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





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