side length of the small square the answer is 5/4.

Problem 7. We can also solve this one with a system of equations.

We have a solution X that is 200 milliliters and we know that is 10% alcohol so 20 milliliters of that is alcohol. We want to get another solution that we’ll call Z that is 25% alcohol when we add the solutions X and Y together so we can model this by the following equation…

.3y + .1(200) = .25z

The 200 is the volume of solution X which we know from the given information so we can simplify this to….

.3y + 20 = .25z

We need to relate the totals now. The last equation models alcohol content. We know that solution X is 200 milliliters so this added to solution Y will give solution Z so we have the following equation…

y + 200 = z so we can solve the following system

.3y + 20 = .25z

y + 200 = z

We are looking for y so it would be quicker to put z in the first equation in terms of y and solve for y. Look at the second equation. We see that z = y + 200 so we can substitute y + 200 for z in the first equation to get

.3y + 20 = .25(y + 200)

.3y + 20 = .25y + 50

.3y – .25y = 50 – 20

.05y = 30

y = 30/.05

y = 600 so the answer is 600 milliliters. We can plug this into the system to check too…

600 + 200 = 800 so z = 800

.3(600) + 20 = .25(800)

180 + 20 = 200

200 = 200 so we’ve also proved our solution.

Problem 9. A computer can perform c calculations in s seconds. How many minutes will it take the computer to perform k calculations?

If the computer does c calculations in s seconds then it does c/s calculations in 1 second and hence does 60c/s calculations in one minute so if a minute is m then k/m is the amount of calculations done per minute which is equivalent to 60c/s so we can make the equation.

K/m = 60c/s

Multiplying both sides by m we get

K = (60c/s)m

Now we can multiply both sides by s now to get

Ks = 60cm

And to finally isolate m we divide both sides by 60c to get.

Ks/60c = m

So the answer is m = ks/60c which is answer choice C.

There won’t be too many of these problems on the SAT that are this difficult nor should they take as long as they look. I just went through them step by step so you could see how they are done.

Got 8 of them correct. Thank you for the questions Chris. And i will give my SAT exam this June. 🙂

Good luck, Ayush 🙂

Thanks and i got 7 out of them all from maths and 1 from English the first question only and i think I need major improvement in English section. I am giving my SAT exam in June this year

Good job, Chirag!

The verbal just takes a little practice. I’m sure you will do well 🙂

I got 7 correct, 2 wrong & left one. I mistook in questions 1 & 4, left 3.

I got all others correct. I’m a 10th grader & going to have my first test next June.

I hope I can get a good score at critical & writing sections, & a full mark in math’s 🙂

Hi Omar,

That’s a great start! With a little more prep you could do very well in all three sections.

Keep up the good work 🙂

Got 10 out of 10… Giving SAT this Saturday I hope a get a decent score 🙂

Hi Yash,

Well, that looks like a good start :). Good luck tmrw.

OMG! Chris you are so hansome. I think im in love lol

Thanks :). As long as my lessons and videos are instructive, I’m happy 🙂

Hi, please can you tell me how the question 6 algebraically solve

thanks

Hi Joud,

This is part of Nick’s comment below. Hopefully that helps 🙂

Problem 6. We know the total area of the square to be 25. Since that’s the case we can also infer that each side is 5. If we call a side of the small square x and of the larger square y then we have x + y = 5 since they span the length of the larger square. The areas of each square are x² and y². We are also given that the area of square y is 9 times the area of square x so y² = 9x². Now we have a system we can solve pretty easily.

x + y = 5

y² = 9x²

Setting each equation equal to y in terms of x we get…

y = 5 – x

y = sqrt(9x²) or y = ±3x

We’re only concerned about y = 3x since the length can only be positive and y = -3x will give a negative answer when setting -3x = 5 – x.

So now we have

y = 5 – x

y = 3x

Setting the right sides equal to one another we have

5 – x = 3x

5 = 3x + x

5 = 4x

5/4 = x

Since x is the side length of the small square the answer is 5/4.

– See more at: https://magoosh.com/sat/2011/10-most-difficult-sat-questions/#comment-200951

I only got the 9th one wrong, but my biggest difficulty in the SAT is the time. Are there any useful tips to do the questions faster so that I don’t run out of time.

That’s a great question, and one that is not easy to answer. For one, not sure how you approached these questions. Secondly, I don’t want to hurt your accuracy by suggesting a strategy–esp. if you are doing okay time-wise on an Official SAT test.

But I don’t want to leave you hanging, so here are a few tips:

1) Assuming that you do math questions the long way (write every step out), get used to doing more math in your head (again, make sure that accuracy doesn’t suffer). Next, revisit the problem and see if there is a shorter way to get the solution.

2) Know your grammatical errors. That way, you are scanning the sentences intelligently, looking for the mistake, instead of hopping that one just kind of “pops out” at you.

Hopefully, that helps somewhat. If not, I’d be happy to give you some more tips (just give me some more specifics on you are currently approaching problems).

Hello! This was very good practice for me! This is a really helpful website! The only question I didn’t understand was number 8; could you please help me with this?

Hi Shelly,

We know that we have to find the lowest common multiple the two numbers have. That way, we can “balance” out the 9 point and 7 point questions to get ‘0’ total points. 63 is the lowest common denominator. That means 7 questions have to be 9 points (7×9 = 63) and 9 questions have to be 7 points (9×7=63). That is a total of 16 questions.

Hope that helps 🙂

Im actually studying for the shsat and i got most of these wrong. I especially want to know how to solve number ten. Thx…

Hi Ny,

Solve for the radius of the triangle to get 3. Double that to get the diameter. Notice how the diameter is the same as altitude of the equilateral triangle. Use 30:60:90 properties to reason that the small side of the triangle is 6/√3. Use (basexheight)/2 to arrive at (3×6√3)2 = (C).

Hope that helps!

How can we exactly assume that the triangle is 30:60:90? I got to 3 and the altitude.. but then what property am I forgetting to know that the triangle is one of those… thanks btw

When you split an equilateral triangle like that in half it becomes 30-60-90.

This is because all the angles in an equilateral triangle are 60. The line that splits it in half bisects the angle into two 30 degree angles. The line is also perpendicular to one side so there are two 90 degree angles. The other two angles are unaffected and remain 60 degrees. You thus get two triangles with degree measures of 30, 60, and 90.

I’m confused, you appeared to multiply by 2 instead of divide. Why?

The post is soo late…

I tried to solve #5. and if the 3 other remaining integers add up to 210 (300-90), the least possible is 1 in my calculations: 209 which is the sum of remaining 2 middle integers is impossible because one of them would have to be greater than 90 to make 209 possible.

So, you have to be careful with ‘1’, because, as you noted, that would give you a number larger than ’90’, and a sum larger than 300. The best way to attack is to maximize the other two integers (giving you 89 and 88), which leaves you with 33: 33 + 88 + 89 + 90 = 300.

Hope that helps resolve any confusion 🙂

I got 9/10 right. I just CANNOT understand problem no. 7. These mixture sums always get me 🙁

Yes, they can be quite frustrating 🙂

Here’s an interesting way to work a problem: just “backsolve” using the answer choices. First off, you can figure out that the number of ml has to be greater than 200, because if you added 200 ml of Y, then that would give you a 20% solution. Since, we need a 25% solution, we know we have to have more of solution Y.

Answer (C) 400 is a good place to start backsolving. Doing so give us:

[(400)30 + (200)10]/ 600 = 12,000 + 2,000 = 14,000/600 = 23 something. Therefore, we have to choose a large number.

By the way, I get 600 by adding the total ml of both solutions (400 + 200).

Backsolving using (E), since it is an easier number to deal with than 480, I get:

[(600)30 + (200)10]/ 800 = (18,000 + 2,000)/ 800 = 25%, the answer.

This might seem long, but I wrote out all the calculation steps, something that only takes a few seconds on the calculator.

Hope that helps provide a helpful approach to a mixture problem 🙂

If 200 ml of solution x is used that means it has 20 ml of alcohol in it because 10% of 200 ml (0.1*200). Let’s say we will use y ml for solution y then 0.3*y of alcohol will be used. Since the total volume of mixture is 200+y, then (20 + 0.3*y)/(200+y) = 0.25. Solving that equation, you will get 600 m for solution y.

Do you need to memorize random words and their meaning in order to pass the reading portion? cause that how i see it. P.S. not a native speaker

Someone,

Yes, memorizing words is indispensable to doing well on the verbal portion–that’s the basic truth. But memorizing words, doesn’t have to be as dull and formulaic as you might think.

Here is a post about how to make learning vocab a little less tedious:

https://magoosh.com/sat/2014/video-post-sat-vocabulary/

Hope that helps 🙂

i got 8 i really don’t get 1 and 3

your questions are very easy,,,,,,,,,,,,its just a speed test for me (!_!)

I took the SAT, and got a 1680 on it,but i got 9/10. The only one i got wrong was number 9 if you could explain it would help me a lot. Also if you know how to improve my writing section score it will help a lot. Thanks

Just use dimensional analysis. If you start with K calculations, set it up so the units cancel.

K calcs * (S sec/C calcs)

Here, the calcs canceled so you’re left with KS/C seconds. Next, convert seconds to minutes.

K calcs * (S sec/C calcs) * (1 min/60 sec)

Now, when you multiply that you, you are left with KS/60C

to be honest i didn’t read through just wanted to give a comment. My SAT is 4 days away and i thought i was ready now it seems like my it on head is empty. sorry i wrote this here i did because i know the wont even put on the site. you did a great work with the questions though , will try to go through it.

I got 10/10 without really getting “stuck” anywhere, but I spent much more time on a few of the math problems than I could have afforded on the test. Time is not my friend; I usually need about 15 extra minutes for each math section to answer all questions. My score on the College Board practice tests has consistently fallen in the 2200-2400 range, but the last time I took the SAT I scored only 2090 because I had to omit quite a few problems. I’m retaking it in October, and I would really appreciate any tips for better time management.

Yayaya…I got all 10 of them right. 😀 😀

Have my SAT in 11 days. Hope I do well.

This was really helpful. Thanks

Hey Sam,

That’s awesome! I’m so glad you found these tough questions helpful. 🙂

Good luck on your SAT! I hope you do well.

Cheers!

Rita

Are we allowed to use trigonometry on the last question?

Hi SAJ,

You are allowed to use any method you care to, as long as you

get the answer :). (Of course, no cheating allowed). With this question

it seems that the non-trigonometry, using 30-60-90 properties, is

a much faster way of doing it.

But if trigonometry works for you, that’s totally valid too!

Only got 7. 🙁

I would have gotten 8 if I wrote down the length of the diameter rather than hoping it would stay in my head.

Well, I still have 4 years to get them right. 🙂

7/10 is pretty darn good, especially if you still have 4 years to prep. 🙂 We’ll be adding 10 new questions soon – stay tuned!

#7 of new sat test has no correct options

Thanks Sol, for catching that! That’s definitely a typo with the cubed root thrown into (C). Not

quite sure how it got there, but I’m removing it now :).

I am in class 10 and its really easy.perfect 10/10

But I liked the questions.Thank you.

Can you please post more questions.

About question 1:

Why “the worst” when only two groups are involved? I always the the superlatvie was reserved only for comparisons of three elements or more.

Hi Frank,

Good question! 🙂

It is true that it used to be assertively taught that a superlative cannot be used in a comparison between just two objects. If we put aside the fact that this is not commonly taught or enforced in style guides anymore, I want to point out some situations when it would be possible to say that there are grounds for ignoring it. Imagine we want to refer to the “most young” individual in a group, but we do not know if there are 2 or 3 people in the group; in this case, we could comfortably use “youngest” despite the potential mismatch. Also, for reasons of style, we might sometimes ignore this idea and say something like, “My father is the tallest of three children, and my mother is the tallest of two.”

Now, all that said, if you want to stick to a rule, you can certainly say comparatives are for 2 objects and superlatives are for 3+ objects, but you might encounter debate with people saying that technically a limit (as expressed by a superlative) exists whether or not you have 3+ objects. (Just because you are younger than your one sibling does not mean you are not also the youngest.) I hope this helps a bit, and thanks for reaching out! 🙂

I don’t understand question 8

the arithmetic mean question

the

Hi Dennis,

Happy to help! If 90 is the largest number, no other number can be 90 or higher. We want to find the smallest possible number one of the four can be. Keep in mind, we need the average (mean) to still be 75 as well. So let’s get the biggest possible numbers and our mystery number x:

90 + 89 + 88 + x

We need to divide this by 4 to get our mean:

(90 + 89 + 88 + x)/4 = 75

Now we do some algebra, first by multiplying both sides by 4:

90 + 89 + 88 + x = 300

267 + x = 300

Now subtract 267 to get x on its own:

x = 33

And that’s our answer. 🙂

I believe the answer for question 7 on the new SAT is incorrect. The answer according to Magoosh is C. However that is because they divided 9x^4 + 4t^4 by both sides. This can’t be done because 9x^4 + 4t^4 may be 0, in which case dividing by 9x^4 + 4t^4 would be 0/0, which is undefined. Am I right?

Hi James,

You raised some very important points! I spoke with our experts and we have adjusted the wording of the problem to take care of the correct issue you brought up. 🙂

Hi, I have researched number 7 from the new SAT questions and I can’t seem to get the answer. Could you please explain how to do it?

Hi Kaitlin,

This is certainly a difficult question! The key to understanding this is to use the difference of squares formula. This is a difficult one to type out so I wrote the problem out and you can access it here. I numbered the steps which I will explain:

Step 1: recognize that the left side of the equation can be simplified using the difference of squares and factor it.

Step 2: Once the left side of the equation is factored, we can see that we have the same term on both sides (9x^4+4t^2). Divide both side by this term.

Step 3: The right side of the equation now simplifies to 1 because we have the same term in the numerator and denominator.

Step 4: Recognize that the term on the left side of the equation can be factored again using the difference of squares.

Step 5: Factor the left side of the equation again and we get our answer.

Does that make sense? Please let me know if you have any more questions 🙂