If a figure has a vertical line of symmetry, then when it is
reflected in (or folded along) this vertical line, the two halves of the
figure are identical to one another. Of the given shapes, such a
vertical line is only possible for (E).
Answer: (E)
As a percent, one-quarter is equivalent to . Thus, Wednesday was chosen by
exactly one-quarter of the students.
Answer: (C)
A square with side length
has area .
Answer: (E)
Since is a
straight angle, its measure is . Thus, the angles with
measures and add to , and so the value of is .
Answer: (E)
Katie’s total time is equal to the sum of minutes and seconds and minutes and seconds. Adding seconds and seconds, we get seconds. Since there are seconds in minute, then seconds is equal to minute and seconds. Thus, Katie’s total time was
complete minutes plus seconds, for a total of minutes and seconds.
Answer: (D)
Solution 1:
The sequence repeats every
symbols, and . Thus,
the sequence of symbols repeats
times, and additional symbols follow. The 3rd
symbol in the sequence is , and thus the 23rd symbol is
.
Solution 2:
If the sequence of symbols is
repeated times, then symbols are written.
Following this, more symbols are
needed to reach , and since the
3rd symbol in the sequence is , the 23rd symbol is .
Answer: (C)
Since , then Olivia cuts her string into 21 pieces.
Since , then Jeff cuts his string into 14 pieces.
Therefore, Olivia has more
pieces of string than Jeff.
Answer: (A)
From the given list, the numbers that are divisible by are , , , .
The numbers that are divisible by
are , , .
The numbers that are divisible by both and are the numbers which appear in both of
the previous two lists. The only number in both lists is .
Thus, the numbers that are divisible by , or by , or by both and are , , , , , .
Since , , , , ,
are of the numbers listed, then the probability
that the chosen number is divisible by , or by , or by both and , is .
Answer: (C)
We rearrange the given subtraction to create the addition
statement .
Next, we consider the units digits.
From the statement, the sum has
a units digit of which means that
(since ). Therefore, we have .
Rearranging this equation, we get , and so .
Therefore, .
(We can check that , as
required.)
Answer: (D)
The perimeter of a rectangle is made up of two widths and two
lengths.
Since each length is twice the width, then two lengths is equivalent to
four widths, and so the perimeter is equal to times the width.
Since the perimeter of the rectangle is , then the width of the rectangle is .
Answer: (A)
Eloise spent a total of and the mean price of each water
pump was .
Thus, Eloise purchased water pumps.
Answer: (C)
Since has a units digit
of , then it is divisible by . Dividing, we get .
Since , then the three
prime factors of are ,
and , and their sum is .
Answer: (D)
A circle with radius has
area .
If the radius is tripled, then the new radius is .
A circle with radius has area
.
The area of the original circle divided by the area of the new circle is
.
Answer: (C)
After Brett pours half of his of water out, he has of water remaining in his glass.
Juanita then pours of her
or of
water into Brett’s glass.
The volume of water now in Brett’s glass is .
Answer: (A)
Since each unshaded section is times the size of each shaded section,
then together the size of shaded
sections is equal to the size of
unshaded section.
Thus, the combined size of all sections is equal to unshaded sections.
We can now imagine the spinner as having equal sized sections of which is shaded.
The probability that the arrow stops in a shaded section is equal to the
fraction of the spinner’s area comprised of shaded sections, which is
.
Answer: (D)
There are different
colours and different numbers,
and therefore
different kinds of robots that may be assembled. (These are R1, R2, R3,
R4, B1, B2, B3, B4, G1, G2, G3, G4, where R, B, G represent the colours red, blue, green.)
Since there are different kinds
of robots, it is possible that the first robots assembled are all different
from one another.
In this case, the th robot
assembled would be the first robot to have the same colour and the same
number as a previously assembled robot, and thus the greatest possible
value of is .
Notes:
It is possible that the first duplicate occurs earlier, but we
want the latest that it can occur.
There must be a duplicate robot among the first assembled, and so .
This solution makes use of the Pigeonhole principle – a
concept worth further investigation.
Answer: (C)
Consider the following list of integers, ordered from smallest to
largest, and having a median of :
.
Since is the smallest integer in
the list and is the largest, and
the list has a range of , then
is more than .
Since and differ by , then to find the smallest possible
value of , we can find the
smallest possible value of and
subtract .
The integers in the list are
different from one another, and so the smallest possible value of is 11 ( must be greater than the median ), and the smallest possible value of
is thus .
Since is more than , then the smallest possible integer in
the list is .
(We note that , where
is greater than and less than , is such a list.)
Answer: (B)
Beginning at height 1 and moving up settings at a time, the desk can stop
at settings , , , , and .
Beginning at height and moving
down 4 settings at a time, the desk can stop at settings , , , , , , and .
The desk originally begins at an odd-numbered height, .
Moving up settings at a time, the
desk can stop at only odd-numbered heights (since an even number added
to an odd number is odd).
Similarly, moving down settings
at a time, the desk can stop at only odd-numbered heights.
Thus, it is not possible for the desk to stop at an even-numbered
setting.
To this point, we have shown that the desk is able to stop at the
settings and is not
able to stop at even-numbered settings.
Next, we will show that it is possible for the desk to stop at the
remaining odd-numbered settings, ,
, , , and .
Since the desk can stop at setting , then it can stop at settings and with one and two presses of the down
button, respectively.
Similarly, since the desk can stop at setting , then it can stop at settings and .
Finally, since the desk can stop at setting , then one press of the up button will
take the desk to setting .
The desk can stop at all odd-numbered settings from to inclusive, and thus is able to stop at
different settings.
Answer: (B)
The three different integers selected from to and whose sum is must be the integers , , . Thus, the vertical column contains the
integers , ,
in some order.
(Can you see why no other combination of three of the given integers has
a sum of ?)
The three different integers selected from to and whose sum is must be , ,
or , ,
or , , .
If the integers in the horizontal row are , , , then there are two integers in common
with those in the vertical column, namely and .
Since there have to be five different integers used in the squares, then
there cannot be two integers in common between the two lists, and so
, ,
cannot appear in the horizontal row.
Similarly, , ,
cannot appear in the horizontal row.
Thus, the horizontal row must contain the integers , ,
with appearing in the centre
square since it is the integer in common between the two lists.
The integer not appearing in any square is .
The figure shows a possible arrangement of the integers.
Answer: (E)
We begin by determining the number of inner toothpicks used to
make a by grid of squares.
A grid containing rows has horizontal lines of inner toothpicks,
each of which contains
toothpicks (since there are
columns).
Thus, the number of inner toothpicks positioned horizontally is .
A grid containing columns has
vertical lines of inner
toothpicks, each of which contains toothpicks (since there are rows).
Thus, the number of inner toothpicks positioned vertically is .
In total, there are
inner toothpicks.
Next, we determine the total number of toothpicks used to make a
by grid of squares.
There are horizontal lines of
toothpicks, each of which contains toothpicks.
There are vertical lines of
toothpicks, each of which contains toothpicks.
Thus, there are a total of toothpicks
used to make a by grid.
(Alternately, we could have determined that there are outer toothpicks, and so there are
toothpicks in
total.)
The percentage of inner toothpicks used is , which is
when rounded to the nearest
percent.
Answer: (E)
All six faces of the prism are painted which means that the by by cubes in the interior of the prism are
the only cubes that have no paint on them.
Each of the three dimensions of the prism (length, width, height) must
be at least , otherwise there are
no by by cubes without paint on them.
The set of interior by by cubes must also be in the shape of a
rectangular prism.
(You should confirm each of these last two sentences for yourself before
reading on.)
There are interior by by cubes, and so the volume of the
interior prism is .
Thus, we are looking for three positive integers, representing the
length, width and height of the interior prism, whose product is .
We may use the positive divisors of (, , , , , ) to help identify the four
possibilities: ,
, , and .
These are the only ways to express as the product of three positive
integers.
Next, we determine the dimensions of the original prisms given each set
of dimensions for the interior prisms.
Consider the interior prism with dimensions .
Recall that this is the prism that remains after all exterior (painted)
cubes are removed.
That is, by by cubes have been removed from the top
and bottom of the
interior prism, from the left and right sides, as well as from the two
ends (the front and back).
This means that the dimensions of the original prism are each greater than the dimensions of the
interior prism. (You should try to visualize this.)
We complete the following table to determine the dimensions and the
volume of the original prism in each case.
Interior prism dimensions
Original prism dimensions
Volume of original prism
Therefore, the mean of all possible values of is .
Answer: (B)
Each Tiny three-digit integer belongs to exactly one of the
following three cases.
Case 1: The units digit is
If the units digit of a Tiny integer is , then the tens digit must also be , otherwise, the units digit and tens
digit can be switched to give a smaller integer.
In this case, there are no restrictions on the hundreds digit and thus
there are 9 such Tiny integers. These are: , , , , , , , , .
Case 2: The units digit is not , but the tens digit is
If the hundreds digit is and
the units digit is , then the
integers in this case are of the form , where . (If is greater than , then switching and creates a smaller integer.)
Integers of this form are Tiny exactly when is greater than or equal to , and is less than or equal to . If , then can be equal to any integer from to inclusive, and so there are such Tiny integers. These are: .
If , then can be equal to any integer from to inclusive, and so there are such Tiny integers. These are: .
Continuing in this way, there are
Tiny integers when , when , when , when , when , 2 when , and finally when .
In this case, there are Tiny integers.
Case 3: The units digit and the tens digit are both not
If the hundreds digit is
(where is greater than or equal
to ), the tens digit is , and the units digit is , then the integers in this case are of
the form . Integers of this form
are Tiny exactly when is less
than or equal to , and is less than or equal to .
For , we count the number of
such Tiny integers in the table that follows.
Value of
Value of
Possible values of
Number of Tiny integers
When , there are Tiny integers in
this case.
For , we may similarly count the
number of Tiny integers.
Value of
Value of
Possible values of
Number of Tiny integers
⋮
⋮
⋮
⋮
When , there are Tiny integers in this
case.
Notice that for each increase in the value of by , the smallest possible value of increases by (to match the value of ), and so the smallest possible value of
also increases by (to match the value of ).
This means that when , for
example, the number of Tiny integers in the first row of the
corresponding table is 1 less than the first row of the table for , and thus is 7.
That is, when , there are Tiny integers, and when
, there are Tiny integers.
Continuing in this way, we summarize the count of Tiny integers for Case
3.
Value of
Number of Tiny integers
The number of Tiny three-digit integers in this case is and so the
total number of Tiny three-digit integers is .
Answer: (E)
We begin by recognizing that is the only even prime number.
If , and are each odd prime numbers, then both
and are even prime numbers (since the sum
of two odd numbers is even).
However, both and are each at least , and therefore each must be an odd
prime number.
This tells us that cannot all
be odd prime numbers, and so exactly one of them is equal to (since they are all different from one
another and is the only even
prime number).
If , then each of and is odd and so is even, which is not possible.
Similarly, if , then each of
and is odd and so is even, which is not possible, and
so we conclude that .
Substituting , the list of different prime numbers becomes: and
we note that becomes
.
Since and are prime numbers that differ by
, next we consider the consecutive
odd prime numbers with less than
.
These are: and ,
and , and , and , and , and .
So then is equal to one of , , , , , or .
If , then which is divisible by and thus not a prime number.
If , then which is divisible by and thus not a prime number.
If , then which is divisible by and thus not a prime number.
If , then which is divisible by and thus not a prime number.
If , then which is divisible by and thus not a prime number.
Finally, if , then and , and both of these are prime
numbers.
Alternately, we may have noted that if has units digit , then has units digit , and if has units digit , then also has units digit , and so each is divisible by , which is not possible since each is a
prime number. We could have then removed as possibilities and
considered only as we did
above.
The table below summarizes what we know about the different prime numbers to this
point.
As shown previously, since and
are consecutive odd prime
numbers (with less than 50), then
is equal to one of 3, 11, 17, 29,
or 41 (recall that and the 8
numbers must all be different).
Since , then .
For which value(s) of is a prime number different from
those already in our list?
If , then which is not possible since .
If , then which is divisible by 13 and
therefore not a prime number.
If , then which is divisible by 5 and
therefore not a prime number.
If , then which is divisible by 7 and
therefore not a prime number.
Finally, if , then which is a prime number.
The final list of 8 different prime numbers is shown below.
The value of is .
Answer: (B)
Grade 8
The number of
coins needed to make is .
Answer: (E)
If a figure has a vertical line of symmetry, then when it is
reflected in (or folded along) this vertical line, the two halves of the
figure are identical to one another. Of the given shapes, such a
vertical line is only possible for (E).
Answer: (E)
Of the given numbers, only and are greater than . Since the tenths digit of , namely , is greater than the tenths digit of
, which is , then is the largest number in the
list.
Answer: (B)
If of is , then half of is , and so is equal to times , which is equal to . (We may confirm that of , or half of , is as required.)
Answer: (D)
Reading from the graph, Ryan ran km on Monday, km on Tuesday, km on Wednesday, km on Thursday, and km on Friday.
Thus, the total distance that Ryan ran over the five days is km.
Answer: (D)
When is increased by
, the result is .
When is then multiplied by , the final result is .
Answer: (B)
If , then and so .
Answer: (B)
Since the measure of a straight angle is 180, then .
Solving this equation, we get or , and so .
Answer: (D)
The ratio of the number of spoons to the number of forks is , which means that for each spoon in
the drawer, there are forks in
the drawer.
That is, the spoons and forks in the drawer can be separated into groups
of ( spoon and forks), and thus the total number of
spoons and forks in the drawer must be a multiple of .
Each of the answers , , , and is a multiple of .
The only answer given that is not a multiple of is , and so the total number of spoons and
forks in the drawer cannot be .
Answer: (D)
The large square with side length has area .
The shaded squares with side lengths ,
and , have areas , and , and , respectively.
The total area of the shaded regions is .
The total area of the unshaded region is equal to the area of the shaded
regions subtracted from the area of the large square, or .
Answer: (C)
Continuing the sequence, we get , , , , , , , , , , and so the smallest number greater
than that appears in the
sequence is .
Answer: (E)
Since 385 has a units digit of , then it is divisible by . Dividing, we get .
Since , then the three
prime factors of are ,
and , and their sum is .
Answer: (D)
We begin by splitting
into three equilateral triangles, as shown. (Confirm for yourself that
this is the only way to do this.)
Since the equilateral triangle in the middle shares a side with each
of the two other equilateral triangles, then the sides of all three
equilateral triangles are equal in length.
The perimeter of is made up of
side lengths of these equilateral
triangles.
Since the perimeter of is
, then each side
length of these equilateral triangles is .
Since is a side of an
equilateral triangle, then .
Answer: (C)
One container of ice cream can make cones.
Thus, containers of ice cream can
make cones.
Since cones were made from containers of ice cream, and it takes
containers of ice cream to make
cones, then containers of ice cream remain
after the cones are made.
One container of ice cream can make sundaes.
Thus, the remaining containers of
ice cream can make
sundaes.
Answer: (C)
Given that has a remainder
of when divided by , then is more than a multiple of .
This means that has a units digit
of , which is more than a units digit of , and thus has a remainder of when divided by .
We may confirm that examples of numbers that are equal to more than a multiple of , such as , , , , do indeed leave a remainder of when divided by .
Answer: (D)
Before the hole was drilled, the volume of the block of wood was
.
The cylindrical hole has radius and height equal to that of the block of wood, .
The volume of the cylindrical hole is thus .
In , the volume of the
block of wood after the hole is drilled is , which when rounded
to the nearest , is
.
Answer: (A)
There are different
colours and different numbers,
and therefore
different kinds of robots that may be assembled. (These are R1, R2, R3,
R4, B1, B2, B3, B4, G1, G2, G3, G4, where R, B, G represent the colours red, blue, green.)
Since there are different kinds
of robots, it is possible that the first robots assembled are all different
from one another.
In this case, the th robot
assembled would be the first robot to have the same colour and the same
number as a previously assembled robot, and thus the greatest possible
value of is .
Notes:
It is possible that the first duplicate occurs earlier, but we
want the latest that it can occur.
There must be a duplicate robot among the first assembled, and so .
This solution makes use of the Pigeonhole principle – a
concept worth further investigation.
Answer: (C)
The angle of one complete rotation measures 360.
Since the spinner is divided into
equal sections, each rotation
moves the arrow to the next dividing line between two sections.
Rotating clockwise, the section labelled is the th section.
This means that if the angle of rotation is greater than and less
than,
then the arrow stops in the section labelled .
Since each of the given answers is greater than and less than , the arrow must have spun
through more than one and less than two complete rotations.
On the second spin around, the arrow will stop in the section labelled
if the angle of rotation is
greater than and less
than .
Only one of the given answers lies in this range, and so the angle of
rotation must be .
Answer: (C)
Solution 1:
In this solution, we work backward from each of the given choices.
Since we are asked to find the smallest possible integer in the list, we
begin with the smallest of the five choices, .
If the smallest integer in the list is , then the largest integer in the list
is (since the three
integers have a range of ).
If the three integers have a mean of , then they have a sum of . Two of the integers are
and , and so the third (the middle) integer
is .
Since is greater than , this is not possible (the range of
these three integers is ,
not ).
If the smallest integer in the list is (the next smallest answer given), then
the largest integer in the list is .
If two of the integers are and
, then the third (the middle)
integer is .
Since is greater than , this is not possible (the range of
these three integers is ,
not ).
If the smallest integer in the list is (the next smallest answer given), then
the largest integer in the list is .
If two of the integers are and
, then the third (the middle)
integer is .
We may confirm that the three integers , , indeed have a range of and a mean of .
We have shown that is the
smallest of the five choices to satisfy the given conditions, and so
is the smallest possible integer
in the list.
Solution 2:
Assume that the list of
integers, ordered from smallest to largest, is .
Since is the smallest integer in
the list and is the largest, and
the list has a range of , then
is more than or .
The three integers have a mean of , and so or .
Substituting , the previous
equation becomes or
.
To find the smallest possible value of , we determine the largest possible
value of , recalling that and and so .
Since is even for all possible
values of , and is even, then must be even (since ).
We begin by choosing an arbitrary value of and using this value to determine and .
If , then and so and .
In this case, the three integers are , , .
We continue to increase the value of in order to determine the smallest
possible value for .
If , then and so and .
In this case, the three integers are .
If , then and so and .
In this case, the three integers are , which is not possible since
.
Continuing to increase the value of will continue to give values of that are less than .
Decreasing the value of will give
values of that are greater than
.
Therefore, the smallest possible integer in the list is .
Answer: (E)
Since pieces of fruit are
not apples, then the number of pears added to the number of bananas is
.
Since pieces of fruit are not
pears, then the number of apples added to the number of bananas is .
Since pieces of fruit are not
bananas, then the number of apples added to the number of pears is .
In adding these three totals , we are counting the number of
pears twice, the number of bananas twice, and the number of apples twice
(since each appears in exactly two of the three statements above).
That is, is twice the
total number of pieces of fruit, and so the number of pieces of fruit in
the box is .
Answer: (D)
Expressing the product in
terms of prime factors, we get This
product has factors of ,
factors of , and factor of , and thus is equal to .
The value of is .
Answer: (B)
The ratio of the number of quarters to the number of dimes to the
number of nickels is .
This means that for some positive integer , the number of quarters is , the number of dimes is , and the number of nickels is .
The value of each quarter is
or cents, and so the value of
the quarters in the jar, in cents, is .
The value of each dime is or
cents, and so the value of the
dimes in the jar, in cents, is .
The total value of the quarters and dimes is or cents, and so or , and so .
The number of nickels is , and
so the value of the nickels is .
Answer: (C)
The smallest five positive integers, each having a divisor of
, are , , , , and .
Thus, the smallest possible sum of five different positive integers
whose greatest common divisor is
is .
We know that the sum is at least and is equal to , which means that , and so or .
Each of the five integers is divisible by , and so the sum of the five integers,
, is divisible by .
Thus, we want the largest possible divisor of that is less than or equal to .
Since , the
divisors of that are less than
or equal to are: , , , , , , , and , and so the largest possible value of
is .
The sum of the digits of the largest possible value of is .
(We note that , , , , and are five such integers whose greatest
common divisor is and whose sum
is .)
Answer: (B)
There are different
locations at which the path splits, and we label these splits to , as shown.
There are six different locations where the path splits. The opening leads to split 1.
From split 1, moving to the left leads to split 2 and moving to the right leads to split 4.
From 2, left leads to 3 and right leads to 5.
From 4, left leads to 5 and right leads to 6.
From 3, left leads to bin A and right leads to bin B.
From 5, left leads to bin B and right leads to split 6.
From 6, left leads to bin B and right leads to bin C.
We begin by determining the probability that a ball lands in the bin
labelled .
There is exactly one path that leads to bin .
This path travels downward to the left at each of the three splits
labelled , and .
At each of these splits, the probability that a ball travels to the left
is , and so the probability
that a ball lands in bin is .
Next, we determine the probability that a ball lands in the bin
labelled .
There are exactly three paths that lead to bin .
One of these paths travels downward to the right at each of the three
splits labelled , and .
Thus, the probability that a ball lands in bin by following this path is .
A second path to bin travels
downward to the right at split ,
to the left at split , to the
right at split , and to the right
at split .
The probability that a ball follows this path is .
The third and final path to bin
travels left at split , and to the
right at each of the three splits ,
and .
The probability that a ball follows this path is also .
The probability that a ball lands in bin is the sum of the probabilities of
travelling each of these three paths or
Finally, we determine the probability that a ball lands in bin .
There are six different paths that lead to bin , and we could determine the probability
that a ball follows each of these just as we did for bins and .
However, it is more efficient to recognize that a ball must land in one
of the three bins, and thus the probability that it lands in bin is 1 minus the probability that it
lands in bin minus the
probability that it lands in bin ,
or
The probability that the two balls land in different bins is equal to
minus the probability that the
two balls land in the same bin.
The probability that a ball lands in bin is , and so the probability that
two balls land in bin is .
The probability that a ball lands in bin is , and so the probability that
two balls land in bin is .
The probability that a ball lands in bin is , and so the probability that
two balls land in bin is .
Therefore, the probability that the two balls land in different bins is
equal to
Answer: (A)
The smallest possible value of is closest to .
On a flat surface, the shortest distance between two points is along a
straight line between the points. The ant must walk on the surface of
the figure, and so to determine a straight line distance between and , we can "flatten" the figure.
In the figures below, we show a straight line path whose distance is
closest to from each of the
two perspectives.
A path from to that moves along the surface of the figure.
First perspective
The first perspective shows the path starting at and moving across the following cube faces, in order:
In the bottom layer, the right face of the rightmost cube in the front row, the right face of the rightmost cube in the middle row, then the top face of the rightmost cube in the middle row, and
in the middle layer, the front face of the rightmost cube.
The path then moves out of view through and behind the arch.
Second perspective
The second perspective shows the start of the path (which is on the front face from this perspective) and the continuation of the path behind the arch (which is on the right face from this perspective). The end of the path moves across the right faces of the back two cubes in the top layer, ending at .
To help see this path more clearly, we strip away all but the 5 cubes
that the ant walks on, below.
To see that this path is along a straight line from to , we draw a partial net of the previous
diagrams. This partial net includes each face that the ant walks on. The
numbers shown in the diagram below indicate that the face is from the
cube with the matching number in the diagrams above.
To determine the length of ,
we position so that is perpendicular to , as shown below.
Triangle is a right-angled
triangle with and , and so by the Pythagorean Theorem,
we get ,
and so the distance is closest to
. Since is the smallest of the five choices
given, and we have shown that there is a path of this length from to on the figure’s surface, then the
smallest possible value of is
.
Each of the other four given answers, , , , and is a result of the ant travelling
along other paths from to . For example, there exists a second
possible straight line path on a net of this figure for which is approximately . Can you determine this path, and
each of the other three paths from to whose lengths are equal to , , and ? There are also paths different from
the one shown above for which . Can you find these?