Exercises on Basic Arithmetic 381
Exercises on Basic Arithmetic
Multiple-Choice Questions
1. For how many positive integers, a, is it true that
a2 ≤ 2a?
(A) None (B) 1 (C) 2 (D) 4 (E) More than 4
2. If 0 < a < b < 1, which of the following is (are)
true?
I. a – b is negative.
1
II.
is positive.
ab
1 1
− is positive.
III.
b a
(A) I only (B) II only (C) III only
(D) I and II only (E) I, II, and III
3. How many of the numbers in the following list are
NOT even numbers?
0 192
64
,
, 6.4, 64 , 642,
, 64 2 ,
64 64
1.6
0.64646464...
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
–64,
4. If a and b are negative, and c is positive, which of
the following is (are) true?
I. a – b < a – c
a
b
II. if a < b, then
< .
c
c
1 1
<
III.
b c
(A) I only (B) II only (C) III only
(D) II and III only (E) I, II, and III
5. At 3:00 A.M. the temperature was 13° below zero.
By noon it had risen to 32°. What was the average
hourly increase in temperature?
19 °
(A) ⎛ ⎞
⎝ 9⎠
19 °
(B) ⎛ ⎞
⎝ 6⎠
(C) 5° (D) 7.5°
8. If p and q are primes greater than 2, which of the
following must be true?
I. p + q is even.
II. pq is odd.
III. p2 – q2 is even.
(A) I only (B) II only (C) I and II only
(D) I and III only (E) I, II, and III
3
1
9. What is the value of 2 2 − 2 2 ?
1
1
(A)
(B)
(C) 1 (D) 2 (E) 2
2
4
Questions 10 and 11 refer to the following definition.
For any positive integer n, τ(n) represents the number
of positive divisors of n.
10. Which of the following is (are) true?
I. τ(5) = τ(7)
II. τ(5)· τ(7) = τ(35)
III. τ(5) + τ(7) = τ(12)
(A) I only (B) II only (C) I and II only
(D) I and III only (E) I, II, and III
11. What is the value of τ(τ(τ(12)))?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 6
12. Which of the following is equal to (78 × 79)10?
(A) 727 (B) 782 (C) 7170 (D) 49170 (E) 49720
13. If x y represents the number of integers greater
than x and less than y, what is the value of
–π 2 ?
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6
14. If 0 < x < 1, which of the following lists the numbers in increasing order?
(A)
x , x, x2
2
(D) x, x ,
x
(B) x2, x,
(E) x,
x
(C) x2,
x, x
2
x, x
(E) 45°
15. If 50100 = k(10050), what is the value of k?
c
6. If (7a)(7b) =
7
, what is d in terms of a, b, and c?
7d
c
(B) c – a – b
ab
c
(E)
a+b
(A)
(C) a + b – c
(D) c – ab
7. A number is “nifty” if it is a multiple of 2 or 3.
How many nifty numbers are there between –11
and 11?
(A) 6 (B) 7 (C) 11 (D) 15 (E) 17
(A) 250 (B) 2550
100
(E) ⎛ 1 ⎞
⎝ 2⎠
(C) 5050
1 50
(D) ⎛ ⎞
⎝ 2⎠
382 Reviewing Mathematics
20. If x is an integer less than
1000 that has a remainder of 1
when it is divided by 2, 3, 4,
5, 6, or 7, what is one possible
value of x?
Grid-in Questions
16. If 25¢ buys 1.3 French francs,
how many francs can be
bought for $1.60?
17. At Ben’s Butcher Shop 99
pounds of chopped meat is
being divided into packages
each weighing 2.5 pounds.
How many pounds of meat are
left when there isn’t enough to
make another whole package?
18. Maria has two electronic beepers. One of them beeps every
4 seconds; the other beeps
every 9 seconds. If they are
turned on at exactly the same
time, how many times during
the next hour will both beepers beep at the same time?
0
0
0
1
1
1
1
0
0
0
2
2
2
2
1
1
1
1
3
3
3
3
2
2
2
2
4
4
4
4
3
3
3
3
5
5
5
5
4
4
4
4
6
6
6
6
5
5
5
5
7
7
7
7
6
6
6
6
8
8
8
8
7
7
7
7
9
9
9
9
8
8
8
8
9
9
9
9
21. What is the value of 24 ÷ 2–4?
0
0
0
1
1
1
1
0
0
0
2
2
2
2
1
1
1
1
3
3
3
3
2
2
2
2
4
4
4
4
3
3
3
3
5
5
5
5
4
4
4
4
6
6
6
6
5
5
5
5
7
7
7
7
6
6
6
6
8
8
8
8
7
7
7
7
9
9
9
9
8
8
8
8
9
9
9
9
22. What is the value of
|(–2 – 3) – (2 – 3)|?
0
0
0
1
1
1
1
2
2
2
2
0
0
0
3
3
3
3
1
1
1
1
4
4
4
4
2
2
2
2
5
5
5
5
3
3
3
3
6
6
6
6
4
4
4
4
7
7
7
7
5
5
5
5
8
8
8
8
6
6
6
6
9
9
9
9
7
7
7
7
8
8
8
8
9
9
9
9
19. If –7 ≤ x ≤ 7 and 0 ≤ y ≤ 12,
what is the greatest possible
value of y – x?
23. For any integer, a, greater than
1, let ↑a↓ be the greatest
prime factor of a. What is
↑132↓?
0
0
0
1
1
1
1
2
2
2
2
0
0
0
3
3
3
3
1
1
1
1
4
4
4
4
2
2
2
2
5
5
5
5
3
3
3
3
6
6
6
6
4
4
4
4
7
7
7
7
5
5
5
5
8
8
8
8
6
6
6
6
9
9
9
9
7
7
7
7
8
8
8
8
9
9
9
9
Answer Key 383
24. If the product of four consecutive integers is equal to one of
the integers, what is the largest
possible value of one of the
integers?
25. If x and y are positive integers,
and (13x) y = 1313, what is the
average (arithmetic mean) of x
and y?
0
0
0
0
0
0
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
9
9
9
9
9
9
9
9
7.
8.
9.
D
E
E
Answer Key
1.
2.
3.
16.
C
D
D
4.
5.
6.
8 . 3 2
17.
0
0
0
1
1
1
1
2
2
2
3
3
4
4
5
D
C
B
1 . 5
0
1
10.
11.
12.
18.
1 0 0
0
0
1
1
1
C
C
C
13.
14.
15.
19.
1 9
0
0
1
1
1
1
0
D
B
B
20.
4 2 1
0
0
0
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8
8
8
8
8
8
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
0
2
3
3
4
4
5
5
5
6
6
6
6
7
7
7
7
7
8
8
8
8
8
9
9
9
9
or 841
21.
2 5 6
1
22.
4
23.
1 1
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
4
2
0
0
24.
3
0
25.
7
0
0
0
0
0
0
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
384 Reviewing Mathematics
Answer Explanations
1. C. Since a is positive, divide both sides of the
given inequality by a: a2 ≤ 2a ⇒ a ≤ 2 ⇒
a = 1 or 2. There are two positive integers that satisfy the given inequality.
2. D. Since a < b, a – b is negative. (I is true.) Since
a and b are positive, so is their product, ab;
and the reciprocal of a positive number is
1 1
a−b
positive. (II is true.) − =
. Since
b a
ab
the numerator, a – b, is negative and the
denominator, ab, is positive, the value of the
fraction is negative. (III is false.)
3. D. Four numbers in the list are not even numbers:
6.4, 64 2 , and 0.64646464..., which are not
192
integers, and
, which equals 3, an odd
64
integer.
4. D. Since b is negative and c is positive,
b < c ⇒ –b > –c ⇒ a – b > a – c.
(I is false.) Since c is positive, dividing by c
preserves the inequality. (II is true.) Since b is
1
1
negative,
is negative, and so is less than ,
b
c
which is positive. (III is true.)
5. C. In the 9 hours from 3:00 A.M. to noon, the
temperature rose 32 – (–13) = 32 + 13 = 45°.
Therefore, the average hourly increase was
45 ÷ 9 = 5°.
7c
= 7c – d. Therefore:
7d
a+b=c–d⇒a+b+d=c⇒
d = c – a – b.
6. B. (7a)(7b) = 7a+b , and
7. D. There are 15 “nifty” numbers between –11 and
11: 2, 3, 4, 6, 8, 9, 10, their opposites, and 0.
8. E.
9. E.
All primes greater than 2 are odd, so p and q
are odd, and p + q is even. (I is true.) The
product of two odd numbers is odd. (II is
true.) Since p and q are odd, so are their
squares, and so the difference of the squares
is even. (III is true.)
3
2
1
2
( )
2 −2 = 2
1
3 2
1
2
−2 =
8− 2 = 4 2− 2 =2 2− 2 = 2.
10. C. Since 5 and 7 have two positive factors each,
τ(5) = τ(7). (I is true.) Since 35 has four divisors (1, 5, 7, and 35) and τ(5)· τ(7) = 2 × 2 =
4, II is true. The value of τ(12) is 6, which is
not equal to 2 + 2. (III is false.)
11. C.
τ(τ(τ(12))) = τ(τ(6)) = τ(4) = 3.
12. C. First, multiply inside the parentheses: 78 × 79 =
717; then raise to the 10th power: (717)10 = 7170.
13. D. There are five integers (1, 0, –1, –2, –3) that
are greater than –3.14 (–π) and less than 1.41
( 2).
14. B. For any number, x, between 0 and 1: x2 < x and
x< x .
15. B. 50100 = k(10050) ⇒ (5050)( 5050) = k(250)(5050) ⇒
50 50
k = 50 = 2550.
2
16. (8.32) You could set up a proportion (see Section
12-D):
cents
25 160
=
=
,
francs 1.3
x
but the easiest way is to multiply by 4 to find
that $1 buys 4 × 1.3 = 5.2 francs, and then
multiply 5.2 × 1.60 = 8.32.
17. (1.5) Divide: 99 ÷ 2.5 = 39.6. The butchers can
make 39 packages, weighing a total of
39 × 2.5 = 97.5 pounds, and have 99 – 97.5 =
1.5 pounds of meat left over.
18. (100) Since 36 is the LCM of 4 and 9, the beepers will beep together every 36 seconds. One
hour = 60 minutes = 3600 seconds, and so the
simultaneous beeping will occur 100 times.
19. (19) To make y – x as large as possible, let y
be as large as possible (12), and subtract
the smallest amount possible (x = –7):
12 – (–7) = 19.
20. (421 or 841) The LCM of 2, 3, 4, 5, 6, 7 is 420,
so 420 is divisible by each of these integers,
and there will be a remainder of 1 when 421 is
divided by any of them. One more than any
multiple of 420 will also work.
21. (256)
2 4 ÷ 2 −4 =
24
= 2 4−( −4 ) = 2 4+ 4 = 28 = 256 .
2 −4
22. (4) |(–2 – 3) – (2 – 3)| = |(–5) – (–1)| = |–5 + 1| =
|–4| = 4.
23. (11) The easiest way to find the
greatest prime factor of 132 is
to find its prime factorization:
132 = 2 × 2 × 3 × 11, so
11 is the greatest prime factor.
132
66
2
6
2
11
3
12-B Fractions and Decimals 385
24. (3) If all four integers were negative, their product
would be positive, and so could not equal one
of them. If all four integers were positive,
their product would be much greater than any
of them (even 1 × 2 × 3 × 4 = 24). Therefore,
the integers must include 0, in which case
their product is 0. The largest set of four consecutive integers that includes 0 is 0, 1, 2, 3.
25. (7) Since 1313 = (13x) y = 13 xy , then xy = 13. The
only positive integers whose product is 13 are
1 and 13. Their average is
1 + 13
= 7.
2
12-B FRACTIONS AND
DECIMALS
Several questions on the SAT involve fractions and/or
decimals. In this section we will review all of the important facts on these topics that you need to know for the
SAT. Even if you are using a calculator with fraction
capabilities, it is essential that you review all of this
material thoroughly. (See Chapter 1 for a discussion of
calculators that can perform operations with fractions.)
When a whole is divided into n equal parts, each part is
1
. For example,
called one-nth of the whole, written as
n
if a pizza is cut (divided ) into eight equal slices, each
⎛ 1⎞
slice is one-eighth ⎝ ⎠ of the pizza; a day is divided into
8
⎛ 1⎞
24 equal hours, so an hour is one-twenty-fourth ⎝ ⎠
24
⎛ 1⎞
of a day; and an inch is one-twelfth ⎝ ⎠ of a foot.
12
• If Sam slept for 5 hours, he slept for five-twenty-fourths
⎛ 5⎞
⎝ 24 ⎠ of a day.
• If Tom bought eight slices of pizza, he bought eight⎛ 8⎞
eighths ⎝ ⎠ of a pie.
8
• If Joe’s shelf is 30 inches long, it measures thirty⎛ 30 ⎞
twelfths ⎝ ⎠ of a foot.
12
5 8
30
, , and
, in which one
24 8
12
integer is written over a second integer, are called
fractions. The center line is the fraction bar. The
number above the bar is called the numerator, and
the number below the bar is the denominator.
Numbers such as
CAUTION: The denominator of a fraction can
never be 0.
5
, in which the numerator is less
24
than the denominator, is called a proper fraction. Its
value is less than 1.
30
• A fraction such as
, in which the numerator is more
12
than the denominator, is called an improper fraction.
Its value is greater than 1.
8
• A fraction such as , in which the numerator and
8
denominator are the same, is also an improper fraction, but it is equal to 1.
• A fraction such as
It is useful to think of the fraction bar as a symbol for
division. If three pizzas are divided equally among eight
3
people, each person gets
of a pizza. If you actually
8
use your calculator to divide 3 by 8, you get
3
= 0.375.
8
Key Fact B1
Every fraction, proper or improper, can be expressed in
decimal form (or as a whole number) by dividing the
numerator by the denominator. For example:
3
= 0.3
10
8
=1
8
3
= 0.75
4
11
= 1.375
8
5
= 0.625
8
48
=3
16
3
= 0.1875
16
100
= 12.5
8
Note: Any number beginning with a decimal point can be
written with a 0 to the left of the decimal point. In fact,
some calculators will express 3 ÷ 8 as .375, whereas
others will print 0.375.
Calculator
Shortcut
On the SAT, never do long division to convert a fraction
to a decimal. Use your calculator.
Unlike the examples above, when most fractions are
converted to decimals, the division does not terminate
after two, three, or four decimal places; rather it goes on
forever with some set of digits repeating itself.
2
= 0.666666...
3
3
5
= 0.272727...
= 0.416666...
11
12
17
= 1.133333...
15
On the SAT, you do not need to be concerned with this
repetition. On grid-in problems you just enter as much of
the number as will fit in the grid; and on multiple-choice
questions, all numbers written as decimals terminate.
Although on the SAT you will have occasion to convert
fractions to decimals (by dividing), you will not have to
convert decimals to fractions.