SAT Math Hard Practice Quiz: Numbers and Operations
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Course
SAT
Institution
SAT
1. A bag contains tomatoes that are either green or red.
The ratio of green tomatoes to red tomatoes in the bag
is 4 to 3. When five green tomatoes and five red tomatoes
are removed, the ratio becomes 3 to 2. How many red
tomatoes were originally in the bag?
(A) 12
(B) 15
(C) 18
(D) 24
(E)...
Numbers and Operations 5. How many integers between 10 and 500 begin and end in
3?
1. A bag contains tomatoes that are either green or red.
The ratio of green tomatoes to red tomatoes in the bag
is 4 to 3. When five green tomatoes and five red tomatoes
are removed, the ratio becomes 3 to 2. How many red
tomatoes were originally in the bag?
(A) 12 6. A particular integer N is divisible by two different prime
(B) 15 numbers p and q. Which of the following must be true?
(C) 18
(D) 24 I. N is not a prime number.
(E) 30
II. N is divisible by pq.
III. N is an odd integer.
2. If each digit in an integer is greater than the digit to the (A) I only
left, the integer is said to be “monotonic”. For example, (B) II only
12 is a monotonic integer since 2 > 1. How many positive (C) I and II only
two-digit monotonic integers are there? (D) I and III only
(E) I, II, and III
(A) 28
(B) 32
(C) 36
(D) 40 7. A perfect square is an integer that is the square of an
(E) 44 integer. Suppose that m and n are positive integers such
that mn > 15. If 15mn is a perfect square, what is the
least possible value of mn ?
a, 2a − 1, 3a − 2, 4a − 3, . . .
3. For a particular number a, the first term in the sequence
above is equal to a, and each term thereafter is 7 greater
than the previous term. What is the value of the 16th
term in the sequence? 8. M is a set of six consecutive even integers. When the
least three integers of set M are summed, the result is x.
When the greatest three integers of set M are summed,
the result is y. Which of the following is true?
(A) y = x − 18
3
4. If p is a prime number, how many factors does p have? (B) y = x + 18
(C) y = 2x
(D) y = 2x + 4
(A) One (E) y = 2x + 6
(B) Two
(C) Three
(D) Four
(E) Five
erikthered.com/tutor pg. 1
, SAT Math Hard Practice Quiz
9. A three-digit number, XYZ , is formed of three different 10. An integer is subtracted from its square. The result could
non-zero digits X, Y , and Z. A new number is formed by be which of the following?
rearranging the same three digits. What is the greatest
possible difference between the two numbers? (For ex-
ample, 345 could be rearranged into 435, for a difference (A) A negative integer.
of 435 − 345 = 90.) (B) An odd integer.
(C) The product of two consecutive even integers.
(D) The product of two consecutive odd integers.
(E) The product of two consecutive integers.
erikthered.com/tutor pg. 2
, SAT Math Hard Practice Quiz
Algebra and Functions 4. Let m and n be positive integers such that one-third of
m is n less than one-half of m. Which of the following is
a possible value of m ?
1. Let m be √
an even integer. How many possible values of
m satisfy m + 7 ≤ 3 ?
(A) 15
(B) 21
(A) One (C) 24
(B) Two (D) 26
(C) Three (E) 28
(D) Four
(E) Five
5. If a and b are numbers such that (a − 4)(b + 6) = 0, then
x+3 what is the smallest possible value of a2 + b2 ?
2. Let x be defined by x = for any x such that
x−1
x 6= 1. Which of the following is equivalent to x − 1 ?
x+2
(A)
x−1
4 6. Let f (x) = ax2 and g(x) = bx4 for any value of x. If a
(B) and b are positive constants, for how many values of x is
x−1
f (x) = g(x) ?
2x + 4
(C)
x−1
2 (A) None
(D) (B) One
x−1
(C) Two
x+2 (D) Three
(E)
x−2 (E) Four
3 2
3. Let a and b be numbers such √ that a = b . Which of the 7. Let a and b be numbers such that 30 < a < 40 and 50 <
following is equivalent to b a ?
b < 70. Which of the following represents all possible
values of a − b ?
(A) b2/3
(A) −40 < a − b < −20
(B) b4/3 (B) −40 < a − b < −10
(C) −30 < a − b < −20
(C) b2 (D) −20 < a − b < −10
(E) −20 < a − b < 30
(D) b3
(E) b4
erikthered.com/tutor pg. 3
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