Example: Let the “kernel” of a number be defined as the square of its greatest prime factor.. So just find the greatest prime factor and square it.. If a number has a kernel of 4, then 4 m
Trang 1330 MCGRAW-HILL’S SAT
7. When the Apex Pet Store first opened, the ratio of
cats to dogs was 4 to 5 Since then, the number of
cats has doubled, while the number of dogs has
increased by 12 If the ratio of dogs to cats is now
1 to 1, how many cats did the store have when it
opened?
8. Hillside High School has 504 students One-quarter
of the students are sophomores, and 3/7 of the
sophomores are boys If one-third of the
sopho-more girls take French, how many sophosopho-more
girls do not take French?
(A) 24
(B) 36
(C) 48
(D) 72
(E) 126
9. A jar contains only red, green, and blue marbles
If it is three times as likely that you randomly pick a red marble as a green marble, and five times as likely that you pick a green one as a blue one, which of the following could be the number
of marbles in the jar?
(A) 38 (B) 39 (C) 40 (D) 41 (E) 42
.
1
2
3
4
5
7
8
9
6
1
0
2
3
4
5
7
8
9
6
1
0
2
3
4
5
7
8
9
6
1 0 2 3 4 5 7 8 9 6
Trang 2CHAPTER 8 / ESSENTIAL ALGEBRA I SKILLS 331
Concept Review 7
1 m = Mike’s current age, d = Dave’s current age;
m = 2(d − 5)
2 a = the population of town A, b = the population
of town B; a = 1.4b
3 n= number of marbles in the jar;
n − (2/3)n = 5 + (1/6)n
4 b = number of blue marbles, r = number of red
marbles; b = 4 + 2r
5 c = cost of one candy bar, l = cost of one lollipop;
3c + 2l = 2.20, and 4c + 2l = 2.80.
– (3c + 2l = 2.20)
c = 60
Plug in to find l: 3(.60) + 2l = 2.20
Simplify: 1.80 + 2l = 2.20
6 n= number of seats in the stadium;
(2/3)n − 1,000 = (3/7)n Subtract (2/3)n: −1,000 = −(5/21)n
Multiply by −(21/5): 4,200 = n
7
Simplify:
Multiply by 4: 2m + 2n = s + t Subtract t: 2m + 2n − t = s
8 b = value of blue chip, r = value of red chip, g = value
of green chip; b = 2 + r, r = 2 + g, and 5g = m, so
Cost of 10 blue and 5 red chips: 10b + 5r Substitute b = 2 + r: 10(2 + r) + 5r
Substitute r = 2 + g: 20 + 15(2 + g)
Substitute g = m/5: 50 + 3m
m n+ = +⎛s t
⎝⎜
⎞
⎠⎟
m n+ = ⎛s t+
⎝⎜
⎞
⎠⎟
2
1
2 2
Answer Key 7: Word Problems
SAT Practice 7
1 C You could test the choices here, or do the
algebra:
Multiply by x: 24 − x = 3x
2 n = Nora’s current age, m = Mary’s current age.
Interpret first sentence: n − 3 = (1/2)m
Interpret second sentence: m = n + 4
Substitute n = m − 4: m − 4 − 3 = (1/2)m
Multiply by −2: 14 = m
3 E p/q = 9/7, q/r = 14/3.
Multiply:
4 40 J = number of books Joan had originally.
E = number of books Emily had originally J = 2E.
After the exchange, Emily has E+ 5 and Joan has
J − 5 books, so J − 5 = 10 + (E + 5).
Simplify: J − 5 = E + 15
Subtract 15: J − 20 = E
Substitute into J = 2(J − 20)
first equation:
Solve for J: J= 40
(Reread and check)
5 C Let x be the cost of living in 1960 In 1970, the cost
of living was 1.2x, and in 1980 it was 1.5x Use the
per-cent change formula: (1.5x − 1.2x)/1.2x = 25 = 25%.
p q
q r
p r
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜
⎞
⎠⎟= =
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜
⎞
⎠⎟ =
9 7
14 3
6 1
24
3
− =x x
6 8 Let w = the number of games won and l = the number of games lost w/l = 7/5 and w + l = 48.
Substitute into 2nd eq.: (7/5)l + l = 48
Plug in to find w: w+ 20 = 48
How many more games won than lost?
w − l = 28 − 20 = 8
7 16 Let c = number of cats originally, d = number
of dogs originally c/d= 4/5 Now the number of
cats is 2c and the number of dogs is d+ 12 If the
ratio of dogs to cats is now 1 to 1, 2c = d + 12.
Cross-multiply: 5c = 4d
Divide by 4: (5/4)c = d
Substitute: 2c = (5/4)c + 12 Subtract (5/4)c: (3/4)c= 12
Multiply by 4/3: c= 16 (Reread
and check)
8 C Number of sophomores = (1/4)(504) = 126
If 3/7 of the sophomores are boys, 4/7 are girls: (4/7)(126) = 72 If 1/3 of the sophomore girls take French, 2/3 do not: (2/3)(72) = 48
9 E r, g, and b are the numbers of red, green, and blue marbles r = 3g and g = 5b Total marbles =
r + g + b.
Substitute r = 3g: 3g + g + b = 4g + b Substitute g = 5b: 4(5b) + b = 21b
So the total must be a multiple of 21, and 42 = 2(21)
Trang 3SPECIAL MATH PROBLEMS
CHAPTER 9
332
✓
Copyright © 2008 by The McGraw-Hill Companies, Inc Click here for terms of use
Trang 4CHAPTER 9 / SPECIAL MATH PROBLEMS 333
For all real numbers a and b, let the expression
a ¿ b be defined by the equation a ¿ b = 10a + b.
Question 3: What is 5 ¿ 10?
Just substitute 5 for a and 10 for b in 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 Δ, 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
32equals 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 132= 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 24= 16
There’s a lot of detail to learn and understand to do well on the SAT For more tools and resources that will help, visit our Online Practice Plus at www.MHPracticePlus.com/SATmath.
Trang 5334 MCGRAW-HILL’S SAT
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.
2 §−1.5 + §1.5 =
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 .
5 9 @ (k− 1) =
7 If q is any positive real number and n is an integer, let q @ n be defined by the equation
If y @ 2 = 64, what is the value of y?
8 For any integer n and real number x, let x ^ n be defined by the equation x ^ n = nx n−1 If y ^ 4 = −32, what is
the value of y?
9 For any integer n, let Ωn be defined as the sum of the distinct prime factors of n For instance, Ω36 = 5, because
2 and 3 are the only prime factors of 36 and 2 + 3 = 5 What is the smallest value of w for which Ωw = 12?
q n@ = q n+1
q n@ = q n+1
§ 15+§ 17
Trang 6CHAPTER 9 / SPECIAL MATH PROBLEMS 335
1. For all real numbers d, e, and f, let
d * e * f = de + ef + df If 2 * 3 * x = 12, then x =
(A)
(B)
(C)
(D) 2
(E) 6
2. If b≠ 0, let If x # y = 1, then which of
the following statements must be true?
(A) x = y
(B) x = |y|
(C) x = −y
(D) x2− y2= 0
(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
represent-ing 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?
(D) 1 (E) 3
3
−1 3
−1 2
x=2 3
1 5
4
5 1 + = 5
4
a b a b
# = 22
8
5
6
5
5
6
5. For x≠ 0, let What is the value of $$5?
6. For all nonnegative real numbers x, let ◊x be
defined by the equation For what
value of x does ◊x = 1.5?
(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, [341] = 3 + 4 + 1 = 8
If a is an integer greater than 0 but less than
1,000, which of the following must be true?
I [10a] < [a]+1
II [[a]] < 20 III If a is even, then [a] is even
(A) none (B) II only (C) I and II only (D) II 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
&=
−
⎧
⎨
⎪
⎩⎪
2 3
if is even
if is odd
◊ =x x
4
$x x
=1
SAT Practice 1: New Symbol or Term Problems
1
2 3 4 5 7 8 9 6
1 0 2 3 4 5 7 8 9 6
1 0 2 3 4 5 7 8 9 6
1 0 2 3 4 5 7 8 9 6
Trang 7336 MCGRAW-HILL’S SAT
Answer Key 1: New Symbol or Term Problems
Simplify and divide by 4: y3= −8 Take the cube root: y= −2
9 If Ωw = 12, then w must be a number whose
dis-tinct prime factors add up to 12 The prime num-bers less than 12 are 2, 3, 5, 7, and 11 Which of these have a sum of 12? (Remember you can’t re-peat 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 5 × 7 = 35 and 2 × 3 × 7 = 42 Since the question asks for the
least such number, the answer is 35.
SAT Practice 1
Translate: (2)(3) + (3)(x) + (2)(x) = 12
2 D If x # y = 1, then (x2/y2) = 1, which means x2= y2
Notice that x = −1 and y = 1 is one possible
solu-tion, which means that
(A) x = y
(B) x = ⏐y⏐
(E) x and y are both positive
is not necessarily true Another simple
solution is x = 1 and y = 1, which means that
(C) x = −y
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 2⁄3and y are complementary, then the sum
of their reciprocals is 1: 3⁄2+ 1/y = 1
Subtract 3⁄2: 1/y= −1/2
Take the reciprocal of both sides: y= −2
5 5 The “double” symbol means you simply per-form the operation twice Start with 5, then $5 = 1/5 Therefore, $$5 = $(1/5) = 1/(1/5) = 5
6 E
Multiply by 4:
Square both sides:
Plug in x= 36 to the original and see that it works
7 C If a is 12, which is even, then [12] = 1 + 2 = 3 is odd, which means that statement III is not necessarily true (Notice that this eliminates choices
(D) and (E).) Statement I is true because [10a] will always equal [a] because 10a has the same digits
as a, but with an extra 0 at the end, which
con-tributes nothing to the sum of digits Therefore,
[10a] < [a] + 1 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 [999] = 9 + 9 + 9 = 27; therefore, [[999]] = [27] = 2 + 7 = 9 It’s possible to get a slightly bigger value for [[a]] if a is, say, 991: [[991]] = [19] = 10, but you can see that [[a]] will never
ap-proach 20
8 C Since 13 is odd, 13& = 13 − 3 = 10 Therefore, 13&& = 10& Since 10 is even, 10& = 2(10) = 20
◊ = =
=
=
x x
4 1 5 6 36
Concept Review 1
1 §−4.5 = −5
2 §−1.5 + §1.5 = −2 + 1 = −1
3
4
5
6
7
Simplify:
Take the cube root:
Square:
y y y
64 4 16
2 1
3
=( ) = ( ) =
=
=
+
x2@0=( )x2 0 1+ =x
9@ k( )−1 =( )9k− +1 1=3k
8 3@ =( )84 =64
§ 15+ 17= + =3 4 7
Trang 8CHAPTER 9 / SPECIAL MATH PROBLEMS 337
Average (Arithmetic Mean) Problems
You probably know the procedure for finding an
average of a set of numbers: add them up and divide
by how many numbers you have For instance, the
av-erage of 3, 7, and 8 is (3 + 7 + 8)/3 = 6 You can
de-scribe this procedure with the “average formula”:
Since this is an algebraic equation, you can
manipu-late it just like any other equation, and get two more
formulas:
Sum = average × how many numbers
This is a great tool for setting up tough problems To find
any one of the three quantities, you simply need to find
the other two, and then perform the operation between
them For instance, if the problem says, “The average
(arithmetic mean) of five numbers is 30,” just write 30 in
the “average” place and 5 in the “how many” place
No-tice that there is a multiplication sign between them, so
multiply 30 × 5 = 150 to find the third quantity: their sum
Medians
How many numbers = sum
average
Average = sum
how many numbers
Lesson 2: Mean/Median/Mode Problems
Just about every SAT will include at least one
question about averages, otherwise known as
arithmetic means These won’t be simplistic
questions like “What is the average of this set
of numbers?” You will have to really
under-stand the concept of averages beyond the basic
formula
Occasionally the SAT may ask you about the
mode of a set of numbers A mode is the num-ber that appears the most frequently in a set (Just
remember: MOde = MOst.) It’s easy to see that not every set of numbers has a mode For instance, the mode of [−3, 4, 4, 1, 12] is 4, but [4, 9, 14, 19, 24] doesn’t have a mode
The average (arithmetic mean) and the me-dian are not always equal, but they are equal whenever the numbers are spaced symmetri-cally around a single number
it splits the highway exactly in half The median
of a set of numbers, then, is the middle number when they are listed in increasing order For
in-stance, the median of {−3, 7, 65} is 7, because the set has just as many numbers bigger than
7 as less than 7 If you have an even number of numbers, like {2, 4, 7, 9}, then the set doesn’t have one “middle” number, so the median is the average of the two middle numbers (So the median of {2, 4, 7, 9} is (4+7)/2 = 5.5.)
All three of these formulas can be summarized
in one handy little “pyramid”:
When you take standardized tests like the SAT, your score report often gives your score as a percentile, which shows the percentage of students whose scores were lower than yours If your percentile score is
50%, this means that you scored at the median of all
the scores: just as many (50%) of the students scored below your score as above your score
Example:
Consider any set of numbers that is evenly spaced, like 4, 9, 14, 19, and 24:
Notice that these numbers are spaced symmetrically about the number 14 This implies that the mean and the median both equal 14 This can be helpful to know, because finding the median of a set is often much easier than calculating the mean
Modes
4 9 14 19 24
A median is something that splits a set into two
equal parts Just think of the median of a
highway:
average howmany
sum
×
÷
÷
Trang 91 Draw the “average pyramid.”
2 Explain how to use the average pyramid to solve a problem involving averages
3 Define a median
4 Define a mode
5 In what situations is the mean of a set of numbers the same as its median?
6 The average (arithmetic mean) of four numbers is 15 If one of the numbers is 18, what is the average of the remaining three numbers?
7 The average (arithmetic mean) of five different positive integers is 25 If none of the numbers is less than
10, then what is the greatest possible value of one of these numbers?
8 Ms Appel’s class, which has twenty students, scored an average of 90% on a test Mr Bandera’s class, which has 30 students, scored an average of 80% on the same test What was the combined average score for the two classes?
Concept Review 2:
Mean/Median/Mode Problems
Trang 101. If y = 2x + 1, what is the average (arithmetic
mean) of 2x, 2x, y, and 3y, in terms of x?
(A) 2x (B) 2x+ 1 (C) 3x
(D) 3x+ 1 (E) 3x+ 2
2. The average (arithmetic mean) of seven
inte-gers is 11 If each of these inteinte-gers is less than
20, then what is the least possible value of any
one of these integers?
(A) −113 (B) −77 (C) −37
(D) −22 (E) 0
3. The median of 8, 6, 1, and k is 5 What is k?
4. The average (arithmetic mean) of two numbers is
z If one of the two numbers is x, what is the value
of the other number in terms of x and z?
(A) z − x (B) x − z (C) 2z − x
(D) x − 2z (E)
5. A set of n numbers has an average (arithmetic
mean) of 3k and a sum of 12m, where k and m
are positive What is the value of n in terms of k
and m?
6. The average (arithmetic mean) of 5, 8, 2, and k
is 0 What is the median of this set?
(A) 0 (B) 3.5 (C) 3.75
(D) 5 (E) 5.5
m
k
k m
4
4k m
4m
k
x z+ 2
7. A die is rolled 20 times, and the outcomes are as tabulated above If the average (arithmetic
mean) of all the rolls is a, the median of all the rolls is b, and the mode of all the rolls is c, then
which of the following must be true?
I a = b II b > c III c= 5 (A) I only (B) II only
(C) I and II only (D) II and III only (E) I, II, and III
8. If a 30% salt solution is added to a 50% salt so-lution, which of the following could be the con-centration of the resulting mixture?
I 40%
II 45%
III 50%
(A) I only (B) I and II only (C) I and III only (D) II and III only (E) I, II, and III
9. Set A consists of five numbers with a median of
m If Set B consists of the five numbers that are
two greater than each of the numbers in Set A, which of the following must be true?
I The median of Set B is greater
than m.
II The average (arithmetic mean) of
Set B is greater than m.
III The greatest possible difference between two numbers in Set B is greater than the greatest possible difference between two numbers
in Set A
(A) I only (B) I and II only (C) I and III only (D) II and III only (E) I, II, and III
SAT Practice 2: Mean/Median/Mode Problems
Roll Frequency
.
1
2
3
4
5
7
8
9
6
1
0
2
3
4
5
7
8
9
6
1 0 2 3 4 5 7 8 9 6
1 0 2 3 4 5 7 8 9 6