Alternative Solutions: Pratice Test #7

DITW = Do it their way

Trial and Error = pick an answer, play with it, see if it fits what they said in the problem…

Back Door = make up numbers for the variables in the problem, work out an answer that is based on the numbers you made up, then put those made up numbers back into the answer choices, ruling out any that don’t produce a matching answer.

More detailed explanations available in the book…


 1. DITW

 2. There way is to distribute and then group terms, which is fine.  But you can also make up a number for x (say x=5), see what you get, and then see which answer choice gives the same value.  In other words, this is actually an entry-level back door problem.

3. If you don’t feel liked doing algebra, just try each answer choice in both equations. 

 4. DITW

 5. DITW

 6. You don’t need to do nearly as much algebra as the official answer suggests.  You do need to think about linear functions:  if a linear function takes you from your current value down to zero in 12 years, then in 4 years it will take you 1/3 of the way to zero!  So you will still have 2/3 of the initial value.  [Shouldn’t we be encouraging this kind of thinking?]

 7. Another back door:  make up a number for x, get an answer, put that made up number into the answers.

 8. Their way is fine, but they say it the slow way.  Also, like many SAT questions, they intentionally give you the info in an inconvenient order.  Start with the fact that 90% of what he earns is $270.  So without doing algebra, think: what number when reduced by 10% lands you on $270?  I would not be surprised if $300 was your first guess.  Then, since he already earned $80, he needs another $220.  At $10/hr, that’s 22 hours.

 Also, just for the record, you could also do this one by trying each answer choice one at a time.

 9. DITW

 10.  This looks like it is going to be much harder than it is!  But it turns out that when they gave you the roots, they also gave you the factors: (x+1), (x+3) and (x-5).  Only one of those is an answer choice.

 11. Well, if this were in the calculator section, you could use the back door.  Alas, it isn’t, so you do need to know your exponent laws…DITW

 12. Really useful parabola factoid: the vertex lies midway between the zeroes (which they gave you).  Amazingly, that is the way they did it too!  DITW

 13. Seriously?  Yet another back door problem: make up a number for x….

 14. Still no algebra required.  The lower limit is obviously zero – as it is in each answer choice.  The upper limit can be found by trial and error with the answer choices.  For example, is the width  20?   Well,  2.5 times 20 give a length of 50.  This gives a perimeter of 140 and we have not even added in the height of 60 yet! So 20 is too big…I’d try b next.  Can you see why?

 15. OK, start by making up a number for x and getting your value for the expression.  Then, you can either solve for the k value that gives the same value – or use trial and error.  So, if you are keeping score, this problem lets you show off both of the two major algebra-avoidance techniques.  Nice.

 16.  When they give you one algebraic expression  and they ask you for another, you should be on the lookout for simple ways to convert one into the other.  In this case, dividing both sides by 2 does the trick.

 17. DITW (and who says there isn’t geometry on the new SAT!)

 18.  If you only know one thing about radians, know that π radians is 180 degrees.  If you know a second thing about radians, know how to take that fact and scale it up or down.  So if π radians is 180 degrees, 2π radians is 360 degrees, and 4π radians is 720 degrees.

 19. All of my students are laughing at this question.  They know that if you just make a really neat drawing, the answer will be staring right at you, no algebra required.

 20. DITW


 1. DITW

 2. Back door – make up a number…

 3. DITW

 4. DITW – and this is a common question-type on the new SAT.  Don’t assume more than what the problem actually states.

 5. DITW

 6. Stick in x = -3. Then, do trial and error with the answer choices.

 7. DITW

 8. Just read the graph!  At 1.2 on the x axis…

 9. As we said earlier, look to transform what they give you into what they ask for.  In this case, you can add 6 to both sides, then divide by 9.  But if you don’t like that, here’s another way: mess around with numbers until you find an a, x and b that make the equation true.  For example, a=2, x=1, b=1…it works!  Then, use those numbers to answer the question!

 10. DITW

 11. DITW

 12. You should learn the fast way to do percent changes: A 6% increase is the same as multiplying by 1.06.  If you knew that, you could just try each answer choice with ease!  Or take the $53 and divide it by 1.06 to get the initial price.

 13. DITW

 14. The relevant geometry fact is that the angles of a quadrilateral add up to 360 degrees.  Then, while the algebra is pretty easy this time, you could also do trial and error if you prefer.

 15. DITW (Actually, they give two solutions, both fine.)

 16.  If you approach this by making up numbers that fit, you will find that you are done in moments.

 17. And here it is: another example of y = b + mx.  The b is the flat fee and the m is the rate per mile.  This is a very common question on the new SAT.

 18. Once you see what they are asking, this is not that bad.  I’ll translate for you: find the point with the highest y-value.  Then, tell me how far that point is vertically from the line of best fit.

 19. They want you to solve for w in terms of A.  But if that gives you trouble, this is also a classic back door problem.  Make up w and A – if you are lazy, try w = 26, A = 1 – then go see which answer works.

 20. DITW

 21. This is another linear modeling question.  Make sure you know that if y=mx + b (or y=b+mx) the b value is the starting value and the m value is the rate of change.  This time, they are asking for the rate of change, so pick two points and find the slope.

 22. This is really just asking if you know how to find a median.  But be careful!  Many students will go right to the middle of the chart and say that the median is 19.5.  That’s a trap: you have to list the data in ascending or descending order and only then can you go to the middle of the list.

 23. DITW

 24. Well, you can debate whether this is fair or not.  It certainly favors the kids who have had physics.  They would know right away that 72 is the starting height.  You can think of it as the height you get when you plug in t=0.

 25. This one is silly.  If you could make up values for x  and k that go together, you could just see which answer works with those numbers. But wait!  You don’t have to make up the numbers – you can pick them right from the chart.  For example, use  x=9 and k=37.7 and see which answer is true.

 26.  If you see the algebraic method, that’s fine.  But if not, use the values in the chart to make up numbers for p, f and c so that the total of the calories comes out to 180.  It will take you just a few moments to find a possible set.  Then you can stick those numbers into each answer choice.

 27. You should know the quick way to do percent increases and how successive equal percent increases leads to exponential growth.  (See pages 181-183) This is a frequently occurring SAT item. You want to be able to do this about as fast as you can read it.

 28. Make up numbers for a point on the line. A line has only one slope.  So far, we know it has risen 6 and run 3.  So keep doing that to find another point.  You could use (6,12).  For that matter, you can even use (3,6). Then, check out the ratio of t to s using your point.

 29. DITW

 30. DITW

 31. DITW

 32. They are hoping to make you use the slope formula.  And they know that some of you will mess up subtracting fractions (even with a calculator).  But you can also carefully find two point that have integer coordinates and then get rise over run by just counting boxes.  Of course, you first check to make sure the boxes are the same scale – and they are.

 33. I suppose you could do algebra for this one.  But why not take a few attempts at trial and error first.  I’ll get you started: could it be 35 right and 5 wrong?  That would be 70 – 5 = 65 points…too high, try again.  I promise you will land on the right answer in just another guess or two.

 34. DITW

 35. Again, a little trial and error before you wade into the algebra. This time, it took me three tries.  But that’s because I didn’t notice that zero works! Also, you TI89 fans are probably laughing about this one.

 36. DITW

 37. It’s nice to see this old classic. They put some data in a chart and they ask you to find the mean.  (In other versions of this question, they ask for the median.) One way to get the answer is to pull the data out of the chart, listing out the individual results:


 If you are wondering how I knew how many of each to list, you need to examine the chart a little more carefully.  I went by what the chart said!

 Once you have the 20 data points listed, you can easily add them up and divide by 20 to get the average.

 38. DITW




Comments are closed.