ALTERNATIVE SOLUTIONS: TEST #8

For more than half of the problems on this test, there is another way to solve the problem, a way that is sneakier, more creative, a little subversive.  It’s good to have more than one way to attack a problem, so read on and learn to play.

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 my book

PRACTICE TEST #8, SECTION 3: NO CALCULATOR

1. As algebra goes, this is not that bad. But you can also just try the answers. Lazy kids try b first  but no luck. Then they try d and their laziness is rewarded.
1. Well, you should REALLY know how to find a linear equation from a graph. So I guess I am saying DITW. But you can also pick a point on the graph,  say (1,2)  and then see which equation works. If you are unlucky and two answers work (as would happen if you chose m=2, d=4) then pick another point and try again.
1. If you can do the algebra to solve for P, go ahead. But this is also a classic back door problem.  Make up #s that fit, get your answer and then plug your #s into the answer choices.
1. DITW
1. And another back door!  Draw a rectangle, pick a value for w, make the length 6 more than that, find the perimeter and then stick your w number into each answer.
1. DITW
1. Why do algebra?  Try the numbers! 5 works,  -1 does not.  No need to check 0 (but it doesn’t work either).
1. And yet another back door! Try x = 4, see what you get.  Then plug x=4 into each answer.
1. You do have to know that this is a circle centered at (6,-5).  Draw the picture, fill in the center and the endpoint that they gave you: P(10,-5).  Then look at the answers and plot them on your graph.  There will only be one that could be anywhere close to the other end of that diameter.
1. The SAT LOVES this kind of problem.  The official solution shows the algebra method.  They also discuss a slick think-around.  But they forget to mention that you can also use trial and error. For example, say there were 20 2-person tents (B). That takes care of 40 people.  The other 40 of the 60 tents would be 4-person and would hold 160.  That adds up to 200 people, but we need 202.  I bet choice C will work. Try it!
1. If I plug in x=-2, the graph tells me to expect an output around almost -20…try the answer choices.
1. Make up an a and a b that fit: for example, a=1, b=4 — in other words, this is yet another back door problem!
1. While I still think you should know how to write a linear function, note that the right answer has to give an output of 1.9 when the input is 13. So check the answers to see which one does what it’s supposed to.
1. DITW
1. DITW
1. Before doing algebra, a lazy kid checks 1,2 3 and maybe even -1, -2 and -3. Oh, look: 3 works
1. DITW
1. DITW (but they say it funny: just add the equations, lined up as they are.  You get 4y=6, so y=1.5
1. They LOVE this question too: y=ax+b and a is the rate of change, b is the initial value.  If you know that, this is kind of easy to be problem 19.
1. DITW

PRACTICE TEST #8, SECTION 4: WITH CALCULATOR

1. Back door  say c=5. That’s \$10. Now put c=5 into the answers…
1. DITW
1. This is a classic ratio problem: set up thing1/ thing 2 = other thing 1/other thing 2.

4. While you COULD plug x=19 into the equation they give you, you could also estimate directly from the graph.  You have to extend the line a little bit but the answer is clearly a little more than 50 and clearly less than 55.

1. DITW
1. Since you don’t actually have to solve the system, it’s better to DITW.
1. Y=mx+b and the slope is 3.
1. If you see that you can multiply both sides by x+1 to get (x+1)^2, then you take the square root of both sides. That’s their way.  You could also do trial and error.  Or, since this is the calculator section, if you have a ti89 or an nspire, you can let the calculator do the algebra for you.
1. You have a calculator!  They give you a formula.  And they give you answers! Try them one at a time.  When you get 473, stop.
1. An odd and interesting question.  But DITW.
1. Division, or just trial and error…
1. DITW
1. This will be an easy back door play as long as you know what a negative exponent means and what ½ as an exponent means. When it says a ^(-1/2) that means “one over the square root of a”.  So if you make a=25, x will be 1/5.  Now go to the answer choices, plug in 1/5 and see which comes out as 25.
1. It will be undefined when the denominator = 0. But you can skip the algebra and just try the answer choices.
1. DITW
1. DITW  but again I have to complain about the wording.  They say “what proportion” when they should really say “what fraction”.  I know I am nitpicking, but still…
1. If you understand their way that’s fine. But many people find it easier to understand if they list out all the data:

18,18,18,18,18,18, 19, 19, 19, 19, 19, 20,20,20,20,20, 21,21,22,23,30

Notice how much easier it is now to find mean, median and mode.

1. DITW
1. DITW
1. DITW
1. Yet another question testing your ability to interpret y=mx+b. The m is the slope which is also the rate of change.
1. DITW
1. I am sure that many students who think they are too advanced to bother making up numbers will actually get this wrong (by putting b). Say t=3. Calculate 1800(1.02)^3.

In 3 years, there are  q=12 quarters.  Put q=12 into each answer choice.  You will see that it is a and not b that’s the answer you were looking for.

24. Well, DITW…but also: you get this wrong by assuming things that the problem never said.  In the math section, as in the reading section, that will burn you every time.

1. Yes, you could find the slope and intercept and then write the equation…but here’s something helpful to notice: when you plug t = 10 into the right answer, you better get 846 back as your answer.  Start calculating!
1. DITW
1. DITW and note that this is a righteously challenging question, as it should be for its location in the section.
1. You do not need to know how to calculate standard deviations. You just need an intuitive feeling that if the data is clustered around the mean, the std. deviation is low and if it is spread out, the std deviation is higher. Then, you can just eyeball this one.
1. First of all, eliminate c and d — they are exponential but the problem says “constant rate” so you know we need linear graphs. And after 20 minutes, 30% of the paper is gone so 70% remains or 3500 sheets. So of the two answers, pick the one that comes out to 3500 when you plug in m = 20.
1. DITW and it’s a good problem!
1. Remember: early grid-in should be easy.  Don’t be psyched out by the chemistry.  You have 51 thingies.  They each contain two smaller thingies of one kind and one of a second kind.  So overall, you have 102 of the first kind and 51 of the second kind.
1. Stick in x=1. Algebra or trial and error to find what works.
1. If you multiply both sides of an equation by a constant, it’s an equivalent equation with the same solutions and the same graph!  It’s easy to see that the constant they multiplied by this time was 3. So 10 x 3 = 30.
1. DITW
1. Yes, you can set the two equations equal and do the algebra.  Or you can let your ti89 do it.  Or with any graphing calculator, you can graph both equations and then trace along until you find the intersection.  But my laziest students draw the g function’s graph by hand! You know it starts at (0,10) and it has a slope of -1. So go to (0,10). Then,go down 1, over 1, down 1, over 1 (watching the scale of the graph). In two steps, you will land on the first intersection point.
1. This is exactly the kind of question that rewards thorough SAT prep. It looks very hard.  But then you notice that the angles they ask about are complementary. And you know the rule for complementary angles: sine of one = cosine of the other. So they subtract to get zero.  If you take 5 practice tests, you will see this concept tested 3 times.
1. Just read the chart and be careful: for the first sample, you multiply by 10, for the second you multiply by 8 and then you subtract the two results. (Which is, to be fair, DITW)
1. And this is DITW as well.

OK, I count more than 30 out of the 58 that have at least one alternative solution!