Alternative Solutions: Practice Test #4

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. Try x=0 and see what happens.

 2. This is a Case of the Missing Constant.  See page 126.

 3. Let’s use trial and error: if y=2 then x has to be 12.  Put them into the second equation and see what happens.

 4. Classic back door – make up a number for x…I used x=2. So -3x = -6 and f(-6)=17.  Now put x=2 into each answer choice.

 5. Another back door.  Try x=0…see what you get and see which answers match. Then try x=1…it’s magic.

 6. And yet another back door.  I made b=7 and a=10 because they were the laziest numbers I could find that fit the equation.  Then it’s off to the answer choices…

 7. DITW

 8. DITW

 9. Use trial and error to show that 6 works and 3 does not.

 10.  Hmm…if your arithmetic is good enough, you can use trial and error.  But otherwise, go ahead and DITW.

 11.  One way to avoid the algebra this time is to make a really neat graph.  You have to solve the first equation for y and you do have to know one of the ways for finding the vertex of a parabola.   But your reward is that you can look at your picture and see that the line crosses the parabola twice.  You may think that my way is no fun.  Look at their way too and see what you prefer.

 12. Classic back door.  I started with x=5…

 13. DITW

 14. DITW – across the country, students are now going to be really motivated to learn how to divide complex numbers! 

 15.  You can back door this one too!  Make up a k and a p…then you can use the quadratic formula or you can complete the square.  But it’s easier with concrete numbers.

 16. DITW

 17.  You can make up numbers for the lengths of the triangle.  But the sine of x has to be .6.  So I used 6 for the vertical leg, 10 for the hypotenuse.  Then, Pythagoras says the horizontal leg is 8 – it’s a 6-8-10 right triangle.  But that makes the cosine of y 6/10 or .6 as well.  THEY REALLY WANT YOU TO KNOW THE COMPLEMENT RULE FOR SINES AND COSINES!

 18.  This is obnoxious.  In general, there is no way to factor a cubic equation.  But there must be a way to factor THIS one or the problem can’t be solved.  Group the first two terms together and the second two together.  You can factor out (x-5) from each of them.  So the whole thing becomes:

(x2 + 2)(x – 5)=0.  The solution is x=5.

 19. DITW

 20. I found it helpful to draw a diagram: a number line labeled below with the numbers 50, 60, 70, and 80.  I wrote a -5 above the 50 and a -80 above the 80.  So what I had looked like this:


50        60        70        80

 Then you can see that in 3 10-kilometer jumps, the temperature went down by 75 degrees so it must be 25 degrees per 10-kilometer jump.



 1. Well, this is easy either way, but you COULD use trial and error if you felt like it.

 2. And this could be done by the back door. But it is quicker if you recognize  y = b +mx when you see it.

 3. DITW

 4. First of all, this is a probability scale-up.  And second of all, you should notice that the percentage is close to 25% or ¼.  Only one answer choice is in the neighborhood of ¼ of 225.

 5. Algebra or trial and error…

 6. Algebra, trial and error or just play around with numbers until you find a pair where one is 11 more than the other and they add up to 59.

 7. Just read the chart to get 4/50.  Also, I believe this question has a minor math vocabulary error.  In regular English usage, the word “proportion”  is sometimes used interchangeably with the word “fraction”.  But mathematically, a proportion is a statement that two (or more) fractions are equivalent.  I wish they had used the word “fraction” in this question.  Nitpicking, I know, and yet…

 8. Draw it! You will see that the only way to avoid quadrant I is to slope down hill.

 9. DITW

10. DITW (just read the graph)

 11. You don’t actually have to calculate the ratios or even estimate them.  The ratio is also the rise over run, starting from the origin.  So imagine lines (or actually draw them if you like) starting from the origin and going to each of the four labeled points.  Pick the line with the steepest slope.

 12. DITW

 13. Classic example of exponential growth: population increases by the same factor (x10 in this case) in equal time increments (5 years in this case).

 14. DITW

 15. Exponential decay is one of the graphs you want to recognize just by looking at it.  (Exponential growth is another. See #20 below)  See pages 181-188.

 16. To start, this is yet another example where the key is to understand the form y = b + mx.  In this case, the b (or starting amount) is the material cost and the m is the rate of change.  But as the given formula tells you, you have to add the two daily rental costs to get that rate of change. 

So at store A you get y = 750 + 80x and at store B you gety = 600 + 105x.

But then what?  You could set up and solve an inequality (DITW) or you can just try numbers.  If you try x = 7 you will see that it is already too many days. 

 17. See above – the slope is the daily cost.

 18. DITW

 19. Trial and error: choice A works. Choice B doesn’t.  So A is the smallest…but you can also do some very quick algebra.  Remember that you don’t have to solve for the variable.  As the official key does point out, you can just add 4 to both sides.

 20. See the note for #15 above

 21. DITW

 22. It drops from 2.25 to 2, a decrease of .25.  And that’s a little more than 10% of the starting number.  (10% is one that you want to know how to do without a calculator – you just divide by 10.)

 23. You are not expected to know how to calculate standard deviation.  They just want you to eyeball the data and notice that for City A, the data is clustered closer to the mean and for City B there are more data points out on the margins. 

 24. DITW

 25. OK, I admit that I never would have come up with their solution to this one! I approached this in a completely different way.  Here goes:

If the polynomial is divisible by (2x+3) then -1.5 must be a root.  That means that when you put -1.5 into the function, you get zero out.  So I put -1.5 into the two given functions…



Then look at those numbers: if you triple the second one and add it to the first one, you get zero!

And the polynomial in choice B does exactly that: p(x)=f(x) + 3g(x)

 26. Make up numbers and draw it on a number line.  You will see that y has to be bigger than x but that x can be positive or negative…it just has to stay between y and –y.

 27. DITW – and it’s y = b + mx again!

 28. While foiling is easy and completing the square is fun, you do have another option: graph the given function on your calculator and then use your calculator to find the minimum value – you should get -25.  That only appears in choice D.  But wait – if they had been nastier, they could have had two answers with -25 in them but one of them not even the right parabola!  If that happens, just graph the answer choices as well to see which one gives the same graph as the original function.  But like I said…foiling is easy and completing the square is fun.

 29.  A classic back door!  Make up a value for m, work out the answer, plug your m-value back into the answer choices.

 30.  This one is supposed to be hard.  But I think the hardest part is figuring out what they are asking you!  Try reading it one more time…ok, now I will translate.  The question is essentially asking this: At what y-value can we draw a horizontal line that will cross this graph three times?

Only one answer works.

 31. DITW – and note: you can think of this one as y = b + mx too.

 32. Make up an x, use it to find P.  Add 1 to your x, find P again.  You will see that P goes up by ½ or .5.  Or re-write the equation as P =(1/2) x + 110.  Then you know that 110 is the starting value and ½ is the rate of change.

 33. DITW (but if you have taken chemistry, this would be a good time to use factor-label unit conversion techniques.)

 34. If 3/5 of the bats are male, then 2/5 are female.  So the male: female ration is 3:2.  And we have 260 female bats.  Set up the proportion, cross multiply and divide and you will see that we need 390 male bats…but we already have 240, so subtract to get 150.

 35. Make up numbers for n and v.  Calculate q.  Then multiply to get 1.5v and recalculate q.  No matter what numbers you make up, the ratio of your new q to your old q will be 2.25.

By the way, if you have taken physics, you have probably done problems just like this but involving kinetic energy.  That formula, K = ½ mv2 works the same way…

 36. The full circumference is 20pi.  You have to multiply that by the fraction x/360  and have the answer come out between 5 and 6.  Why not use trial and error? 

 37.  The quick way to find a 28 percent decrease (see page 74) is to multiply by .72.

 38.  And the quick way to find successive percent decreases: $360 × .72 × .72 × .72 or if you prefer: $360 × (.72)3

 And this time, we found alternative solutions to about 40 out of the 58!

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