CHAPTER 9 / SPECIAL MATH PROBLEMS 333
New Symbol or Term Problems Example:
For all real numbers aand b,let the expression
a¿bbe defined by the equation a¿b= 10 a+b.
Question 3: What is 5 ¿ 10?
Just substitute 5 for aand 10 for bin the given equa-
tion: 5 ¿ 10 =10(5) + 10 =60.
Question 4: If 2.5 ¿ x=50, what is the value of x?
Just translate the left side of the equation:
2.5 ¿ x=10(2.5) +x= 50
Then solve for x: 25 +x= 50
x= 25
Question 5: What is 1.5 ¿ (1.5 ¿ 1.5)?
According to the order of operations, evaluate what is
in parentheses first:
1.5 ¿ (1.5 ¿ 1.5)
Substitute: 1.5 ¿ (10(1.5) +1.5)
Simplify: 1.5 ¿ (16.5)
Substitute again: 10(1.5) +16.5
Simplify: 15 +16.5 =31.5
Lesson 1: New Symbol or Term Problems
Don’t be intimidated by SAT questions with
strange symbols, like ∆, φ, or ¥, or new terms
that you haven’t seen before. These crazy sym-
bols or terms are just made up on the spot, and
the problems will always explain what they
mean. Just read the definition of the new sym-
bol or term carefully and use it to “translate” the
expressions with the new symbol or term.Example:
Let the “kernel” of a number be defined as the
square of its greatest prime factor. For instance, the
kernel of 18 is 9, because the greatest prime factor
of 18 is 3 (prime factorization: 18 = 2 × 3 ×3), and
32 equals 9.
Question 1: What is the kernel of 39?
Don’t worry about the fact that you haven’t heard of
a “kernel” before. Just read the definition carefully. By
the definition, the kernel of 39 is the square of its
greatest prime factor. So just find the greatest prime
factor and square it. First, factor 39 into 3 ×13, so its
greatest prime factor is 13, and 13^2 =169.
Question 2: What is the greatest integer less than 20
that has a kernel of 4?
This requires a bit more thinking. If a number has a
kernel of 4, then 4 must be the square of its greatest
prime factor, so its greatest prime factor must be 2.
The only numbers that have a greatest prime factor of
2 are the powers of 2. The greatest power of 2 that is
less than 20 is 2^4 =16.