# Alternative Solutions: Practice Test #3

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…

PRACTICE TEST #3, SECTION 3 – NO CALCULATOR

1. DITW

2. OK to use algebra, but be lazy about it.  Once again, there is no reason to solve for r.  Look at what they ask for and think about how to get there from where you are starting:  take the original equation, double both sides and then add 3…

3. This is NOT a back door problem.  I mean, it could be, but even after you make up a number for a, you still have to know what it means to raise the number to an exponent of 2/3.  It’s the cube root of the square (or the square of the cube root – either way is fine.)

4. This is supposed to be easy algebra, but in fact it is also easy to mess up!  So before you do algebra, read and think:  30 is twice as big as a number, x.  So x is 15.  And now that you know the value of x, checking to see which equation has it right is pretty straight-forward.

5.  Their way: cross multiply, distribute, solve for x, divide by 5.  But you can also do this by trial and error.  For example, if the answer is A then 10 is  x/5.  So x would have to be 50.  Put 50 into the original equation and you see it does not work.  So A is wrong.  Eventually, you will try C: x/5=2 and that makes x=10.  And if you put x=10 into the original equation, it does work!

6. You COULD solve the system, but this problem is actually a throw-back to the kind of thing that used to be on the SAT all the time.  Try adding the equations exactly as they appear.  Look at what you get and compare it to what they asked for…divide both sides by 5 and you will be done.  (To be fair, they do in fact mention this approach in the official solution.)

7. DITW – This is a very basic application of the remainder theorem, disguised in chart form.  If f(4) = 0, then x-4 is a factor.  See page 227 for more detail.

8. Here are two alternatives to their way…

Quickest: apply the slope formula to the points (c,d) and (0,4).

Back door:  make up a k-value to get a line.  I used y=3x+4.  Draw a neat graph to find a point on that line.  Starting from (0,4) I went over 1 and up 3 to get to (1,7).  Use those coordinates as your c and d values, plugging into each answer choice to see which one matches your line’s slope.

9. If the coefficients have the same ratio, then the lines are parallel (or they are the same line).  So set up the proportion  k/4=3/5, cross multiply and divide.

10.  DITW

11. You might discover the relationships more quickly if you make up numbers that fit.  And if it goes wrong on your first try, just try again – you will be solving the problem.  As you fiddle around with the numbers, eventually you discover that x, z, w and t all have to be the same.  And y and u have to be the same as each other but they don’t have to be the same as the others.

12. DITW – they actually give the quickest solution!  I was sure they would take you down a longer path…

13.  You can DITW.  But you can also just let x=1and see what you get.  I think the equation you land on is not that bad, but if you don’t want to solve it, you can try the answer choices and see which one works.  Be lazy…try 3…nope… -3…yep!

14. DITW or complete the square.

15. While you COULD use made up numbers, with no calculator you are probably better off DITW.

16.  This is a clever little problem.  They could have made it a lot harder than they did…before wading in to the factoring needed, why not play with some numbers?  Try x = 1.  It probably won’t work because there is no way they would make it that easy – oh, wait.  It worked.  Never mind.  Oh and look at that: 2 works as well.  How about that.

17.  DITW – and yes, start by clearing the fractions.

18.  DITW

19.  This is a classic two-variable word problem.  So you can DITW if you are comfortable.  But as we discussed (pages 156-158) there are always alternatives.

If you are patient, you can use trial and error even here in a grid-in section.  For example, say the burgers have 600 calories each.  Then the fries have 550 each.  In that case, 2 burgers and 3 fries have 600 +  600 + 550 + 550 + 550 = 2850 calories.  That’s too high.  Lower your guess and repeat.  I know this takes a while, but if you don’t know the algebra, this is one way around it.

But as I have mentioned, you can often think your way out of these.  The burgers have 50 more calories than the fries.  So if we replace the 2 burgers with  2 orders of fries, that means the 5 orders of fries will have 1600 calories.  (That’s the original 1700 minus the 100 we saved by replacing burgers with fries.)  Now we know that each order of fries must have 1600/5=320 calories.  And the burgers have 50 more = 370.

20.  I’m still hungry from solving the last problem.  So for this one, DITW.

PRACTICE TEST #3, SECTION 4 – WITH CALCULATOR

1. DITW

2. DITW

3. DITW

4. Pick an input n-value…say 3.  When you plug it into the function, you are supposed to get 4 as your answer.  Check the answer choices to find the one that works.  If more than one of them work, check another input-output pair.

5. DITW

6. No doubt, you could add these and combine the terms.  That’s probably best.  But you could also use the back door…make up an x value, work out the two expressions, add them.  Then put your x-value into the answer choices.  I’m just saying that would work too.

7. Trial and error…

8.  Once again, we see y=mx + b and we are asked to show that we know that the m-value is the rate of change…

9. DITW

10. DITW – and don’t worry if you have not had physics yet.  All of the information you need is right there in the problem.

11. DITW

12. DITW

13. Again, don’t be intimidated by the physics.  This is a classic back door problem.  Make up t, v and k, then calculate h.  Then stick h, k and t back into your answer choices to see which one gives you the matching v-value.

14. And another classic back door…say h = 3 hours….that’s 180 minutes…at \$.20 per minute, you are looking at a cost of \$36.  Now put h=3 into each answer…

15.  DITW

16. DITW, but they say it in a more complicated way than we need:  you are just looking for the point on the x-axis where the f- function and the g-function add up to zero. So one of them will be above the axis and the other will below the axis by the same distance. That happens at x= -2.

17.  You can look at the supply equation and ask yourself what happens if p increases by 10.  Or you can actually try it: make up a P value.  I’ll try P=20.  I get S(P)=(1/2)×20+40=50.  Now increase the P value by 10 to P=30.  S(P)=(1/2)×30+40=55.  So now we know what happens when P increases by 10.

18. Trial and error –take an answer, put it into both the supply equation and the demand equation.  If the two equations give the same result, you’ve found the answer.

19.  DITW

20. DITW

21. This is just asking if you know what it means to compound interest and that this leads to exponential growth.  Review pages 181-188.

22. OK, I admit that I used algebra on this one — but less messy algebra than their way.  I let 2y be the sum of the other two numbers.  That may seem like an odd choice, but 50% more than 2y is 3y.  So now the sum, 5y=855.  Divide by 5 to get y=171.  And 3y=513.  Also, you could do this by trial and error, but it isn’t really easier that way.

23. Remember, after SOHCAHTOA, it seems that the most important trig fact is the rule about sines and cosines of complementary angles.  Saying that sin(a)=cos(b) is another way of telling you that a and b add up to 90 degrees.  If you realize that, you can find k either by algebra or by trying their answers one at a time. But wait!  Even if you don’t know any of this…you can try each of k-values in the answer choices to find the a-value and the b-value and then just use your calculator to check if sin(a)=cos(b)!

24. They intend for you to set up and solve equations.  But trial and error works too.  For example, if there are 16 students:  3×16=48…and 48 + the 5 leftover = 53 milliliters.  Then, 53 + 21 = 74.  But 74 divided by 4 is not 16.  So move on to another answer choice until you find the one that works.

25.  DITW

26.  You can use trial and error to make sure you get the same slope going from the origin to each of the two given points.  But really, as long as you are doing trial and error, why not just draw a neat diagram?  The one where all three points line up on the same line will be obvious.

27.  Make up dimensions for the rectangle.  10 x 10 is a lazy choice…  Then, increase the length by 10% to get 11.  Try decreasing the width by the percent given in each answer choice.  If the area of the resulting rectangle is 12% less than the rectangle you started with, you have found your winner.  Note: the advantage of choosing initial values that give an area of 100 is that it makes it easier to recognize when the area has dropped 12% — the new area will be 88.

28.  While it would be great if you just looked at this and recognized the correct expression for exponential decay, you can also use the back door.  Say we let t = 20.  The population will drop 10 percent from that original 50,000 to 45,000.  And when you put t=20 into each answer, only one comes out to 45,000.

29.  This is a tough one…but look where you are in the section.  If you knew the numbers in the chart, you could take the number of right-handed females and divide it by the total number of right-handed students.  But they make you work to get the numbers in the chart.  To DITW, you set up a system of 2 equations and two unknowns.  And once again, there are alternatives to that approach:

You could do trial and error, playing with numbers that add up to 18.  Multiply the females by 5 and the males by 9.  If the numbers you get add up to 122, you are all set.

Then there is that think-about it approach.  Suppose the group was all female.  There would be 18 female lefties and 18×5=90 righties.  But we are supposed to have 122 righties.  We are missing 32.  So we need to replace some of the female lefties with male lefties.  Since there are 9 times as many male righties as lefties (instead of 5 times as many) every time we replace a female with a male, we gain 4 more righties.  So to make up that deficit of 32, we replace 8 female lefties with male lefties, leaving 10 female.  10×5 = 50, 8×9=72 and 50 + 72 = 122.

I know this method is tricky.  You would have to practice it to do it with confidence.  But it DOES work!  And if you work through all of the released practice tests, you will certainly find plenty of chances to practice.

30. DITW

31.  Just play with numbers: \$3 + 2 times however many student tickets must be at least \$11 and not more than \$14.  Try numbers until you find one that works.

32. DITW

33. DITW

34. In the “Advanced Basics” section (page 207) I told you that the harder material would have easier questions.  This one is asking you if you know that a full circle is 2pi radians.  You find the fraction we are talking about by dividing:  (5pi/4)/(2pi) and you get 5/8 or .625.

35. DITW

36. DITW – or graph the boundary lines on your calculator and then trace along to find the y-value where the lines meet.

37. DITW  – But once again, stay calm and read.  You are not supposed to already know this law.

38. DITW

OK, this time we were able to find alternative solutions to about 29 out of the 58 problems.