# Alternative Solutions: Practice Test #1

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 #1, SECTION 3 – NO CALCULATOR

1. You can just try the answer choices one at a time…

2. DITW

3. Back door!  Make up your own m and p

4. Recognize that this is y = b + mx…the b is the starting value.

5. This looks like a back door problem and I guess you could do that, but this time, I agree that it is actually much faster to do the algebra: distribute the minus sign and group the terms…so DITW

6. Either recognize that y = b + mx, and m is the rate of change…or to see how much the boy grows each year, just make up a number for a, plug it in and then plug in a second number, one year older.  For example, try a=10 and then a=11.  The difference is how much he grew in a year.

7. DITW

8. If you see the algebra, it’s quick.  But making up numbers is even quicker. I used a=4, and b=2…but any numbers that fit will do.

9.  Use trial and error!  But be lazy: check the easier equation first.  Only one choice works.

10.  This is an example of “the case of the missing constant” (see page 126). But if you do notice that the function is an even function, you can go right to the answer: g(4) = g(-4) = 8.  So “their” way is quicker but requires more insight.

11. Trial and error would work, but faster to DITW

12. Draw a neat diagram with a line through the origin that rises one and runs seven.  It will be obvious that only one of the given answers could possibly lie on that line.

13. This is a gift to students who know the back door!  Say x = 6…work out the fraction, common denominator and all that…then put  x = 6 into each answer.

14.  Faster if you know the laws of exponents.  But if you are shaky…back door!

I tried x=4, y=0.  Took a little while to show that 212 = 84, but it works.

15. This is a tricky one whether you DITW or mine…First notice that if you start to foil out the expression, you find that ab=15.  And they gave you a + b = 8.  You could solve that system algebraically, but I prefer to look at the equations and try to make up numbers.  Seriously, the first numbers I tried were 5 and 3!  (It can be a=5, b=3 or the other way around.)  If you put a=5 and b=3 into the original expression and then go ahead and foil it all the way, you get 41.  You don’t need the other possible value to know that the answer is D.

16. Algebra or just a quick trial and error

17. These are two similar triangles – it’s one of the classic examples.  Fill in the lengths they give you and you will see that the bigger triangle’s sides are twice the length of the smaller…

18. DITW

19.  The sine of an angle is also the cosine of the complement.  So the answer is 4/5!  And by the way, if you have a hard time remembering that rule, it may help to think of “cosine” as short for “complement’s sine”!  Works for tangent and cotangent as well…

20.  DITW

PRACTICE TEST #1, SECTION 4 – WITH CALCULATOR

1. DITW

2. Like a “missing constant” problem, you could plug in their values for x and y, solve for k and then plug in the new x value.  Or you could recognize that if y=kx, then x and y are directly proportional – set up ratios, cross multiply and divide.

3. Know the rules about parallel lines and transversals: pairs of angles that are not congruent are supplementary.  180 – 35 = 145.  (See page 95 for a review)

4. Algebra is quick here, but this can also be solved with trial and error:  Be lazy though – start with the choices that are easily divisible by 8.  D does not work, but check out C:  If 8x = 16, then x = 2…and 16 + 4x comes out to 24 which is 10 more than 14.  How about that.

5. DITW – what else could it be?  There is only one data set with a clear downward trend.

6. DITW

7. This is not really algebra.  Just add up the heights of the bars on the graph.  It adds up to 27.5.  But they tell you that the total is actually 27,500.  So the numbers on the side of the graph must be “thousands.”  The “official” solution says the exact same thing but makes it look more algebraic.

8. You can do this by trial and error:  carefully try 0, 1 and 2.  None of them work.  So it has to be D.  You can also do this by thinking:  the smallest value you can have for the part of the expression with absolute value bars is zero.  Then they add 1 to that value.  So it can never add up to zero.

9. Back door!  Say t = 10  Plug in and get a=1062.8.  Put that into each answer – only A comes out to 10.

10.  Trial and Error – put each answer into the equation they gave you…

11. Trial and Error.

12.  Really helps if you just list out all of the data:  3,3,5,5,5,5,6,7,7,9,9,9 – now finding the average is just a matter of adding them up and then dividing.  And by the way, had they asked for the median, a list like this would make that very easy too.

13. DITW

14. DITW

15. This is another example of y = b + mx.  The only difference is that the vertical axis is labeled with a C instead of the customary y.  But the intercept is still the starting amount…

16.  …and the equation is still y = starting amount + slope  × run.  That the slope is 3 can be determined just by counting rise over run on the graph.  But don’t just count boxes – look at the scale of the graph!  Otherwise, you get ¾ which is wrong.

17.  They are just asking you to find the INPUT (x-value) that gives the lowest OUTPUT (y-value).

18.  DITW

19.  Yes, you could do algebra here.  Trial and Error also works:  for example, if they sold 77 salads, that would be 77 × \$6.50 = \$500.50.  So they would need to take in another \$336 from drinks.  At \$2 each, that means 168 drinks.  But 168 + 77 is more than the 209 items they sold.  Now you try it with choice B.  You are in for a happy surprise.

20. Classic back door.  And like most back door problems involving percents, it’s a good choice to make the starting value \$100…The 20% discount knocks the price down to \$80.  With 8% sales tax, what you pay is p = \$86.40.  Put that value into each answer choice…wait for the magic.

21. DITW

22. DITW

23. DITW – annoying and kind of tedious, but what are you gonna do?  Read the chart, find the ratios, pick the one closest to human resources.

24.  OK, this one makes me laugh.  You are supposed to use the center and endpoint of the radius to find the length of the radius and then write the equation of the circle in standard form.  But there is a lazy work-around:  that endpoint they give you does have to be a point on the circle.  So you can just put that point’s x and y values into the equations given in each answer choice.  Only one choice works.

25.  Trial and error – plug in the choices, pick the one that gives a height closest to zero. (When you are on the ground, your height above the ground is zero.  That’s the only physics knowledge you needed to answer this.)

26.  If you don’t want to do the algebra, go to each answer choice, find 20% and add it on.  When you get to choice B, you will see that you end up with the correct number of pears.

27.  This is like a scale-up probability problem.  The sample regions have an area of 1 square meter.  The full region is 100 square meters.  So you just need the answer that is 100 times the average of the data points.  That average is in the neighborhood of 150…

28. DITW

29. DITW – but review the section on the remainder theorem (page 227).

30.  Well, in theory you are supposed to complete the square.  But look at the graph: the vertex is clearly in the neighborhood of (1,-16) and there is only one answer choice where both of those appear directly as constants.

31. Just pick a number between 12 and 18 – let’s say he does 15 per hour.  Divide 72 by 15 and move on.

32. DITW

33. DITW

34. DITW

35. DITW

36. You are looking for the value that makes the denominator zero.  This is one of those questions where you can make use of your calculator.  If you have a TI89, just set the denominator to zero and use the F2 command to solve that equation.  Or with any graphing calculator at all, graph the equation and then trace along the graph to find where the function equals zero.

If you decide to do this one by algebra, a neat trick is to replace x-5 with a new variable, say m.  If you do this, you get an expression that is easy to factor.  But remember to go back and solve for x.

37.  Review page 74 to see the quick way to calculate percent changes:  the base in that expression is 1.02 – that gives you a 2% increase each year.

38.  Review percents and exponential growth (page 181).  Then you will see that all you need here is to calculate (100)(1.02)10 and 100(1.025)10 and then subtract.

OK, that’s all for the first practice test. If you are keeping score, I believe that for 36 out of the 58 problems, there was an alternative method available.