## SAT Test Prep

**CHAPTER 9**

SPECIAL MATH PROBLEMS

SPECIAL MATH PROBLEMS

__1. New Symbol or Term Problems__

__2. Mean/Median/Mode Problems__

__3. Numerical Reasoning Problems__

__4. Rate Problems__

__5. Counting Problems__

__6. Probability Problems__

**Lesson 1: New Symbol or Term Problems**

**New Symbol or Term Problems**

Don’t be intimidated by SAT questions with strange symbols, like Δ, ϕ, or ¥, or new terms that you haven’t seen before. These crazy symbols 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 symbol 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: , and 3^{2} 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 .

**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 .

**Example:**

For all real numbers *a* and *b*, let the expression be defined by the equation .

**Question 3:** What is

Just substitute 5 for *a* and 10 for *b* in the given equation: .

**Question 4:** If , what is the value of *x*?

Just translate the left side of the equation:

Then solve for *x*:

**Question 5:** What is 1.5 ¿ (1.5 ¿ 1.5)?

According to the order of operations, evaluate what is in parentheses first:

**Concept Review 1: New Symbol or Term Problems**

For questions 1–6, translate each expression into its simplest terms, using the definition of the new symbol.

The following definition pertains to questions 1–3:

For any real number *x*, let §*x* be defined as the greatest integer less than or equal to *x*.

The following definition pertains to questions 4–6:

If *q* is any positive real number and *n* is an integer, let *q* @ *n* be defined by the equation

__7.__ If *q* is any positive real number and *n* is an integer, let *q* @ *n* be defined by the equation

If , what is the value of *y*?

__8.__ For any integer *n* and real number *x*, let *x* ^ *n* be defined by the equation . If , what is the value of *y*?

__9.__ For any integer *n*, let Ω*n* be defined as the sum of the distinct prime factors of *n*. For instance, , because 2 and 3 are the only prime factors of 36 and . What is the smallest value of *w* for which

**SAT Practice 1: New Symbol or Term Problems**

**1**__.__ For all real numbers *d, e*, and *f*, let . If , then *x* =

(D) 2

(E) 6

**2**__.__ If , let If , then which of the following statements must be true?

(A)

(B)

(C)

(D)

(E) *x* and *y* are both positive

**3**__.__ On a digital clock, a time like 6:06 is called a “double” time because the number representing the hour is the same as the number representing the minute. Other such “doubles” are 8:08 and 9:09. What is the *smallest* time period between any two such doubles?

(A) 11 mins.

(B) 49 mins.

(C) 60 mins.

(D) 61 mins.

(E) 101 mins.

**4**__.__ Two numbers are “complementary” if their reciprocals have a sum of 1. For instance, 5 and are complementary because

If *x* and *y* are complementary, and if what is *y*?

(A) –2

(E) 3

**5**__.__ For , let What is the value of $$5?

**6**__.__ For all nonnegative real numbers *x*, let be defined by the equation For what value of *x* does

(A) 0.3

(B) 6

(C) 12

(D) 14

(E) 36

**7**__.__ For any integer *n*, let [*n*] be defined as the sum of the digits of *n*. For instance, . If *a* is an integer greater than 0 but less than 1,000, which of the following must be true?

III. If *a* is even, then [*a*] is even

(A) none

(B) II only

(C) I and II only

(D) I and III only

(E) I, II, and III

**8**__.__ For all integers, *n*, let What is the value of 13&&?

(A) 10

(B) 13

(C) 20

(D) 23

(E) 26

**Answer Key 1: New Symbol or Term Problems**

**Concept Review 1**

__9.__ If , then *w* must be a number whose distinct prime factors add up to 12. The prime numbers less than 12 are 2, 3, 5, 7, and 11. Which of these have a sum of 12? (Remember you can’t repeat any, because it says the numbers have to be *distinct*.) A little trial and error shows that the only possibilities are 5 and 7, or 2, 3, and 7. The smallest numbers with these factors are and . Since the question asks for the *least* such number, the answer is 35.

**SAT Practice I**

**2**__.__ **D** If , then , which means . Notice that and is one possible solution, which means that

(E) *x* and *y* are both positive is *not* necessarily true. Another simple solution is and , which means that

is not necessarily true, leaving only

(D) as an answer.

**3**__.__ **B** All of the consecutive “double times” are 1 hour and 1 minute apart except for 12:12 and 1:01, which are only 49 minutes apart.

**4**__.__ **A** If and *y* are complementary, then the sum of their reciprocals is 1:

**5**__.__ **5** The “double” symbol means you simply perform the operation twice. Start with 5, then . Therefore, .

Plug in *x* = 36 to the original and see that it works.

**7**__.__ **C** If *a* is 12, which is even, then is odd, which means that statement III is not necessarily true. (Notice that this eliminates choices (D) and (E).) Statement I is true because [10*a*] will always equal [*a*] because 10*a* has the same digits as *a*, but with an extra 0 at the end, which contributes nothing to the sum of digits. Therefore, is always true. Notice that this leaves only choice (C) as a possibility. To check statement II, though (just to be sure!), notice that the biggest sum of digits that you can get if *a* is less than 1,000 is from 999. ; therefore, . It’s possible to get a *slightly* bigger value for [[*a*]] if *a* is, say, , but you can see that [[*a*]] will never approach 20.

**8**__.__ **C** Since 13 is odd, . Therefore, ;. Since 10 is even, .