These solutions are for the 2008-2009, 2009-2010 and 2010-2011 SHSAT Handbook Form A tests, which were all identical. Here are my fully worked out solutions to the math section. The test can be found here.
Lesson Links: The Triangle Inequality
Solving Scrambled Paragraphs
Scrambled paragraphs are some of the trickiest problems on the SHSAT. The scrambled paragraph section in the Barron’s SHSAT book gives you a pretty good idea of what to look for in these problems. I will not try to replicate that full lesson here. The purpose of this post is to give an idea of the thought process you should go through when you’re solving scrambled paragraphs. We will do two examples: paragraphs 1 and 2 from Form A of the 2015-2016 Department of Education SHSAT handbook.
Lesson Links: Systems of Equations
Regents Prep– Pay particular attention to the Solving Linear Systems Algebraically Using Addition or Subtraction part
Relevant question from SHSAT handbook:
2012-2013 Handbook Form A
57) If 
, what is the value of 
in terms of 
?
A) 
B) 
C) 
D) 
E) 
If / then questions part 2
In part 1 we learned how to solve basic if/then questions. A few key points: Put all statements into if/then form. The contrapositive of a true if/then must also be true. The converse and inverse do not have to be true and are often wrong answer choices. Now we will learn some techniques for dealing with trickier questions where you are given more than one if/then.
The Chain Rule
When we have two if/then statements, sometimes we can connect them. For example, suppose these two statements are true:
If I go to Roma’s restaurant, then I get pizza. If I get pizza, then I get a soda.
This is a chain of events: when I go to Roma’s, I get pizza, and when I get pizza I also get a soda. So whenever I go to Roma’s I end up getting a soda! We can get a new if/then by skipping over the middle step: If I go to Roma’s restaurant, then I get a soda. This must be true!
Lesson Links: Solving for a in terms of b
Relevant questions from the SHSAT handbook:
2012-2013 Handbook Form A
68) If 
, what is the value of 
in terms of ?
F) 
G) 
H) 
J) 
K) 
If / then questions part 1
On the SHSAT, you may see a logical reasoning question like this:
q) Suppose the following statement is true: If it is Monday, then Bob goes to the park. Which of the following must also be true?
A) If it is not Monday, then Bob does not go to the park
B) If Bob goes to the park, then it is Monday
C) Bob’s favorite place is the park
D) If Bob doesn’t go to the park, then it is not Monday
E) Bob doesn’t like going to the beach
Lesson Links: Percents
Lesson Links: Ratios and proportions
Ratios and proportions show up a lot on the SHSATs! Mastering them will give you a nice score boost.
Links:
Regents Prep lessons and practice
Take special notice of this hint from the Regents Prep ratios lesson: When working with ratios such as 2 : 3 : 5, use 2x, 3x, and 5x to write an equation!
Lesson Links: Mean, median and mode averages
Basic lesson and practice on averages:
http://www.regentsprep.org/Regents/math/ALGEBRA/AD2/indexAD2.htm
There are a couple types of tricky problems related to averages that often show up on the SHSATs:
Finding the average from a frequency table:
Practice 1:
http://www.bbc.co.uk/bitesize/ks3/maths/handling_data/measures_average/revision/7/
Practice 2- don’t worry about the second half of the page, which is more advanced:
https://www.mathsisfun.com/data/mean-frequency-table.html
Missing numbers in averages:
Lesson/ practice 1:
https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-data-statistics/cc-6th-mean-median-challenge/v/using-mean-to-find-missing-value
Practice 2:
http://www.softschools.com/math/topics/determining_a_missing_data_value_given_the_mean/