Wednesday, May 18, 2022
(in North America and South America)
Thursday, May 19, 2022
(outside of North American and South America)
©2022 University of Waterloo
Arranging the five answers along with 10 from smallest to
largest, we get
Since 10 is 2 more than 8 and 10 is 3 less than 13, then 8 is the
closest number to 10.
Answer: (C)
Reading from the graph, the greatest number of hours that Gabe spent riding his bike is 4, and this occurred on Tuesday.
Answer: (B)
Of the given answers, 0 is the only value of
Answer: (B)
Since
Answer: (C)
The faces shown are labelled with
Answer: (D)
Since
Answer: (A)
The largest height of the singers in Saura’s choir is 183.5
cm.
The smallest height of the singers in Saura’s choir is 141 cm.
Thus, the range of their heights is
Answer: (A)
Beginning at the origin
In the diagram, the point
Answer: (E)
When Emily jumps for 75 seconds, she jumps for
Jumping at the rate of 52 times in 60 seconds, Emily jumps
Since Emily jumps 52 times in 60 seconds and 13 times in 15 seconds,
then Emily jumps
Answer: (C)
In $1.00 worth of dimes, there are
In $1.00 worth of quarters, there are
The jar contains 10 dimes and a total of
If Terry randomly removes one coin from the jar, the probability that it
is a dime is
Answer: (E)
Since 42 is an even number, then 2 is a factor of 42.
Since
Each of
Thus, the sum of the prime factors of 42 is
(We note that
Answer: (C)
The sum of the angles in
Since
Since these two angles are equal, they each measure
Since
The value of
Answer: (D)
A two-digit number has at least one digit that is a 4 if its tens
digit is a 4 or if its ones digit is a 4.
There are 10 two-digit numbers whose tens digit is a 4.
These are
There are 9 two-digit numbers whose ones digit is a 4.
These are
The number of two-digit numbers that have at least one digit that is a 4
is
Answer: (C)
The side lengths of each of the three identical squares are
equal.
The perimeter of
Thus, the length of each side of the three identical squares is
The area of each of the three identical squares is
Therefore, the area of
Answer: (B)
The first Wednesday of a month must occur on one of the first 7
days of a month, that is, the 1st, 2nd,
3rd, 4th, 5th, 6th, or 7th of the month.
The second Wednesday of a month occurs 7 days following the first
Wednesday of that month, and the third Wednesday of a month occurs 14
days following the first Wednesday of that month.
Adding 14 days to each of the possible dates for the first Wednesday of
the month, we get that the third Wednesday of a month must occur on the
15th, 16th, 17th, 18th, 19th, 20th, or 21st of that month.
Of the given answers, the public holiday cannot occur on the 22nd of that month.
Answer: (B)
Solution 1:
When a standard fair coin is tossed three times, there are 8 possible
outcomes.
If H represents head and T represents tail, these 8 outcomes are: HHH,
HHT, HTH, THH, HTT, THT, TTH, and TTT.
Of these 8 outcomes, there are exactly 2 whose outcomes are all the same
(HHH and TTT).
Therefore, the probability that the three outcomes are all the same is
Solution 2:
The three outcomes are all the same exactly when each of the three
tosses is a head or when each of the three tosses is a tail.
When a coin is tossed, the probability that the coin shows a head is
Thus, the probability that each of the three tosses is a head is
Similarly, the probability that each of the three tosses is a tail is
Therefore, the probability that the three outcomes are all the same
is
Answer: (C)
If the value of
Since the given sum is 2022, then the value of
When
Therefore
Answer: (C)
We begin by recognizing that moving a block from Box A to Box B,
and then moving the same block from Box B back to Box A has does not
change the total mass in each box.
Since such a pair of moves increases the number of blocks that Jasmine
moves from Box A to Box B, then all such pairs of moves will not occur
in counting the fewest number of blocks that Jasmine could have moved
from Box A to Box B.
(Similarly, moving a block from Box B to Box A, and then moving that
same block from Box A back to Box B is a pair of moves that will not
occur.)
That is, once a block has been moved from one box to another, that block
will not be moved in the opposite direction between the two boxes.
Next, we consider what total masses of blocks can be moved from Box B
to Box A.
Box B contains one 50 g block and three 10 g blocks, and so Jasmine can
move the following total masses of blocks from Box B to Box A: 10 g,
20 g, 30 g, 50 g, 60 g, 70 g, and 80 g. (Can you see how to get each of
these masses using the blocks in Box B and why no other masses are
possible?)
If Jasmine moves 10 g from Box B to Box A, then she must move blocks
whose total mass is
For each of the other masses that may be moved from Box B to Box A, we
summarize the total mass of blocks that Jasmine must move from Box A to
Box B. Note that in each case, the mass moved from Box A to Box B must
be 65 g greater than the mass moved from Box B to Box A.
Mass moved from Box B to Box A | Mass moved from Box A to Box B |
---|---|
10 g | 75 g |
20 g | 85 g |
30 g | 95 g |
50 g | 115 g |
60 g | 125 g |
70 g | 135 g |
80 g | 145 g |
Next, we determine if it is possible for Jasmine to choose blocks
from Box A whose total mass is given in the second column in the table
above.
Recall that Box A originally contains one 100 g block, one 20 g block,
and three 5 g blocks.
Is it possible for Jasmine to choose blocks from Box A whose total mass
is exactly 75 g?
Since the 100 g block is too heavy, and the remaining blocks have a
total mass that is less than 75 g, then it is not possible for Jasmine
to choose blocks from Box A whose total mass is exactly 75 g.
In the table below, we determine which of the exact total masses Jasmine
is able to move from Box A to Box B.
For those masses which are possible, we list the number of blocks that
Jasmine must move from Box A to Box B.
Total mass moved from Box B to Box A | Total mass moved from Box A to Box B | Is it possible to move this mass from Box A to Box B? | Number of blocks moved from Box A to Box B |
---|---|---|---|
10 g | 75 g | No | |
20 g | 85 g | No | |
30 g | 95 g | No | |
50 g | 115 g | Yes; one 100 g, three 5 g | 4 |
60 g | 125 g | Yes; one 100 g, one 20 g, one 5 g | 3 |
70 g | 135 g | Yes; one 100 g, one 20 g, three 5 g | 5 |
80 g | 145 g | No |
From the table above, the fewest number of blocks that Jasmine could
have moved from Box A to Box B is 3. In this case, Jasmine moves a total
mass of
This confirms that Box B contains
Answer: (A)
The original ratio of red candies to blue candies is
For example, there could have been
We continue these possibilities in the table below, and consider the
number of red and blue candies and the resulting ratio after three blue
candies are removed from the dish.
Possible numbers of red and blue candies originally | 3 red, 5 blue |
6 red, 10 blue |
9 red, 15 blue |
12 red, 20 blue |
15 red, 25 blue |
18 red, 30 blue |
---|---|---|---|---|---|---|
Numbers of red and blue candies after 3 blue are removed | 3 red, 2 blue |
6 red, 7 blue |
9 red, 12 blue |
12 red, 17 blue |
15 red, 22 blue |
18 red, 27 blue |
New ratio of the number of red candies to blue |
If the dish originally contained 18 red candies and 30 blue (note
that
Therefore, there were
Answer: (B)
Let Anyu, Brad, Chi, and Diego be represented by
When rearranged,
For each of these 3 cases, we count the number of ways to arrange
Case 1:
Since
If
If
If
Thus there are exactly 3 possible rearrangements when
Case 2:
Since
In a manner similar to Case 1, it can be shown that there are 3 possible
rearrangements in this case:
Case 3:
Since
Similarly, there are 3 possible rearrangements in this case:
So that each person is not in their original position, the four
friends can rearrange themselves in
(Such a rearrangement of a list in which no element appears in its
original position is called a derangement.)
Answer: (B)
We begin by constructing the three diagonals inside the smaller squares that were missing from the diagram given in the question, as shown.
The two diagonals inside each of these 4 smaller squares divide each
smaller square into 4 identical triangles having equal area.
Thus, square
Since 7 of these triangles are shaded, then the fraction of
Answer: (C)
Solution 1:
Since the sum of
Similarly, since the sum of
It can similarly be shown that
Using the above observations, the sequence
Since the sum of
Since the sum of
The value of
Since
Therefore, the largest possible value of
(We note that
Solution 2:
The sum of the values of each group of four consecutive letters is
35.
Thus,
Rearranging the sum of these eight letters, we get
Substituting, we get
The value of
Since
Substituting, we get
(We note that
Answer: (C)
Katharina placed the 8 letters in a mixed-up order around the
circle.
We number the position of
Jaxon begins at position 1, writes down the letter
Thus, the first 3 letters that Jaxon writes down are the letters in
positions 1, 4 and 7.
Continuing to move clockwise around the circle, the next three positions
at which letters have not yet been written down are positions 8, 2 and 3
(the letter in position 1 was already written down), and so Jaxon writes
down the letter in position 3.
The next three positions at which letters have not yet been written down
are 5, 6 and 8 (the letters in positions 4 and 7 were already written
down), and so Jaxon writes down the letter in position 8.
At this point, Jaxon has written the letters in positions 1, 4, 7, 3,
and 8.
The next three positions at which letters have not yet been written down
are 2, 5 and 6 (the letters in positions 1, 3 and 4 were already written
down), and so Jaxon writes down the letter in position 6.
At this point, only the letters in positions 2 and 5 have not been
written.
Since Jaxon left off at position 6, he skips the letter at position 2,
skips the letter at position 5, and then writes down the letter at
position 2.
Finally, Jaxon writes the final letter at position 5.
Thus in order, Jaxon writes the letters in positions 1, 4, 7, 3, 8, 6,
2, and 5.
Since Jaxon’s list is
Therefore, Katharina’s clockwise order is
Answer: (C)
A palindrome greater than 10 000 and less than 100 000 is a
5-digit positive integer of the form
A positive integer is a multiple of 18 if it is a multiple of both 2 and
9 (and a positive integer that is a multiple of both 2 and 9 is a
multiple of 18).
A positive integer is a multiple of 2 if it is even, and thus the digit
A positive integer is a multiple of 9 exactly when the sum of its digits
is a multiple of 9, and thus
Next we consider four possible cases, one case for each of the possible
values of
Case 1:
When
Since
Since
Thus,
Next, we determine the possible values of
Thus when
Case 2:
When
Since
Since
Thus,
Next, we determine the possible values of
Thus when
Case 3:
When
Since
Thus when
Case 4:
When
Since
Thus when
Therefore, the number of palindromes that are greater than 10 000 and
less than 100 000 and that are multiples of 18 is
Answer: (D)
After all exchanges there are 4 balls in each bag,
and so if a bag contains exactly 3 different colours of balls, then it
must contain exactly 2 balls of the same colour and 2 balls each having
a colour that is different than all other balls in the bag.
Among the 8 balls in the two bags, there are 2 red balls and 2 black
balls and each of the remaining balls has a colour that is different
than all the other balls.
Thus after all exchanges, one bag must contain the 2 red balls and the
other bag must contain the 2 black balls.
Since Becca’s bag initially contains both black balls, and Becca moves
only 1 ball from her bag to Arjun’s bag, it is not possible for Arjun’s
bag to contain the 2 black balls after all exchanges.
This tells us that if each bag contains exactly 3 different colours of
balls after all exchanges, then Arjun’s bag contains the 2 red balls and
Becca’s bag contains the 2 black balls.
We let the first letter of each colour represent a ball of that
colour.
Then initially, Arjun’s bag contains
For the first ball chosen to be moved, there are exactly two cases to
consider:
Case 1: The first ball moved from Arjun’s bag to Becca’s bag is
Case 2: The first ball moved from Arjun’s bag to Becca’s bag is not
We begin with Case 1 and determine the probability that after all
exchanges, each bag contains exactly 3 different colours of balls.
Case 1: The first ball moved from Arjun’s bag to
Becca’s bag is
Since Arjun’s bag initially contains 5 balls, 2 of which are
After
Since Arjun’s bag must contain both
Becca’s bag contains 4 balls, 1 of which is
After the first two balls are moved, Arjun’s bag contains
Finally, a ball is chosen from Arjun’s bag and moved to Becca’s.
Since Arjun’s bag must contain both
If the first ball moved from Arjun’s bag to Becca’s bag is
Case 2: The first ball moved from Arjun’s bag to
Becca’s bag is
Since Arjun’s bag initially contains 5 balls, the probability that
After one of
For the second ball chosen to be moved, there are exactly two cases to
consider and so we split Case 2 into these two separate cases.
Case 2a: The first ball moved from Arjun’s bag to Becca’s bag is
Case 2b: The first ball moved from Arjun’s bag to Becca’s bag is
We begin with Case 2a and determine the probability that after all
exchanges, each bag contains exactly 3 different colours of balls.
Case 2a: The first ball moved from Arjun’s bag to
Becca’s bag is
We previously determined that the probability that
At this point, Becca’s bag contains 4 balls, 2 of which are
After
Since Becca’s bag must contain both
Arjun’s bag contains 5 balls, 1 of which is
If the first ball moved from Arjun’s bag to Becca’s bag is
Case 2b: The first ball moved from Arjun’s bag to
Becca’s bag is
The probability that
At this point, Becca’s bag contains 4 balls, 2 of which are
After this ball (
Since Arjun’s bag must contain both
Arjun’s bag contains 5 balls, 2 of which are
If the first ball moved from Arjun’s bag to Becca’s bag is
of choosing each of the 3 balls or
Finally, the probability that each bag contains exactly 3 different
colours of balls after all exchanges is
Answer: (A)
The regular pentagon shown has 5 sides, each with length 2
cm.
The perimeter of the pentagon is
Answer: (E)
The faces shown are labelled with
Answer: (D)
If the number is
Answer: (C)
Reading from the graph, the approximate number of bobbleheads
sold in the years 2016, 2017, 2018, 2019, 2020, and 2021 were 20, 35,
40, 38, 60, and 75, respectively.
Beginning with 2016 and 2017, the increases (or decreases) in the sale
of bobbleheads between consecutive years are approximately
Thus the greatest increase in the sale of bobbleheads was approximately
22 and this occurred between 2019 and 2020.
Answer: (D)
Continuing to count down by 11, we get
Answer: (C)
Since
Answer: (B)
Since
Since
Answer: (D)
Solution 1:
The value of 4 quarters is $1.00.
Thus, the value of
Since the jar initially contains 267 quarters, then
Solution 2:
The value of 267 quarters is
Thus, the number of quarters that must be added to the jar is
Answer: (D)
Each package of greeting cards comes with
Thus, 3 packages of greeting cards comes with
Kirra began with 7 cards and no envelopes.
To have more envelopes than cards, Kirra must buy enough packages to
make up the difference between the number of cards and the number of
envelopes (which is 7).
Thus, the smallest number of packages that Kirra must buy is 4.
Note: We can check that if Kirra buys 3 packages she has
Answer: (B)
The horizontal distance between the point
Since
Let us consider why each of the other statements is false.
The points
The point
Since
Further, the vertical distance between the point
Since
Similarly, the points
The point
Since
Since
Answer: (E)
In the sequence, the letters of the alphabet repeat in blocks of
26 letters.
Thus, 10 of these blocks gives a sequence that contains
Each block of 26 letters ends with the letter
Moving backward in the sequence from this
Answer: (C)
The first Wednesday of a month must occur on one of the first 7
days of a month, that is, the 1st, 2nd,
3rd, 4th, 5th, 6th, or 7th of the month.
The second Wednesday of a month occurs 7 days following the first
Wednesday of that month, and the third Wednesday of a month occurs 14
days following the first Wednesday of that month.
Adding 14 days to each of the possible dates for the first Wednesday of
the month, we get that the third Wednesday of a month must occur on the
15th, 16th, 17th, 18th, 19th, 20th, or 21st of that month.
Of the given answers, the public holiday cannot occur on the 22nd of that month.
Answer: (B)
The probability that the arrow stops on the largest section is
50% or
The probability that it stops on the next largest section is 1 in 3 or
Thus, the probability that the arrow stops on the smallest section is
Answer: (C)
A positive number is divisible by both 3 and 4 if it is divisible
by 12 (and a positive number that is divisible by 12 is divisible by
both 3 and 4).
The positive two-digit numbers that are divisible by 12 (and thus 3 and
4) are
Of these,
Thus, there are 4 positive two-digit numbers that satisfy the given
property.
Answer: (A)
Solution 1:
The area of the walkway is equal to the area of the pool subtracted
from the combined area of the pool and walkway.
That is, if the area of the walkway is
The combined area of the pool and walkway is equal to the area of the
rectangle with length 22 m and width 10 m.
The length, 22 m, is given by the 20 m pool length plus the 1 m wide
walkway on each end of the pool.
Similarly, the 10 m width is given by the 8 m pool width plus the 1 m
wide walkway on each side of the pool.
Thus the area of the walkway is
Solution 2:
We begin by extending each side of the pool 1 m in each
direction.
This divides the area of the walkway into four 1 m by 1 m squares (in
the corners), and two 20 m by 1 m rectangles (along the 20 m sides of
the pool), and two 8 m by 1 m rectangles (along the 8 m sides of the
pool), as shown.
Thus the area of the walkway is
Answer: (B)
Reading from the Venn diagram, 5 students participate in both
music and sports, 15 students participate in music (and not sports), and
20 students participate in sports (and not music).
Thus, there are
Of the 50 students,
Answer: (C)
If the number of golf balls in Bin G is
In this case, the total number of golf balls is
Multiplying both sides of this equation by 3, we get
Dividing both sides by 5, we get
The number of golf balls in Bin F is
(Alternately, we may have begun by letting there be
Answer: (B)
Figure 1 is formed with 7 squares.
Figure 2 is formed with
Figure 3 is formed with
Figure 4 will be formed with
Figure 5 will be formed with
Thus, the number of groups of 5 squares needed to help form each figure
is increasing by 1.
Also, in each case the number of groups of 5 squares needed is one less
than the Figure number.
For example, Figure 6 will be formed with 5 groups of 5 squares plus 7
additional squares. In general, we can say that Figure
Since
Thus,
Answer: (C)
There are 60 minutes in an hour, and so the number of minutes
between 7 a.m. and 11 a.m. is
Since Mateo stopped for a 40 minute break, he drove for
Thus, the average speed for Mateo’s 300 km trip was
Since there are 60 minutes in an hour, if Mateo averaged 1.5 km per
minute, then he averaged
Answer: (C)
Since
The midpoint of
The midpoint of
Since
Triangle
Similarly,
Since
Since
Substituting, we get
Answer: (A)
A perfect square is a number that can be expressed as the product
of two equal integers.
By this definition, 0 is a perfect square since
Since the product of 0 and every positive integer is 0, then every
positive integer is a factor of 0, and so 0 has an infinite number of
positive factors.
The next three smallest perfect squares are
Each of these has at most three positive factors.
The next largest perfect square is
The positive factors of 16 are 1, 2, 4, 8, and 16, and so 16 is a
perfect square that has exactly five positive factors.
The remaining perfect squares that are less than 100 are 25, 36, 49, 64,
and 81.
Both 25 and 49 each have exactly three positive factors.
The positive factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
The positive factors of 64 are 1, 2, 4, 8, 16, 32, and 64.
Thus, both 36 and 64 each have more than five positive factors.
Finally, the positive factors of 81 are 1, 3, 9, 27, and 81.
The two perfect squares that are less than 100 and that have exactly
five positive factors are 16 and 81, and their sum is
Answer: (E)
The sum of the values of each group of three consecutive letters
is 35.
Thus,
Since each of these equations is equal to 35, then the left sides of the
two equations are equal to each other.
That is,
Since
Since
Finally, we get
Answer: (D)
Consider folding the net into the cube and positioning the cube
with the face labelled
Beginning at
That is, the ant can walk from
From each of these vertical faces, the ant can walk to
We call these two possibilities Case 1 and Case 2, and for each case we
consider the number of possible orders in which the ant can visit the
faces.
Case 1: The ant’s 2nd move is to
We begin by recognizing that there is only 1 choice for the ant’s 2nd
move in this case.
From
The ant cannot return to the vertical face that it has already
visited.
Also, the ant cannot move to the vertical face that is opposite the
vertical face it has already visited. Why?
Consider for example that the ant visits, in order,
If the ant walks to
However, once at
Thus from
From this vertical face, the ant must walk to another vertical face
(since it has already visited
One such adjacent face has already been visited and the other has not,
and so the ant’s 1 choice is for its 4th move to be to the adjacent
vertical face it has not visited.
For example, if the order is
Finally, the ant’s final move must be to the adjacent vertical face that
it has not visited.
Summarizing Case 1, there are 4 choices for the first move from
Case 2: The ant’s 2nd move is to an adjacent vertical face
There are two vertical faces adjacent to each vertical face, and so
the ant has 2 choices for its 2nd move.
For its 3rd move, the ant can walk to the bottom face
We call the first of these Case 2a and the second Case 2b.
Case 2a: The ant’s 3rd move is to
We begin by recognizing that there is only 1 choice for the ant’s 3rd
move in this case.
From
The ant cannot return to either of the two vertical faces that it has
already visited, and so there are 2 choices for the ant’s 4th
move.
For example, if the order is
The ant’s final move is to the final vertical face and thus there is
only 1 choice.
Summarizing Case 2a, there are 4 choices for the first move from
Case 2b: The ant’s 3rd move is to the adjacent face that has not been visited
We begin by recognizing that there is only 1 choice for the ant’s 3rd
move in this case.
At this point the ant has visited three vertical faces.
The ant’s 4th move can be to
That is, the ant has 2 choices for its 4th move.
If the ant’s 4th move is to
If the ant’s 4th move is to the final vertical face, then its last move
is to
That is, once the ant chooses its 4th move, it has only 1 choice for its
final move.
Summarizing Case 2b, there are 4 choices for the first move from
Thus, if the ant starts at
Answer: (E)
Solution 1:
We begin by recognizing that numbers with the given property cannot
have two digits that are zero. Can you see why?
Thus, numbers with this property have exactly one zero or they have no
zeros.
We consider each of these two cases separately.
Case 1: Suppose the number has exactly one digit that is a zero.
Each of the numbers greater than 100 and less than 999 is a
three-digit number and so in this case, the number also has two non-zero
digits.
Since one of the digits is equal to the sum of the other two digits and
one of the digits is zero, then the two non-zero digits must be equal to
one another.
The two non-zero digits can equal any integer from 1 to 9, and thus
there are 9 possible values for the non-zero digits.
For each of these 9 possibilities, the zero digit can be the second
digit in the number or it can be the third digit (the first digit cannot
be zero).
That is, there are 9 possible values for the non-zero digits and 2 ways
to arrange the three digits, and thus
For example, numbers of this form are 101 and 110, 202 and 220, and so
on.
Case 2: Suppose the number has no digits that are equal to zero.
Let the three digits of the number be
Assume that
If
Similarly, if
Thus
If
Therefore,
If
In this case, the 3 ways to arrange these digits give the numbers 112,
121 and 211, each with the desired property.
If
In this case, the 6 ways to arrange these digits give the numbers 123,
132, 213, 231, 312, and 321, each with the desired property.
From these cases, we recognize that if
However, if the three digits are different from one another, then there
are 3 choices for the first digit, 2 choices for second digit and 1
choice for the third digit, and thus
We consider all possible values of
Values for |
Values for |
Total number of arrangements |
---|---|---|
2 | 1,1 [3] | 3 |
3 | 6 | |
4 | 9 | |
5 | 12 | |
6 | 15 | |
7 | 18 | |
8 | 21 | |
9 | 24 |
Thus, there are
Solution 2:
We begin by recognizing that numbers with the given property cannot
have three equal digits. Can you see why?
Thus, numbers with this property have exactly two digits that are equal
or all three digits are different.
We consider each of these two cases separately.
Case 1: Suppose the number has exactly two digits that are equal.
The two equal digits cannot be 0 since then the third digit would
also be 0.
If for example the two equal digits are 1s, there are two possibilities
for the third digit.
The third digit can be 2 (since
In the cases for which the third digit is equal to the sum of the two
equal digits, the equal digits can be
(We note that the equal digits cannot be greater than 4 since their sum
is greater than 9.)
For each of these 4 possibilities, there are 3 ways to arrange the
digits.
For example, when the equal digits are 1s and the third digit is 2, the
numbers 112, 121 and 211 have the desired property.
Thus, there are
In the cases for which two of the digits are equal and the third digit
is 0, the equal digits can be any integer from 1 to 9 inclusive.
For each of these 9 possibilities, there are 2 ways to arrange the
digits.
For example, when the equal digits are 1s and the third digit is 0, the
numbers 101 and 110 have the desired property.
Thus, there are
In total, there are
Case 2: Suppose all three digits are different from one another.
Let the three digits of the number be
Since
If
Similarly, if
Therefore,
If
In this case, the 6 ways to arrange these digits give the numbers 123,
132, 213, 231, 312, 321 and each has the desired property.
We consider all possible values of
Values for |
Values for |
---|---|
3 | |
4 | |
5 | |
6 | |
7 | |
8 | |
9 |
For each of these 16 possibilities in the table above, there are 6
ways to arrange the three digits, and so there are
Thus, there are a total of
Answer: (B)
Of the 4200 samples to test, let the number of samples that
contain blueberry be
Since each sample either contains blueberry or it does not, then there
are
Student A reports correctly on 90% of the
Thus, Student A reports
Student A reports correctly on 88% of the
incorrectly reports that
(That is, when a student is wrong when reporting “no blueberry", it
means that there is blueberry.)
Therefore, Student A reports
In total, Student A reports that
Similarly, Student B reports that
reports that
Student B reports 315 more samples as containing blueberry than
Student A, and so
Together, the three students report
Simplifying this expression, we get
If
Since
Similarly, if
Since
Thus, we want all integers
An integer is a multiple of 5 exactly when its units (ones) digit is 0
or 5.
The units digit of the total number of samples,
Thus,
The value of
The value of
The values of
We can confirm that when
The sum of all values of
Answer: (B)