Warm-Up 5

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					                              Warm-Up 5
1. ________ The number 169 can be written as the sum of three positive perfect squares. What
            is the greatest of these perfect squares?

2. ________ The world‛s tallest mountain,
      vehicles                                           Greatest Depth Attained by Submersibles
               measured from its base, is Mauna                        36000

               Loa. Only 13,680 feet of this

                                                      Number of Feet
               33,356-foot mountain are visible
               above the water; the rest of the                        24000
                                                                                                              19700 19700
               mountain is under the Pacific Ocean.                     18000
               Of the seven submersible vehicles in                                                   13000

               the graph, how many could get to the                    12000                  9775

               bottom of Mauna Loa on the ocean                         6000
                                                                               3000   3300
                                                                           re V-1 ana lvin Tow son ste
                                                                        he                      a   ie
                                                                    y sp DSR Cy      A ep
                                                                                             - J Tr
                                                                 th                  De rgo
                                                              Ba                        A

3. ________ If x = −2, what is the value of 2x − 7?

4. ________ The ratio of boys to girls in a class of 35 students is 3:4. How many more girls than
              boys are in the class?
                                                                                    (x, y)
5. ________ On a coordinate grid, the triangle is translated from
              position A to position B. What is the value of x + y ?                  B
                                                                             (2, 3)        (9, 4)
                                                                                             (0, 0)           (4, 0)

6. ________ The diagonals of a rhombus measure 10 meters and 24 meters and are perpendicular
              bisectors of each other. What is the perimeter of the rhombus?

7. ________ The Fibonacci sequence is 1, 1, 2, 3, 5, 8, … . Each number after the first two
            numbers is the sum of the preceding two numbers. What is the first perfect square
            greater than 1 to occur in this sequence?

8. ________ How many toothpicks would you need in order to
             build the sixth figure in the pattern? The first
             three figures of the pattern are shown here.
             The first figure is made from four toothpicks.

9. ________ When four standard six-sided dice are rolled, what is the most likely sum of the
            four numbers?

10. _______ A rectangular frame measuring 10” by 15” has five square
            holes for photos. Four of the holes are each 2” by 2”, and
            one hole is 3” by 3”. What portion of the frame‛s area was
            not cut away for the photo holes? Express your answer as
            a common fraction.

MATHCOUNTS 2005-2006                                                                                                                29
                                   Warm-Up 5

1.   144             (E, G)       5. 14              (M, P, T)       8. 64                 (P, T)

2. 3                    (E)       6. 52                 (F, M)       9. 14                 (S, T)
3. –11                  (F)       7. 144                   (T)       10.                       (F)
4. 5                 (F, G)

Solution - Problem #8
    We have been given the first three figures of a pattern, and we now
must determine how many toothpicks would be needed to create the
sixth figure. We could certainly continue to draw the figures until we
get to the sixth figure in the pattern, but perhaps there is a quicker,
more efficient way to proceed. We can see that for each successive
figure, a square region is added to the top, bottom, right and left “arms” of the figure. However,
notice that though a square region with four sides is formed, only three toothpicks are necessary
to create each new square region. Since four square regions are added each time, and each one
requires three toothpicks, we can see that we are adding 4 × 3 = 12 toothpicks at each successive
stage of the pattern. The pattern starts with four toothpicks. The second figure requires us to
add 12 toothpicks. The third figure requires us to add 12 more toothpicks. To get to the sixth
figure of the pattern, we will have to add 12 toothpicks a total of five times, which is a grand total
of 4 + 12(5) = 64 toothpicks. How many toothpicks would be required to create the 100th figure of
the pattern?
                                                                                    Figure 1
Representation - Problem #4
     We are told that the ratio of boys to girls in a class of 35 students
is 3:4. This means that for every three boys in the class, there are
four girls. We could draw boys and girls, but to make this a little                 Figure 2
easier, we will use common symbols to represent a boy (arrow) and
a girl (plus). Figure 1 shows our ratio of three boys to four girls.
The glitch, though, is that we have only seven students. We need
35 students. Let‛s add on three more boys and four more girls. We
still have only 14 students, as shown in Figure 2. Notice that to keep
our ratio correct, we must add on three boys and four girls each time,              Figure 3
which is seven students. It will take five groups of these seven students
to arrive at our class total of 35 students (Figure 3). So there are five
groups of these three boys, which is a total of 5 × 3 = 15 boys. There are
also five groups of these four girls, which is a total of 5 × 4 = 20 girls.
There are then 20 − 15 = 5 more girls than boys in the class. A shortcut
would have been to see that each group of seven students has one more
girl than boy. Since there are five groups of seven, there are five more
girls than boys.

    Many times when we are working with ratios we can skip right to an algebraic representation of
the situation. A ratio of three boys to four girls in a class of 35 students translates to
3x + 4x = 35. (We don‛t know by which factor to increase the number of boys and girls, but we
know that we need to increase them both by the same factor.) Our equation simplifies to 7x = 35,
which is x = 5. Notice this relates to the previous solution in that x is the number of groups of
seven students needed to make up the class of 35 students. Continuing to use 3x for the number
of boys and 4x for the number of girls, there are 4x − 3x = x more girls than boys, and we have
already determined x = 5.
30                                                                               MATHCOUNTS 2005-2006
                                   Warm-Up 6
1. ________ The set {2, 3, 4, 5, 9} contains five of the factors of which one of the following four
            integers?       480      1260 1890 3888

2. ________ The table below shows the average amount paid to winners of the Kentucky Derby
            races from 1976 through 2004. For example, there were four years that
            15 horses started the race, and the first-place winners for those years averaged
            $527,175. For the 29 races shown, what is the average winnings amount associated
            with the median number of starters?
                      No. of       Frequency    First Prize     No. of       Frequency     First Prize
                     Starters                                  Starters
                         9             1         $165,200         16                 3     $688,467
                        10             1         $228,650         17                 3     $680,600
                        11             1         $186,900         18                 2    $1,299,900
                        13             3         $452,450         19                 6     $777,750
                        14             1         $628,800         20                 3     $605,967
                        15             4         $527,175         21                 1     $317,200

                                                                       −3 3 −3
3. ________ What is the 11th term in the geometric sequence 3,           , , ,
                                                                       2 4 8     … ? Express your
            answer as a common fraction.

4. ________ Goran has a standard deck of 52 cards. He considers an ace to have a value of 1, a
            jack has a value of 11, a queen has a value of 12, a king has a value of 13, and all other
            cards are their face value. What is the probability that a randomly selected card
            will have an even value? Express your answer as a common fraction.
5.        sq ft
     ________     Jen and Jerry need to carpet their L-shaped room, having dimensions
                  as shown. The carpet they chose is a solid color and is sold in rolls        10’
                  that are 12 feet wide. How many square feet of wasted carpet will         4’
                  they have if the minimum length from the roll is used and only one
                  seam is permitted?

6. ________ What is the 100th term of the sequence 2, 3, 5, 6, 7, 10, 11, … which consists of all of
            the positive integers that are neither perfect squares nor perfect cubes?

7. ________ What is the integer value of x that minimizes the absolute value of (6x 2 – 2x )?

8. ________ Cheryl, Don, Oleg, Shandra and Hans each have exactly one pet. Their five pets are
             a ferret, parrot, dog, cat and fish, but not necessarily in that order. In how many
             different ways can the five friends be paired with the five animals?

9. ________ The mean of a collection of five positive integers is 7. If the collection has a unique
            mode, what is the greatest possible member of the collection?

    _______ Maria sees these first four figures of a sequence in her textbook. The first figure
10. Figure
            is made from 6 toothpicks. Maria has 41 toothpicks. What is the largest figure of
            this sequence Maria can create if the sequence were to continue forever?

                        Figure 1   Figure 2         Figure 3              Figure 4

MATHCOUNTS 2005-2006                                                                                       31
                                    Warm-Up 6

1.   1260        (E, F, G, P)      5. 8                  (F, M)        8. 120             (F, M, T)

2. 680,600               (T)       6. 112                 (P, T)       9. 31              (G, M, T)
3.                 (F, P, T)       7. 0                   (F, G)       10. 8            (F, M, P, T)
4.                   (M, T)
Solution - Problem #1
    We know 2, 3, 4, 5 and 9 are factors of the number we need to find. We also know that the
number we need to find is either 480, 1260, 1890 or 3888. Immediately, you may notice that the
first three options each end with a zero for a units digit. This is possible only if the number has
a factor of 10, which is really a factor of 2 and a factor of 5. We see that the number we are
searching for must have factors of 2 and 5, so we know it must end in a zero. Therefore, we know
3888 is not our answer. We also know 9 must be a factor of our answer. A number has a factor
of 9 only if its digit-sum is divisible by 9. Looking at 480, which has a digit-sum of 4 + 8 + 0 = 12,
we see that 12 is not divisible by 9, so neither is 480. Let‛s perform the same test on our last
two possibilities: 1260 and 1890. The integer 1260 has a digit-sum of 1 + 2 + 6 + 0 = 9, and that is
divisible by 9. The integer 1890 has a digit-sum of 1 + 8 + 9 + 0 = 18, which also is divisible by 9. So
far we have not been able to rule out 1260 or 1890 as our answer. Notice that by passing the tests
for divisibility by 10 and 9, we are ensured that these two integers are divisible by 2, 3 and 5, also.
That leaves the factor of 4. An integer is divisible by 4 if the two-digit number formed by the tens
and units digit is divisible by 4. Looking at 1260, we just need to see if 60 is divisible by 4, which it
is. For 1890, we need to check 90 for a factor of 4, and we see that it is not divisible by 4. Now we
know that only 1260 is divisible by 2, 3, 4, 5 and 9.

Representation - Problem #3
     For a geometric sequence, a common ratio is multiplied with each term to determine the next
term. If this common ratio is positive, then the terms will stay positive or stay negative. If the
common ratio is negative, the terms will alternate between positive and negative values. If we are
multiplying each term by a common ratio with an absolute value greater than one, the absolute value
of the terms will get larger. However, if we are multiplying each term by a common ratio with an
absolute value less than one, the absolute value of the terms will get smaller. For the geometric
sequence in problem #3, we go from 3 to −3 . What ratio was multiplied with 3 to get −3 ? We can
                                            2                                            2
solve 3x = −3 . Dividing both sides by 3 yields x = −1 . From here we can see that each term has
             2                                      2
been multiplied by −1 to get the next term. Because this common ratio is negative, our terms will
alternate between positive and negative values. Because this common ratio has an absolute value
less than 1, the absolute values of our terms will get smaller.
Here is a visual representation of the terms of the sequence.
The vertical line represents 0, and the terms of the sequence                                      3
are shown from top to bottom. Notice how the segments                  2
representing the terms are each half the size of the previous                         4
segment, and the values are alternating from one side of zero               -3
to the other. If we continue this pattern, we can determine the                    3
11 term of the sequence. Notice, also, the numerator is always

3 while the denominators are successive powers of 2. The 11     th

term will be on the right side of 0 (positive) and will have a
numerator of 3 and a denominator of 210. This is 1024 .

    A shortcut for determining our 11th term is to see that we will have to multiply our first term by
this common ratio of −1 a total of 10 times, which is 3 × ( −1 )10 = 3 × ( −1) = 3 × 1024 = 1024 .
                     2                                      2              210
                                                                                       1      3

32                                                                                 MATHCOUNTS 2005-2006
                               Workout 3
1. ________ The surface of Lake Michigan is 22,300 square miles. If the surface of Lake
              Michigan were a perfect circular region, what would its circumference be? Express
              your answer to the nearest ten.

2. ________ The six-person Droz family spends $14.27 per person weekly at The Corner Grocers.
            If The Corner Grocers makes a net profit of 5% of the amounts customers pay, what
            is the net profit they will make on the Drozes‛ purchases over a six-week period?

3. ________ Lo has 300 yuan, Maria has 175 euros and Jorge has 500 pesos. If the three of
            them line up in order of
            wealth, such that the person         Currency        In USD          Per USD
            holding the greatest value         Chinese Yuan       0.120            8.333
            in currency is at the front,
                                                   Euro           1.320            0.758
            in what order will they be
            standing, front to back?          Mexican Peso        0.090            11.111

4. ________ At the Surf‛s Up! Restaurant, a 15% tip is automatically added to Jarett‛s bill
            because there are more than six people in his party. On his check, the included tip
            is $27. What is the total cost of the check including the tip?

5. ________ In how many different positions can a 2 by 1 rectangle be placed on
             the 8 by 8 square board so that the rectangle covers exactly two
             squares of the square board?

6. ________ If x < −4.38, what is the greatest possible integer value of x ?

7. ________ A 75-pound bag of feed for livestock contains 30 pounds of bone meal, 30 pounds
              of limestone and 15 pounds of salt. If Mandelbrot Feed Co. wishes to sell this same
              mixture in 100-pound bags, how many pounds of salt should be in each bag?

8. ________ The graph shows the daily
         lbs                                                             6                                Up to
             amount of food needed to feed
                                                Number of Cups of Food

                                                                         5                                2 Months
             a puppy based on the puppy‛s
             maximum adult weight. For                                   4
             which maximum adult weight                                                                   2-4 Months
             does a puppy have the greatest                              3
             percent increase of food intake
             from “Up to 2 Months” to                                    2
                                                                                                          4-6 Months
             “4-6 Months?”                                               1

                                                                               30       50        80
                                                                             Maximum Adult Weight (lbs)

9. ________ Water weighs approximately 62.4 lbs/ft3. How much does the water in a full 5‛ by 3‛
             by 2‛ tank in the shape of a rectangular prism weigh?

10. _______ To get from one corner of a 90‛ by 90‛ square plaza to the diagonally opposite
            corner, what percent shorter is it to walk diagonally across than to walk along two of
            its sides? Express your answer to the nearest whole percent.

MATHCOUNTS 2005-2006                                                                                                 33
                                    Workout 3

1.   530                 (F)       5. 112             (M, P, S)       8. 50                    (E)

2. 25.69                (M)        6. −5              (E, G, M)       9. 1872                  (F)

3. Maria, Jorge, Lo      (C)       7. 20              (F, M, P)       10. 29                (F, S)

4. 207                (F, M)

Solution - Problem #5
     We need to be sure we do not leave any positions for the 2 by 1 rectangle out of our count,
while also ensuring that we do not count any positions more than once. (For our picture, arrows
will be easier to see than rectangles.) Our arrows may only be placed in a horizontal position or a
vertical position if they must cover two unit squares. Starting with the upper left square of our
grid, we can place only two arrows that cover it while also remaining
fully within the grid. One is vertical and one is horizontal. If we move
to the second square of the top row, again, we have only two arrows
that cover it, remain fully within the grid and do not go back to cover
the first square. Remember, we do not want to recount an arrow that
we have already counted. If we continue this process, we will see that
each of the first seven squares in the first row yields two positions for
a 2 by 1 arrow. However, the last square of the row will give us only
one new arrow position. This is a total of 15 positions for arrows that
cover a square in the first row. These are shown in the figure to the
right. If we move to the second row, we will see that again we get 15
new positions for the 2 by 1 arrow. Remember that none of these positions may cover a square in
the first row because those positions have already been counted. Each of our first seven rows will
yield 15 new positions with this counting technique. However, the eighth row is different. There
is only one position for the arrow that will cover the first square of the bottom row and no squares
from the row above. The same is true for the second square in this row. There is only one position
for a 2 by 1 arrow that does not cover any squares above it or to the left of it. Each of the first
seven squares of this final row yields one new position. However, there are no new positions for
the final square of the grid. This is a grand total of 7(15) + 7 = 105 + 7 = 112 positions for a 2 by 1
arrow (rectangle) in the 8 by 8 square board.

Solution - Problem #10
    To walk from one corner of the 90‛ by 90‛ square to the opposite corner, along the sides of the
square, we would walk a total of 180‛. If we walk across the square, this distance will be shorter.
The diagonal path across the square plaza splits our square into two isosceles right triangles with
the diagonal path as the hypotenuse of both triangles. We can use the Pythagorean Theorem to
determine the distance of this path. Let d be the length of our diagonal path. We have
d2 = 902 + 902, which is d2 = 16,200 or d ≈ 127.28 feet. This is a difference of 180 − 127.28 = 52.72
feet, which is 52.72 ÷ 180 = 29% off the initial length. (Since we were working with an isosceles
right triangle, we also could see that the diagonal path was equal to 90 2 ≈ 127.28 feet since the
hypotenuse of a 45-45-90 triangle is the product of the length of the leg and 2 .)

    Because we have a formula for the diagonal path, we could check if our answer would remain
the same for a square plaza of any size. Letting s be the side length of a square plaza, the long way
from corner to opposite corner is 2s feet, and the diagonal route is s 2 feet. This is shorter by
2s − s 2 feet, which is a decrease of (2s − s 2 ) ÷ 2s = (2 − 2 )s ÷ 2s = (2 − 2 ) ÷ 2 = 29% for
any value of s.

34                                                                               MATHCOUNTS 2005-2006
                              Warm-Up 7
1. ________ Annie estimated that her household recycles five cubic feet of materials per
        cu ft
              collection day. There are 26 collection days per year in her town of 16,000
              households. If every household is similar to Annie‛s, how much recycled material is
              collected in one year in Annie‛s town?

2. ________ The distance from the earth to the sun is 93,000,000 miles, and light travels at
             186,000 miles per second. How many seconds does light from the sun take to reach
             the earth?

3. ________ Jorge bought a TV at a 15% discount. Later he found that there should have been
            a 20% discount, so the store gave him the $17 he was owed. What was the price of
            the TV before any discount?

4. ________ The whole numbers are arranged in an array as shown. If the pattern continues
            going down one column and up the next, what is the number at the bottom of the
            19th column?

                                1      8       9      16      ...
                                   2        7     10      15
                                   3        6     11      14
                                   4        5     12      13

5. ________ A bracelet is made by stringing together
      patterns                                                    R                        R
               four beads. Each bead is either red or
               green. How many different color patterns      R          G            G             R
               are possible for the bracelet, where patterns
               are considered the same if turning one will        G                        G
               produce the other, as shown here?
                                                                                4              4

6. ________ A rectangular 7-inch by 12-inch picture is framed with a                  7
            mat so that four inches of the mat are seen on the top and
            on each of the sides. Five inches of the mat are seen on
            the bottom. What fraction of the finished product‛s area                        12
            (picture and mat) is the picture? Express your answer as a
            common fraction.


7. ________                                     The numbers 1, 3, 6, 10, … are called triangular
                                                numbers, as shown geometrically here. What is the
                                                20th triangular number?
               1       3     6         10

8. ________ If a # b = a 2 + b and a @ b = b − a , what is the value of ((1 # 3) @ 2)?

9. ________ If y = 2x + 1, which of the following equations is true?
            A) x = 2y + 1       B) x = (½)y + 1      C) x = (½)y − ½        D) x = (½)y + ½

10. _______ A special deck of cards consists of green cards and yellow cards. The odds of
       yel cds
               randomly selecting a green card from the deck are 3:5. If there are 40 cards in the
               deck, how many yellow cards are in the deck?

MATHCOUNTS 2005-2006                                                                                   35
                                   Warm-Up 7

1.   2,080,000        (F, M)      5. 6             (E, M, P, T)      8. −2                    (F)
2. 500                   (F)      6.                    (F, M)       9. C                (E, F, G)
3. 340                (F, M)      7. 210           (F, P, S, T)      10. 25                   (G)

4. 76            (F, P, S, T)

Solution - Problem #4
     We could continue writing out the numbers and filling in the entries through the 19th column,
but perhaps there is a more efficient way of solving this problem. If we follow the progression
of the consecutive whole numbers with our pencil, we can
see that the whole numbers are written consecutively and
seem to go down one column, then up the next column, then        1 8 9 16
down the following column, and this pattern of alternating       2 7 10 15
up and down columns is continued. Since we want to know          3 6 11 14
what number is at the bottom of the 19th column, let‛s look      4 5 12 13
at the numbers at the bottom of the columns we already
know. They are 4, 5, 12 and 13. These four numbers don‛t seem to make a very obvious pattern, but
we can see that there is a difference of 8 between the first and third numbers and between the
second and fourth numbers. Notice this also is true with the numbers at the top of the columns
(1, 8, 9 and 16). We can see that this would continue since every grouping of two columns has eight
consecutive whole numbers, with the greatest one appearing at the top of the second column.
The number at the top of the sixth column, which will be the end of the third grouping of eight
consecutive integers, will be 3 × 8 = 24. The number at the top of the 18th column, which will be the
end of the ninth grouping of eight consecutive integers, will be 9 × 8 = 72. Starting down the 19th
column, we‛ll then have the numbers 73, 74, 75 and finally, 76.

Representation - Problem #7
    To generate the triangular numbers, we continue to add on consecutive integers to our growing
total. So the triangular numbers are 1, (add 2) 3, (add 3) 6, (add 4) 10, (add 5) 15, (add 6) 21, (add
7) 28, etc. It could take a long time to get to the 20th one, and there are many opportunities to
make a mistake in our calculations. Noticing the visual representation of the triangular numbers
on the preceding page, we see that the third triangular number is the number of dots needed for
a right triangle made from three rows of dots, with three dots in the bottom row. The fourth
triangular number would be the number of dots needed for a right triangle made from four rows
of dots, with four dots in the bottom row. The 20th triangular number would then be the number
of dots needed for a right triangle made from 20 rows of dots, with 20 dots in the bottom row.
Notice there will be 20 dots in the bottom row and 1 dot in the top row, for a total of 21 dots for
these two rows. The total number of dots in the second row (2 dots) and in the second-to-last row
(19 dots) is also 21 dots. This pairing continues with the third row and third-to-last row, the fourth
row and fourth-to-last row, etc. Since there are 20 rows, there will be 10 of these pairings of rows
that each have a total of 21 dots. This is a grand total of 10(21) = 210 dots.

    Another way to look at the representation of the triangular numbers is
to take the triangular representation and turn it into a rectangle. Notice
the identical triangle has been flipped and moved up to complete a rectangle.
The second triangular number produces a rectangle with two columns of
three rows, or 6 dots. Half of this is 3. The third triangular number produces a rectangle with
three columns and four rows, or 12 dots. Half of this is 6. The 20th triangular number would
produce a rectangle with 20 columns and 21 rows, or 420 dots. Half of this is 210.
36                                                                               MATHCOUNTS 2005-2006
                               Warm-Up 8
1. ________ A resort hotel charges a regular rate of $100 per night, but Wednesday nights
            are $150, and Saturday nights and Sunday nights are each $200. What is the
            price difference between the most expensive and cheapest possible stays of four
            consecutive nights?

2. ________                 There are three times the number of orange fish as blue fish in a tank
                            at the pet store, and there are no other fish. If Kaya randomly pulls
                            out one fish from the tank, what is the probability that it is orange?
                            Express your answer as a percent.

        sq cm
3. ________ A large square region is divided into four congruent, non-
              overlapping square regions. If the perimeter of one of these small
              square regions is 60 cm, what is the area of the large square?

4. ________    Mary will pick a positive integer less than 80 that is a multiple of 7. Susan will pick
               a positive integer less than 80 that is a multiple of 9. What is the probability that
               they both will pick the same number? Express your answer as a common fraction.

5. ________ A certain amount of work can be done by seven identical machines in 30 hours. How
              long will it take 10 of these same machines to do the same amount of work?

6. ________ How many 1” cubes will pack into a rectangular box with dimensions 8” by 6” by 6.5”?

7. ________ Changee is scheduling
         a.m.                                                                   Avge Time of Event
              use of the pool for the                                                (min:sec)
              2008 Summer Olympic             Women‛s Indiv Freestyle                   2:23
              Games. He has decided to
              allow 15 minutes between        Women‛s Indiv Butterfly                    3:20
              events, and he will use the     Women‛s Indiv Backstroke                   3:10
              historic average time for
                                              Women‛s Indiv Breaststroke                2:56
              each event rounded up to
              the next whole minute for       Women‛s Relay                             9:27
              the length of each event.       Men‛s Indiv Freestyle                      2:10
              He needs to schedule
              these 10 events, in order,      Men‛s Indiv Butterfly                       3:01
              beginning at 8 a.m. At          Men‛s Indiv Backstroke                    2:54
              what time will the men‛s
                                              Men‛s Indiv Breaststroke                  2:45
              relay begin, according to
              his schedule?                   Men‛s Relay                                8:15

          °F                                                 9
8. ________ Each degree in the Celsius temperature system is 5 of a degree in the Fahrenheit
             system, and 32°F = 0°C. What is the Fahrenheit temperature when the Celsius
             temperature is 45°?
9. ________ What number should be added to both the numerator and the denominator of            5   to
            get a fraction equivalent to 5 ?

10. _______ A rectangle has a perimeter of 38 cm. Three-fourths of its width is equal to one-
            fifth of its length. What is the measure of the length?

MATHCOUNTS 2005-2006                                                                                     37
                                    Warm-Up 8

1.   150           (E, G, T)        5. 21                 (F, S)        8. 113                        (F)

2. 75                 (F, M)        6. 288               (M, S)         9. 15                  (G, M, T)

3. 900                (F, M)        7. 10:52                 (T)        10. 15                  (F, G, M)
4.           (C, F, G, M, T)

Solution - Problem #5
    We can see that seven machines each working 30 hours is equivalent to 7 × 30 = 210 hours of
machine work. (It‛s the same as one machine working for 210 hours.) We need to keep 210 hours of
machine work, but now we are going to have 10 machines to do the work. The 210 hours would need
to be spread evenly over the 10 machines, which results in 210 ÷ 10 = 21 hours of work for each of
the 10 machines.

Representation - Problem #9
    We are asked to find the number that should be added to both the numerator and denominator
   1                                 4
of 5 to get a fraction equivalent to 5 . We can apply algebra to the situation and see where that
                                                         1+ x
gets us. Let x be the number we are looking for. Then 5 + x = 5 ; 5(1 + x) = 4(5 + x); 5 + 5x = 20 + 4x;

and x = 15. To check our work, we would see that our numerator would become 16 while our
denominator would become 20, and 20 = 5 .4

     Is there a way to do this without all of the messy algebra? Is there a way to see what is
happening throughout the process? Let‛s take a look at a representation of 5 (row 1), and what

happens as we add numbers to the numerator and denominator.
Row one is showing 5 of the squares as shaded. If we add

1 to both the numerator and denominator of 5 , we get
 6 , which is represented in the second row. If we add a 2 to
both the numerator and denominator of 5 , we get 7 . The

representation of this is in row 3. Notice what never changes in
the representations: There are always four squares not shaded.
This means that when we have added on enough squares to get
to a representation that is equivalent to 5 , there will still be
four squares that are not shaded. These four squares will then
represent 5 of that final row. If 4 squares are 5 of the final
             1                                     1

row, how many squares are in the final row? There must be 4 × 5 = 20. If there are 20 squares in
the row, we must have added on 15 squares to the original five squares in row 1.

Representation - Problem #10
     This one sounds very messy, and we have lots of fractions to take into consideration. However,
let‛s not go right to an equation. Instead, let‛s take a look at the following representation. We are
told that one-fifth of the length, or x, is equal to three-fourths of the width. That leaves one-
fourth of the width, and it must be equal to a third of x. Since the perimeter is 38 cm, we know
length + width = 19 cm, and we can perform a fairly simple calculation
for x: 5x + 1 3 x = 19; 6 3 x = 19 or 19 x = 19; ( 19 ) 19 x = 19( 19 ); x = 3.
              1           1                         3               3                    Length
                                       3                 3
Remember that we are looking for the length, which is 5x or 15 cm.                  x x x x x

                                                                                 3   x

38                                                                                       MATHCOUNTS 2005-2006
                                Workout 4
1. ________ Thirty-two square 8” tiles are laid next to a 12‛ by 18‛
            rectangular pool in the design shown. What percent of
            the perimeter of the pool does not have tile next to it?
            Express your answer to the nearest whole number.

2. ________ John‛s piggy bank had $1.20 when he went to bed on Monday. On Tuesday morning
              he put three coins in the bank. He put in three more coins in the afternoon and then
              three more in the evening. That night there was a total of $2.20 in his bank. If no
              coin is worth more than 25 cents, what is the greatest amount John could have put
              in the bank on Tuesday evening?
                                                                                New York Survey
3. ________ Ryan made the doughnut graph shown here using the results
        sq cm
                                                                                  Soda vs. Pop
               of a survey. If the radius of the inner circle is 3 cm and                SODA
               the radius of the outer circle is 6 cm, what is the area of
               the region representing Pop? Express your answer as a                   Terms for
               decimal to the nearest tenth.                                          Soft Drinks

                                                                              OTHER                 COKE
                                                                                2%       POP         2%

4. ________ A new motorcycle costs $6000. It loses 20% of its value by the end of the first
            year. Each year thereafter, it loses 10% of its remaining value from the end of the
            previous year. How much is the motorcycle worth after five years?

5. ________ Some farmers use circular irrigation systems, creating green circles in their fields.
            If a circular sprinkler at the center of a square field sprays as much of the field as
            possible, without spraying past the boundaries of the field, what percent of the field
            is not watered? Express your answer to the nearest tenth.

6. ________ A player has a batting average of exactly .250. What is the
            greatest amount his average could increase with one more hit?
            Express your answer as a decimal to the nearest thousandth.

7. ________ The amount of $72 is to be divided among Mr. Abbondanzio, Ms. Barta and Ms.
            Conders in the ratio 6:2:1, respectively. How much should Ms. Barta receive?

8. ________ What is the least positive fraction whose numerator is two less than a perfect
            square and whose denominator is one more than the same perfect square?

                                         6 9
                                    8                    6
9. ________ Jared earned grades of 10 , 15 , 11 , 10 and 8 on his quizzes in history. What was the
            average of his five scores on these quizzes? Express your answer as a percent to
            the nearest whole percent.

10. _______ A ½-mile long circus train is traveling at the rate of 10 mph when it reaches a tunnel
             two miles long. How many minutes was it from the time when the front of the engine
             entered the tunnel until the rear of the
             caboose left the tunnel?

MATHCOUNTS 2005-2006                                                                                 39
                                    Workout 4
1.   84              (F, P)       5. 21.5              (F, M)        8.              (G, M, P, T)
2. 60                (G, T)       6. 0.150             (G, M)        9. 69                   (C)

3. 24.6                 (F)       7. 16                (M, T)        10. 15               (F, M)

4. 3149.28        (M, P, T)

Solution - Problem #10
    This problem is a little tricky because we have to figure out how far the train has traveled
from the time the front engine entered the tunnel to the time the rear of the caboose left the
tunnel. The tunnel is two miles long, but the train has actually traveled farther than this. The
front of the train entered the tunnel, went two miles, exited the tunnel, and then went another
half-mile before the end of the caboose was completely out of the tunnel. We need to find the
number of minutes it took for this train to travel 2.5 miles. Using the equation Rate × Time =
Distance, and knowing that the rate was 10 mph and the distance was 2.5 miles, we can set up an
equation for the time (T) needed to complete the
distance: 10 × T = 2.5 or 10T = 2.5. Dividing both
sides of the equation by 10, we have T = 0.25.
Remember that our rate was in mph, our distance                ½ mile                2 miles
was in miles, and so the time is in hours. The
problem asked for the number of minutes, and
0.25 hours is a quarter of an hour, which is 15 minutes.

Representation - Problem #4
    We can approach this problem by figuring out the value of the motorcycle for each year until
we reach the end of the fifth year. We know the motorcycle loses 20% of its value after the
first year, so it keeps 80% of its value. This is (0.80)($6000) = $4800. After this first year, the
motorcycle loses 10% of its value each year, so it will keep 90% of its value. After the second
year, the motorcycle will be worth (0.90)($4800) = $4320. After the third year, its value will be
(0.90)($4320) = $3888. After the fourth year, the value is (0.90)($3888) = $3499.20. And finally,
after the fifth year, the value of the motorcycle is (0.90)($3499.20) = $3149.28.

    In this case we can see that we are performing the same multiplication process four times
over the five-year period. (Remember that the motorcycle keeps a different portion of its value
the first year.) Because of this repeated multiplication, we could save a couple of steps by using
exponents. The long procedure in the solution above is equivalent to performing the following
calculation: ($6000)(0.80)(0.90)4. With the help of a calculator, we see this is again $3149.28.

    Again remembering that we start with a value of $6000, go to $4800 after the first year, and
then keep decreasing by 10%, we could come up with an equation that we could use for any year
after the first year. Using the same logic that resulted in the expression ($6000)(0.80)(0.90)4 for
the value after the fifth year, we can see that if we use the equation y = 4800(0.90)x−1, we can plug
in any number of years (x) and determine the value of the car (y). This is helpful for answering
questions like, “At the end of which year would the value of the car first be below $2500?” We
could enter the equation Y1 = 4800(0.90)x−1 into a graphing calculator, see the ordered pairs on the
TABLE screen, and see that y is first less than $2500 when x is 8. After eight years, the value of
the car is about $2296, which is the first time it is less than $2500.

40                                                                              MATHCOUNTS 2005-2006

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