Numbers & Operations
1. Explanation for Correct Answer B :
Choice (B) is correct. If a positive integer that is a multiple of
remainder will be
If the positive integer is not a multiple of
leave a remainder that is an integer greater than
remainder is
is divided by
For example, when
then dividing it by
and less than
is divided by
the
will
The least possible
the quotient is
and the remainder is
2. Explanation for Correct Answer B :
Choice (B) is correct. Since
lies within all three circles,
purchased all three flavors. Since
is the number of people who
people purchased strawberry ice cream, it follows that
Solving this equation yields
purchased chocolate ice cream. Since
purchased chocolate ice cream.
From the diagram,
people
it follows that
people
3. Explanation for Correct Answer E :
Choice (E) is correct. In order for
is an even integer, then
less than
are
which
is an integer.
to be an integer,
has to be an even integer. If
must be an odd integer. The odd positive integers that are
and
. Thus, there are five positive integers
less than
for
4. Explanation for Correct Answer A :
Choice (A) is correct. The ratios provide two equations:
to
which can be written as
and
you can multiply
by
Therefore, the ratio of
. To find the ratio of
. That yields
to
is
5. Explanation for Correct Answer A :
Choice (A) is correct. Jamal ran
feet, which is equal to
Jamal's distance to Lonnie's distance is
yards to
yards. Thus, the ratio of
yards, which is the same as
6. Explanation for Correct Answer D :
Choice (D) is correct. For a fraction to be positive, the numerator and denominator must have
the same sign, either both positive or both negative. The arrow corresponding to
the value of
is
and
points to
Numbers & Operations
7. Explanation for Correct Answer D :
Choice (D) is correct. The
in the place labeled
value of
The
value of
is in the thousands place, so it represents a
in the place labeled
is in the tens place, so it represents a
Comparing the two values, we see that
is
times
8. Explanation for Correct Answer E :
Choice (E) is correct. The product of any odd number of negative integers is always negative.
Thus, if all five of the integers are negative, their product will be negative. Since this is all of
the five integers, five is the maximum.
9. Explanation for Correct Answer C :
Choice (C) is correct. Since Pat removed
removed
of the cartons and Lee removed
of the cartons. Let
in the truck. Since
left, it follows that
were originally
together they
represent the number of cartons originally
of the cartons were removed,
Solving this equation for
remain. Since there are
gives
cartons
Thus, there
cartons in the truck.
10. Explanation for Correct Answer A :
Choice (A) is correct. This sum of
successive integers is equal to
Therefore, this example shows that the sum of
which is not divisible by
successive integers is not always divisible
by
11. Explanation for Correct Answer B :
Choice (B) is correct. You can rewrite this number as
since
integer
and
Thus,
is the fourth power of the
Numbers & Operations
12. Explanation for Correct Answer C :
Choice (C) is correct. The first term is
or
Since
is
or
Since
is even, the third term is
Since
is even, the second term is half of
or
The third term is odd, so the fourth term
is odd, the fifth term is
or
13. Explanation for Correct Answer C :
I.
A number is even if it is divisible by
is divisible by
II.
The product of an even integer with an even integer
so this product is also even. Thus, the set I has the given property.
A prime number is a positive integer that does not have any factors other than
and itself.
The product of two primes will have those two primes as factors. (For example, the product
of
and
is
The number
is not prime, because it can be factored as
) Thus,
set II does not have the given property.
III.
The product of two positive numbers is positive. Thus, set III has the given property.
Therefore, only sets I and III have the given property.
14. Explanation for Correct Answer D :
Choice (D) is correct. It appears from the figure that
Therefore,
If this is true, then
could correspond to
15. Explanation for Correct Answer D :
Statement I does not have to be true. It is possible that this year none of the students’
birthdays fall on a Sunday.
Statement II must be true. There are only
days in a week and
students. No matter
how the
students' birthdays are distributed into the days, at least two students (in fact,
at least three students) must have their birthday on the same day of the week.
Statement III must be true. There are only
months in a year, so no matter how the
students' birthdays are distributed into the
months, at least two students must have
their birthday in the same month.
Therefore, only statements II and III must be true.
16. Explanation for correct answer C:
use the calculator 5/9=0.555 every other choice is equal to 0.555 except for 30/45
=0.666
17. Explanation for correct answer A:
Write a fraction as a decimal 1/4 = 0.25 < 3/10 = 0.3
Numbers & Operations
18. Explanation for correct answer C:
Use the correct number of operations (17-3) /7-2 (-4-8)
14/7-2(-12)=
2+24=
26
19. Explanation for correct answer D:
Multiply by the recipricol 3/4 /3/8=3/4 8/3=38/4*3=2
20. Explanation for correct answer 512
Multiply by 4 to find the next term in the sequence therefore 128*4=512
21. Explanation for correct answer D:
Use the rules of exponents 81=3 squared so 3 squared=3x+2
22. Explanation for correct answer B:
=59,049
23. Explanation for correct answer c:
Divide 4/5 by 2/5
4/5/2/5=4/5*5/2=2
24. Explanation for correct answer 2/3
70/105+14/21=2/3
25. Explanation:
The correct answer is
expect
. Since
cartons to be sold in
cartons of juice were sold in
weeks, where
Therefore,
weeks, one would
Numbers & Operations