I. Symbols
On
some questions the test will create new functions. You can identify
these questions by the symbols that are used--triangles, squares,
ampersand, etc.). These questions are generally easy as long
as you don't get confused by the symbols. Simply take the function
and plug in the numbers.
Example 30
If a
# b = a + b, then what is 2 # 3?
Solution
2 # 3 would equal 2 + 3, or 5.
Example 31
If a
# b = a + b, then what is (2 # 3) # 2?
Solution
Solve inside of the parentheses first. 2 # 3 would equal 2 +
3, or 5. Then (5) # 2 = 5 + 2 or 7.
Example 32 (harder)
If for numbers x, y, z the function # is defined as
x # y = xy - x
then
x # (y # z) =
Solution
The first
step to solving x # (y # z) is to solve inside the parenthesis
(y # z), then after we have solved what is in the parenthesis
the second step is to do x # (what is in the parenthesis), then
the third step is to solve the equation using the symbol.
Step 1 (solve the parenthesis-- y # z)
1a) if x # y = xy - x (as stated in the question stem)
1b) then y # z = yz - y ( you get this by substituting
y for x and z and y)
Step 2 (insert
the parenthesis value)
the original
question asks x # (y # z), we have already solved y # z, which
according to 1b above
y # z = yz - y
So, in the original equation x # (y # z), substitute yz -
y for y # z.
Now, x # (y # z) = x # (yz - y)
(NOTE:
WHEN SOLVING QUESTIONS WITH SEVERAL NUMBERS AND OPERATIONS, ALWAYS
MULTIPLY AND DIVIDE BEFORE YOUR ADD AND SUBTRACT, FOR EXAMPLE
7 + 3 x 2 = 13, NOT 20. DO NOT SIMPLY GO
FROM LEFT TO RIGHT, MULTIPLY AND DIVIDE BEFORE YOU ADD AND SUBTRACT.)
So, we are now dealing with
x # (yz - y)
Step 3 (apply the # to the final equation)
The # symbol means x # y = xy -x.
(Essentially, take the first number--here x--, multiply it by
the second number--here y--and then subtract the first number.
Let's apply that
to the equation at the end of step 2
x # (yz - y) = x(yz-y) - x
then factor it out the x's.
=
xyz - xy - x
=
x(yz - y - 1)
