PSAT/SAT Practice on Special Symbol Problems
Name: __________________
Period: ____
To test your ability to learn and apply mathematical concepts, a special symbol is
sometimes introduced and defined.
These symbols generally have unusual looking signs (∂, τ ) so you won’t confuse
them with real mathematical symbols.
The key to these questions is to make sure you read the definitions carefully and
substitute the numbers correctly in the definition.
Example: If
a
c
b
d
is defined as
a
c
b
d
= ad + bc, what is the value of
2
6
3
4
*Follow the pattern given: substitute 2 for a, 3 for b, 6 for c, and 4 for d
Giving you: 2(4) + 3(6) = 8 + 18 = 26
Practice Problems: Show ALL work
1. Let x ∂ y be defined as x 
2.
1
, where y  0 . What is the value of 5 ∂ 4?
y
Let ( be defined for all non zero numbers a and b by a ( b =
the value of x?
Questions 3 & 4 refer to the following definition:
Let r  s = rs + s for all integers r and s
3. What is the value for 4  5?
4. If r  s = 0 and s  0 , what must r equal?
10a
. If x ( 7 = 200, what is
b
5. In the figure below, the sum of the three numbers in the horizontal row is equal to the
sum of the three numbers in the vertical column. Which of the following is equal to c?
7
a
b
c
4
(A) a + b
(B) a – b
Questions 6 & 7 refer to the following definition:
and c.
6.
2
a
b
(D) 11 – a
= ac + bc for all numbers a, b,
c
3
4
(A) = 9
7.
(C) 11 – b
=
(B) = 14
x
y
z
(C) = 18
(D) = 20
(E) = 24
= z and z  0 , what is the value of x + y?
(A) = 0 (B) = 1 (C) = 2 (D) = 3 (E) = cannot be determined.
8. For all non negative numbers a let @ be defined by @ 
a
. If @ = 2 what is the value
3
of a?
9. For all number a and b, let the operation ☻ be defined by a ☻b = b2 + ab. What is the
value of (1☻2)☻3?
10. Let # be defined by z # w = zw. If x = 5 # a, y = 5 # b and a + b = 3, what is the value of
xy?