Alternative Solutions: Practice Test #6

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. Linear modeling again!  When y=mx + b (or b + mx) the b is the starting value.  In this case, that’s the service fee.
  2. DITW
  3. DITW
  4. You don’t have to FOIL this out all the way.  Just notice that the x^2 term has to be p^2…
  5. Yes, you should be able to match a graph and its equation.  But if you can’t…pick any x value,,,say x=2.  Find the y-value that goes with that x value.  (You get y=-1) .  Then, check which graph contains the point you have just found, (2,-1).
  6. DITW
  7. If you need to, use the back door: make up numbers, then put your numbers into the answer choices to see which one works.
  8. No real trick to this one once you understand what they are asking.   In other words, when you add entry in the second column to the one in third column,  do you get the value in the first column?
  9. Just try the answers…
  10. DITW
  11. Know your basic parabola facts and your shifts…see pages 165-166
  12. They want you to do polynomial division.  But seriously, this one cries out for the back door.  Try letting x=1.  Work out the answer.  Then put x=1 into each answer choice.  It’s much easier my way.
  13. This is actually pretty subtle.  You could try each value and see if it factors (or see if the discriminant is negative). That’s kind of a pain.  There is another way to get the answer but I admit it is sneaky.  Consider the quadratic function y=2x^2-4x-t.  It opens upward.  So if it has no real solutions, if you shift it upward it will still have no real solutions.  But the problem can only have one answer!  So it has to be A.  (If this doesn’t make sense to you, post a question and I will go into it in more detail.  Also, I predict that very few test-takers will get this one right.)
  14. DITW
  15. Try numbers!  I used a=1 and b=4 so I got 9.  But oh, no!  A and D both come out to 9.  No worries – just do it again with new numbers. I tried a=1, b=6 and got 16.  This time, only D worked.
  16. Other than playing around with numbers, how else are you going to do this?
  17. DITW
  18. This is a Pythagorean Theorem question.  But first, when you see this shape you should automatically be on the alert for similar triangles and ratios…
  19. DITW – even I admit that algebra is the fastest way.  But if you are comfortable with the idea of weighted averages and inverse proportions, there is another way to think about this one.

The goal is a 15% solution. Since that is closer to the 10% stuff than it is to the 25% stuff, I expect to need less of the 25% solution than the 3 liters I already have.  In fact, if you look at how far each solution is from the goa, the 10% stuff is 5% away, the 25% stuff is 10% away.  Like weights balancing on a see-saw, the farther away, the greater the effect.  In fact, it is an inverse proportion.  So I know I’ll need half as much of the 25% solution.

OK, I admit that if I were taking this test, I would never have come up with that method on time.  So in this case, do the algebra if you can.  If not, take a guess and move on.

  1. DITW



  1. Combine like terms if you want.  Or try a number!
  2. DITW
  3. You are looking for shallow, then level, then steep.  Slope is rate of change.  Also, the snow was always getting deeper so C and D are silly.
  4. DITW
  5. They are just checking if you know that you can divide both sides of an inequality, in this case by 3.
  6. DITW
  7. It seems from the released tests that the SAT would really like you to know that to make accurate predictions, a survey must involve RA NDOM sampling.
  8. If this is your sixth practice test, then you have seen this before.  But read carefully so you know if they are asking about everybody or just a subset.  This time, they only want to know about vanilla lovers.
  9. DITW
  10. By algebra, if x is the second voyage, then x + x + 43 = 1003.  But why not just try the answer choices?
  11. DITW
  12. You don’t have to calculate the average growth rates to see which is greatest.  Average growth rates are also slopes of the segments that connect the two points.  So draw them and then pick the steepest one.
  13. And here we have yet another linear model: y= mx + b but with some of the letters changed.  The a value is still the rate of change, or in this case, the amount of daily growth.
  14. Pick a value from the graph or chart . For instance, on day 28 the height was 98.  Try t = 28 in each answer choice.  You will see that only one answer is even close.
  15. Really?  The very next question?  I tried x = 4…
  16. These are similar triangles.  It might help you to see that better if you redraw the one on the left, turning it and flipping it so that it is oriented the same way as its neighbor…
  17. Make up a set of numbers so that 2h + d = 25.  Then go to the answers…by now you should know that I call that “the back door play”.  In basketball, the back door play gets you an easy layup.
  18. DITW
  19. Trial and error!  For example, if d=7.2 inches then 2h = 25-7.2=17.8 inches so h=8.9 inches.  No good.  If it had fallen in the right range, we would then check if the number of steps came out odd.  Keep going until you find the one that works…
  20. DITW
  21. DITW
  22. Many people find it helpful to list out the data:

10,10,10,10, 11,11,11,11,12,13,15,15,15,17,20,21,21,27,31,34,55

Once you do that, finding the median is just a matter of finding the middle number.  Also, once you have done a few this way, you won’t need to any more.  You can just think: there are 21 states on the list so 10 of them will be greater than or equal to the median, 10 will be less than or equal to the median and the 11th one will be the median.  Start at the top of the chart and keep a running tally of how many states you have accounted for.  By the fourth row, you will have counted 10 states.  The median will be in the next row.

  1. DITW
  2. You can DITW or you can use trial and error with each answer choice.
  3. Linear functions are “evenly spaced”.  So to start,  figure out what goes between the first row and the second row.  Then you will know how much you have to go up by to get to f(3).
  4. DITW
  5. DITW
  6. This is yet another example where you are better off finding your own answers first and then checking which equation matches them.  Just draw a number line and you can easily see that the two points they are talking about are -1 and -7.  So check each answer to see which one works with both of those values.
  7. Let’s make up numbers again.  For example, if t= 25, then s = 80 and the average speed is 80/25=3.2 inches per second.  Now, it’s off to the answer choices!
  8. Hmm..if we drew a curve through these points it would be a downward parabola.  So you know it is B or D.  When x=4, it looks like y= 800 or so.  Plug in x=4 into B and D.  It won’t be a close call.
  9. Just try numbers…easy numbers.
  10. DITW
  11. DITW…remember the formulas are given in the front of the section
  12. If you don’t want to do the algebra, you can graph both functions on a graphing calculator and then trace along until you come to the intersection point.
  13. DITW
  14. If the average is .1 greater and there are 6 items then the total is .6 greater…
  15. Make a little chart, fill in the 480 dollars at the bottom,  then work your way back up, cutting the value in half each time.
  16. Here’s a hint: the 6 students represent 15% of the committee.  You can set up ratios to find the rest of the committee members.  For example, to find the administrators:  6/15 = x/25


Alternative Solutions: Practice Test #5

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. While it would be best if you can immediately make the connection between an equation in slope-intercept form and its graph, you do have a backup plan here: pick some point on the graph, say (2,3) and check which answer choice fits. 

 2. Look at the central angle to see what fraction of the circle we are talking about.

 3. If you are quick about it, factoring is fine.  But trial and error with the answer choices works pretty quickly too.

 4. DITW

 5. Trial and error

 6. OK, algebra is pretty quick.  But this is also a classic “back door” problem.  Make up a value for ‘a’, use it to get a^2 – 1 and a + 1 and their sum.  Then put that same ‘a’ value into each answer choice…

 7. DITW

 8. This is an example (yet another) of y=mx+b where they want to see if you know what the y-intercept means…it’s the “starting value” or in this case, the speed at 0 degrees.

 9. This LOOKS like you are going to have to do algebra – substitution and whatnot.  But before you jump in and do that, it’s worth playing around with some easy numbers to see if you can find a solution that way…I won’t ruin the fun by giving away numbers that work.  I’ll just say that you will be surprised to find them with ease!

 10.  Again, if you don’t want to do algebra, use the back door: make up a and b, calculate z, y and 4z+8y.  Then go check the answers to find the match.

 11. DITW

 12. DITW

 13. Make up numbers!  Say n=5…that makes t = 7.  For an additional cup, n=6 and now t=8. So one more cup meant one more tea bag.

 14.  You should recognize a graph of an exponential function (see page 182 ) and you should know what happens when you shift it up 1 unit (See page 129) and then what happens when you flip it vertically (see page 130).  Update 3/18/17  — even if you don’t know what the graph should look like, you know that the function is f(x)=-(2^x +1).  You can make up an x value and see what why value you get. So for instance, if x=0 you get y=-2.  Then look at the graphs to see which one has (0,-2) as a point on the graph.

 15.  I don’t like this kind of question.  It attempts to force you not just to think algebraically but to do it in precisely the way they expect you to.  But there is an alternative:  solve for m using any method you can come up with.  Then plug your m into each answer.  Only one equation will be true.  Here’s one way to do it:  gas costs $4 per gallon.  So to save $5, I need to use a little more than 1 fewer gallon per week – how much more?  The “extra” dollar is ¼ of $4 so the extra amount of gas  is ¼ of a gallon.  So I need to use 1.25 gallons fewer gallons.  (We could have set up a ratio too.)  At 25 mpg, that makes m= 25 times 1.25 = 31.25 miles.  Now put that value into each answer…

16.  DITW

 17. DITW

 18. This time, with the ugly fraction, I could not find numbers just by playing…oh, well – sometimes you have to DITW

 19.  Well, they WANT you to put the fractions over common denominators and clean up.  That will work.  But there is a neat alternative:  they said the expressions are equivalent for ANY value of x other than -2.  So pick a lazy value!  I went with x=0…a little bit of algebra (easier than what they wanted you to do) leads to a = 2.  If you don’t believe me, try it yourself with some other x value.  It will still lead you to a=2.

20.  DITW – and I admit, I am not sure how this question is worthy of being the last one in a section.


 1. DITW

 2. Just pick a row of the chart and check which answer fits. 

Pro tip: don’t start with the first row – they cleverly make it so that multiple answers work.  Try the second or third row…

 3. Simple ratio problem, but don’t forget the units conversion.

 4. Well, the algebra is literally one step: divide both sides by 3.  That’s faster than making up numbers.  In fact, to make up numbers, you would probably end up doing that same algebra anyway.

 5. DITW

 6. No need to set up a system of equations!  You can just try each answer one at a time.  But there is also a “think-about-it” method (see page 159 ). Suppose all she bought were magazines. That would cost $11, leaving $9 unaccounted for.  Novels cost $3 MORE than magazines.  So how many must she have purchased?

 7. No kidding, walk into the SAT knowing about models for linear growth (see pages 146-147).

 8. They want you to FOIL the term on the left and then regroup, combining like terms.  If you are comfortable with that, go ahead.  But you do have a calculator and this is actually a classic back door problem.  Make up an x value and take it from there…

 9.  If you are not sure how to convert the 2 kilometers back to miles, you could also use trial and error, multiplying each answer choice by 1.6 to see when you get closest to 2 miles.

 10. You could make up numbers for this one too…

 11. DITW

 12. Algebra is easiest this time: you just add the equations.  The y terms drop out and you are done!

 13.  Trial and error! But another pro tip: check the easier inequality first.  You will see that just the one inequality rules out all but one of the answer choices.

 14. DITW – and if you have already worked through tests 1 – 4, this should look like a rerun.

 15. DITW

 16. DITW

 17. They expect you to draw the line of best fit and then find its slope.  That will work, but there is an alternative:  pick a diameter, say 14.  Read the age: it’s about 105 years.  Now go to the chart with the growth factors and check which one has a factor where 14 times that factor is in the neighborhood of 105.  There’s only one that is even close…

 18. DITW

19.  So many ways to do this…fastest is to recognize that the outer triangle is equilateral and that the little triangles cut it in half.  (Often very useful to see a 30-60-90 triangle as half of an equilateral triangle.)

 20.  An odd little question.  But you can rule out A and C – they stay constant.  And you can rule out B – the distance from the starting point increases steadily.

 21.  Just make up numbers and see what happens.  But be careful to make a negative and b positive.  And if you are shaky on your arithmetic with negative numbers, let your calculator take care of it.

 22. DITW – but if you don’t feel like doing percents, notice that the answer choices are pretty widely spaced and that 34.6 % is a little more than a third.  But they want the kids with FEWER than 2 siblings, so roughly 2/3 of the class.  There are 1800 classes of that size.  So you could multiply 1800 by 26 and then take 2/3 of that answer.  That will give you a close enough result.  (But you wouldn’t use this rough estimate if the answer choices were tightly spaced.

 23.  Trial and error – pick a purchase price (from column one, in thousands) and see which formula gives the right answer…

 24.  It helps if you know the quick way to find a 40% discount followed by a 20% discount: multiply by .6 and then by .8.  Then you have to notice that the purchase price is the price AFTER the discounts.  So to “undo” these discounts, take $140000 and divide by .6 and by .8 – or start from each answer choice and take the discounts, picking the answer that leads you to $140,000.

 25.  Again, if we can find the answer our own way, we can check which of their answers agrees with ours.  Since 20% of 150 is 30, the first batch of people gave us 6 more than we need.  So the second batch can give us 6 less than we need, which is 24.  So 24 is the lower bound for p.  Stick that into each answer…only d makes sense.

 26.  DITW, though making up a number for a might help you to see what’s going on quicker.

 27. DITW

 28.  You don’t need to write equations but you do need to understand slopes.  The function f has slope = ½ (as you can see directly from the graph) so the function g has slope = ½ × 4 = 2.  You have been given the point (0,-4).  You could actually draw a picture, neatly counting as you rise 2, run 1 all the way to (9, 14).  Or you could notice that rise = slope times run so you have to rise 18 units.  Starting from y=-4, that takes you to y=14.

 29. DITW

 30.  You can think of this as the difference of two squares and then factor it.  But you can also just make up numbers for x and a, calculate y and then go check each answer choice.  In other words, this is another classic back door problem.

 31. DITW – it’s a classic ratio problem.

 32. DITW

 33. DITW

 34. Another “Case of the Missing Constant (see page 126).  They tell us that (2,5) is a point on the graph.  That means that when you use 2 as the x-value, the function output is a 5….

 35.  Jump in and play with numbers.  Make the length 5 more than the width and keep trying until you find a pair that multiplies out to 104…it won’t take long.

 36.  This is a very old trick that still shows up on the SAT.  Draw the segment from A to P.  You have just made two little triangles, each of which is ISOCELES.  How do we know that?  It’s because P is the center of the circle!  All of the radii are equal.  From there, finding the angles is much easier.

 37. DITW

 38. DITW