For this blog post, every member of Group 2 will be required to post a solution to certain problems on the page that says "Ten Essentials" to this blog post. The link is http://www.agmath.com/media//DIR_57727/10$20Lessons.pdf. I made Group 1 do all of these problems but you need to just do the assigned problems. Try to put all the assigned problem solutions in one blog post. Make these solutions easy to understand so everyone in group 2 can fully understand every one of these 50 problems. These problems will be due on Wednesday, November 16th by 10:00 p.m. I will be checking these thoroughly and will give NO snacks for the members of Group 2 who doesn't do this blog assignment and this time I mean it. For those who do all the problems they are assigned and the effort seems apparent, this will be an easy way to earn double snacks.
Here are the problems assigned to each person. To differentiate problems of different pages with the same number, I am using the position of the problem (e.g. Problem # 3 on page 5 will be labeled as 23):
Vignesh : 1, 15, 29, 43
Adiyan : 2, 16, 30, 44
Melinda : 3, 17, 31, 45
Geonil : 4, 18, 32, 46
Shivam : 5, 19, 33, 47
Mihir : 6, 20, 34, 48
Minhoo : 7, 21, 35, 49
Aviral : 8, 22, 6, 50
Sanjana : 9, 23, 37
Julia : 10, 24, 38
Akshara : 11, 25, 39
Lucy : 12, 26, 40
Jeffrey : 13, 27, 41
Gautham : 14, 28, 42
Mihir Khambete-Solutions to #6, #20, #34, and #48.
ReplyDeleteMihir Khambete
#6 solution: If two distinct members of the set {2, 4, 10, 15, 20, 50} are randomly
selected and multiplied, what is the probability that the product is a multiple of
100? Express your answer as a common fraction.
For this problem, the probability will be multiples of hundred/total combinations.
Here are the pairings for each number.
Number Possible Pairings
50 20,12,10,4,2
20 50,15,10
15 20
12 50
10 50,20
4 50
2 50
as you see there are 14 possible pairings.
There are 21 combinations(7c2)
so, the probability is ⅔.
#20 Solution: two-row triangle is created with a total of 15
pieces: nine unit rods and 6 connectors, as shown.
What is the total number of pieces that would be used
to create an eight-row triangle?
we’ll use a table to illustrate the connections.
Rows
Mihir Khambete
#6 solution: If two distinct members of the set {2, 4, 10, 15, 20, 50} are randomly
selected and multiplied, what is the probability that the product is a multiple of
100? Express your answer as a common fraction.
For this problem, the probability will be multiples of hundred/total combinations.
Here are the pairings for each number.
Number Possible Pairings
50----- 20,12,10,4,2
20----- 50,15,10
15----- 20
12------50
10------50,20
4------ 50
2------ 50
as you see there are 14 possible pairings.
There are 21 combinations(7c2)
so, the probability is ⅔.
#20 Solution: two-row triangle is created with a total of 15
pieces: nine unit rods and 6 connectors, as shown.
What is the total number of pieces that would be used
to create an eight-row triangle?
we’ll use a table to illustrate the connections.
one row: 3 connectors, 3 rods
two rows: 6 connectors, 9 rods
three rows: 10 connectors, 18 rods
four rows:15 connectors, 30 rods
five rows: 21 connectors, 45 rods
six rows:28 connectors, 63 rods
seven rows: 36 connectors, 84 rods
eight rows: 45 connectors, 108 rods
In the 8 rows category,.....
45+108=153 pieces.
#34 solution: The interior of a right circular cone is 8 inches tall with a 2-inch radius at the
opening. The interior of the cone is filled with ice cream, and the cone has a
hemisphere of ice cream exactly covering the opening of the cone. What is the
volume of the ice cream? Express your answer in terms of Pi.
We’ll need the formulas for finding volume of a sphere and volume of a cone.
Volume formula for cones: V= 1/3π r squared x height.
Volume formula for sphere: V= 4/3 π r cubed.
Now substitute with the values..
⅓ π(4)(8) + (1/2)4/3π(8).
=32/3π + 16/3π.
=16π inches cubed.
#48 solution: For positive integer n such that n<10,000, the number n+2005 has exactly
21 positive factors. What is the sum of all the possible values of n?
1. to fulfil the conditions...
2006≤ (n+2005)≤7995
2006 is 2005+1;n can be at least one.
7995 is 10000-2005. (for our trials we use ≤ 10000 because you can subtract 2005)
(Revision:To find the # of factors in a number, prime factorize.
Ex: 8=2 cubed.
The number of factors is not 3, but 4 since one is also a factor, so 8 has 4 factors)
For a number with 21 factors,...
21=3 x 7.
21 includes the two factors (3 to the power of 0) and (7 to the power of 0)
So, it becomes [x to the power of 6 x y to the power of 2.]( we’ll recap on this later.)
Let’s try to find a number which has 21 factors and in our range.
A case with 7 distinct primes. 2 x 3 x 5 x 7 x 11 x 13 x 17 is way greater than 10,000.No
A case with one prime factor. 2 to the power of 13 is 8192, so it has 13 factors.
13 factors is insufficient(No)
Now we’ll go back to the (x to the power of 6)(y squared)
2 to the power of 6(64)x 7 squared(49)
64x49=3136.
2 to the power of 6(64) x 11 squared(121)
64x121=7744.
3 to the power of 6(729) x 2 squared(4)
729x4=2916.
Now add. It equals 13,706.
But we’re not finished! Subtract 2005(3). = 7691.
by Mihir
ReplyDelete# 3 is incorrect. It should be 1/2. I overcounted. It becomes 7P2(thanks Ashish)
#48 a detail. 8,192 has 14 factors, since one is also a factor.
sorry again-it's 14/7p2 =1/3
ReplyDeleteThis comment has been removed by the author.
ReplyDelete~Sanjana Kothuri~
ReplyDeleteI am submitting Sanjana's post for her because she has some technical issues.
#9) A fair, six-sided die is tossed eight times. The sequence of eight results is recorded to form an eight-digit number. What is the probability that the number formed is a multiple of eight? Express your answer as a common fraction.
When you choose the 8 numbers, the last three digits have to be divisible by 8 (only 2,4,6 can be last digits). Since you are rolling a six side die, the numbers 0,7,8,9 cannot appear. Then I listed out the last three digits that are divisible by 8:
112
136
144
152
216
224
232
256
264
312
336
344
352
416
424
432
456
464
512
536
544
552
616
624
632
656
664
27 combinations.
Last 3 digits can be arranged in 27 ways. First 6 digits can be any number so each can be arranged in 6C6 = 6 ways.
total no. of ways = 6X6X6X6X6X6X27
Total possibilites for 8 digits 6^8
Probability = (6^5 X 27) / (6^8) = 1/8
#23)The square with vertices (-a, -a), (a, -a), (-a, a) and (a, a) is cut by the line 2
y ? x into congruent quadrilaterals. What is the number of units in the perimeter of each
quadrilateral? Express your answer in simplified radical form in terms of a.
When you draw the square and the line y = x/2, there are 2 congruent quadrilateral. Since the slope of the line is positive, the line passes through the origin, cutting only through the first and third quadrents. Using the x axis, the side of the square, the line y = x/2, and the the pythorgorean theorm I calculated the length of the line in the square. Since the Two quadrilaterals are congruent, the other 3 sides add up to 4a. Therefore the perimeter is
a ( sqrt5 + 4 )
# 37) The price of a stock decreased by 20% in 2001 and then decreased by 20% in 2002. What percent increase in 2003 will restore the price to its value at the beginning of 2001? Express your answer to the nearest hundredth.
Let the price of the stock be $100
The % decrease in 2001 is 20%
The price of the stock is $80
The % decrease in 2002 is 20%
The price of the stock is $64
Original price-new price= $36
The % increase in 2003 is 36/64*100
= 56.25%
13. There are 7 numbers (points). The first five numbers average 86, the last three numbers average 95, and the total average is 88.
ReplyDeleteYou really don’t need to know the last to numbers; you just need to know their sum. To find the sum, you can do:
88(7)-86(5)
There are 7 numbers and their average is 88; multiplying them together finds the sum of all the numbers. 86(5) is used to find the sum of the first 5 numbers. This leaves us with the sum of the last 2 numbers, which is 186.
Thus, the average of the last 2 test scores is 93.
27. Triangle ABC and triangle DEF are congruent, isosceles right triangles. The area of the square in ABC is 15, so one side of the square is √15. If one side of the square is √15, one leg of the triangle is 2√15. If we use the Pythagorean Theorom (a2 + b2 = c2), we know that the hypotenuse of the triangle is √120 because:
a2 + b2 = c2
(2√15) 2 + (2√15) 2 = c2
4(15) + 4(15) = c2
120 = c2
√120 = c
One side of the square in DEF is 1/3 of DF, so the area of the square is:
(√120/3)2
120/9
40/3
The area of the square inscribed in DEF is 40/3.
41. Using the given information, we can use algebra to solve this problem. g=girl b=boy
g+b=1200
2/3g+1/2b=730
1/3g+1/2b=470
1/3g=260
2/3g=520
520 girls attended the picnic.
11. There are 6 classes with 25 so 6*25=150 students. There are 4 classes with 20 students so 4*20=80 students. Lastly, there are 2 classes with 35 so 2*35=70 students. Altogether that is 300 students. Out of that 150 students will say there are 25 in their class (150 25s (150*25=3750), 80 will say 20(80 20s (80*20=16000, and 70 will say 35(70 35s (70*35=2450). So when you add all this up, you get the total for all the scores(7800). To find the average, we divide by the number of questionnaires turned in, which is 300. 7800/300=26
ReplyDeleteThe average of the numbers turned in is 26 (students).
10) Chris and Ashley are playing a game. First, Chris removes one number from the set {1,2,3,4,5}. Then Ashley picks two of the remaining numbers at random without replacement. Ashley wins if the sum of the two numbers are not prime. Otherwise, Chris will win. What number should Chris remove to maximize his chance of winning?
ReplyDeleteSo basically, you will try to help Chris by removing a number that is common to make a sum that is composite. Composite sums are like 4,6,8, and 9.
List them out:
1+3=4
1+5=6
2+4=6
3+5=8
4+5=9
Then, we'll list out how much times each number appears in the list.
1 2 3 4 5
2 1 2 2 3
So, if Chris chooses 5, he'll have a better chance of winning against Ashley.
Hey Ashish this is Melinda[: Just saying, my work does not show itself on blogger for some reason so I'll just email you. Thanks! [:
ReplyDelete24) The interior of a right circular cone is 8 inches tall with a 2-inch radius at the opening. The interior of the cone is filled with ice cream, and the cone has a hemisphere of ice cream exactly covering the opening of the cone. What is the volumne of the ice cream? Express your answer in terms of π.
ReplyDeleteSo you have to know that in order to find out the volumne of a circular cone, the equation is simply:
1/3 x h x b
So the formula for this problem is really:
1/3 x 8 x 4π
...which leads to 32/3π.
But, there's a hemisphere of ice cream covering the top of the ice cream cone. The formula for the volumne of a sphere is:
4/3 π x r^2
So really for half sphere it'sL
1/2 x 4/3 x 8/1 x π...
(16/3 + 32/3)π = 16π.
The answer's 16π.
25. The triangle was divided equally into eight pieces. To be equal, each triangle must have an equal base of 1 inch because they all have a height of 10inches so in order to have equal areas, bases need to have the same measure. The outer triangles would have the greater perimeter because when the area has to be the same, the lines of the triangle have to reach from farther out to the top (center). Now, you can use the Pythagorean Theorem to find the length of the outer most line of the small triangle. You can split big the triangle in half to make 2 equal right triangles with base 4 and height 10. So to find the hypotenuse(which is the outermost segment of the small) one you square both measures and add them up (4^=16, 10^=100, 16+100=116). Now you need to find the square root of that (√116=10.770329... round to 10.77). Now, you know the measure of 2 sides of the triangle: 1inch, 10.77inch. Now to find the second outermost line's measure, you apply the Pythagorean Theorem again. This time, though, another inch is not counted because it is outside the triangle formed by the line you are trying to find the length of so this time it is 3squared plus 10squared (height stays the same)(3^=9, 10^=100, 100+9=109, √109=10.440306...round to 10.44) Now you have the measures for all three sides of the triangle. Add them up to get the perimeter. 10.44+10.77+1=22.21
ReplyDeleteThe perimeter of the triangle with the largest one is 22.21inches.
39. Each animal lives for 1,000,000,000 heartbeats. If a shrew's heartbeat is 800 beats/minute. To find out how many minutes a shrew will live, you divide 800(beats a minute) by 1billion. You get 1,250,000minutes. You do the same thing for the elephant's life. 1,000,000,000beats/25beats per minute=40,000,000minutes in life. Now you can subtract 1,250,000 from 40,000,000. 40,000,000-1,250,000=38,750,000. An elephant will live for 38,750,000 more minutes than a shrew, but the question is asking for the answer in years. First, 60 minutes are in an hour so divide by 60 to get the number of hours. 38,750,000/60=645,833.3333....This is how much longer an elephant will live than a shrew in hours. There are 24 hours in a day so divide by 24 to get how many more days. 645,833.333.../24=26,909.722.... This is in days so divide that by 365 days in a year. 26,909.722.../365=73.725265...This is in years. Round this to the nearest whole number to get 74 years. An easier way to do the division would have been to multiply all the numbers you had to divide by and then divide by the product.
ReplyDeleteAn elephant will live for 74 years longer than a shrew.
I forgot to post the questions along with my solutions so here they are:
ReplyDelete11.The math department at a small high school has six classes with 25 students each, four classes with 20 students each, and two classes with 35 students each. No student takes more than one math class. If each student correctly fills out a questionnaire asking for the total number of students in his/her math class, what is the average of all the numbers turned in on the questionnaires?
25.Steve has an isosceles triangle with base 8 inches an height 10 inches. He wants to cut it into eight pieces having equal areas, as shown. To the nearest hundredth of an inch, what is the number of inches in the greatest perimeter among the eight pieces?
39.The average heart rate of a shrew is 800 beats per minute, as compared to an elephant with a heart rate of 25 beats per minute. If 1 billion heartbeats is a natural life span for each animal, on average, how many more years do el- ephants live than shrews? Assume each year is 365 days. Express your answer to the nearest whole number.
My solutions are labeled by numbers above.
Problems: 1, 15, 29, and 43
ReplyDelete1. A math conference is presenting a lecture series with six different lecturers. If Dr. Smith’s lecture depends on Dr. Jones’ lecture, so that Dr. Smith must be scheduled at some time after Dr. Jones, in how many orders can the six lectures be scheduled? (2006 National Team #9
First I listed the ways that Dr. Smith and Dr. Jones can be arranged without the other people. 1,2 1,3 1,4 1,5 1,6 2,3 2,4 2,5 2,6 3,4 3,5 3,6 4,5 4,6 5,6= 15 ways. Then I multiplied that by the ways the next 4 people can be arranged 4P4=24. 15x24 is Answer:360 ways.
15. Brad bicycles from home at an average speed of 9 miles per hour until he gets a flat tire. With no way to fix the tire, brad walks his bike back home by the same route, averaging 3 miles per hour. If the entire round trip of biking and walking took a total of 6 hours, what was Brad’s average speed in miles per hour for the entire round trip? Express your answer as a decimal to the nearest tenth. (2003 National Sprint #23)
First I found that when you bike 1.5 hours and walk 4.5 hours, the total time equals six and the miles biked (13.5) is the same as miles walked using guess and check. I then found the average by adding 4 hours x3 miles, 1.5 miles in 1/2 hour for walking and 13.5 miles in 1 1/2 hours walking. Together they equal 27 miles. You then divide it by the 6 hours and get Answer:4.5 mph.
29. A triangle has vertices at (-3,2), (6, -2) and (3,5). How many square units are in the area of the triangle? Express your answer as a decimal to the nearest tenth. (2003 National Sprint #11)
First I plotted the points on a coordinate plane that I drew on my graph paper. I then surrounded it with a rectangle that touche the vertices of the triangle. The rectangles area was 63 sq. units. I then found the areas of the triangles that the space left by the original triangle inside the rectangle form and subtracted them from the total area. The 3 triangles in empty space had areas of 10.5, 9, and 18. When subtracted they equal Answer: 25.5 sq. units.
43. If 2x - 9y=14 and 6x=42+y , what is the value of the product xy? (2005 National Sprint #11)
First I simplified 6x=42+y to x=7+y. Then I used guess and check to find that 2x-9y=14 is solved by 7=x and 0=y. Since x is still 7 more than y, this works. Since it says what is xy, 7x0 equals Answer :0
38) The line y = b-x with 0<b<4 intersects the y-axis at P and the line x=4 at S. If the ratio of the area of triangle QRS to the area of triangle QOP is 9:25, what is the value of b?? Express the answer as a decimal to the nearest tenth.
ReplyDeleteSo really, having the actual graph helps a lot, but I'm just going to try my best to explain with words only.
So what I did was I knew that 4-QR = OQ. Then, I knew also that (OQ^2)/2 equals the area of triangle QOP. But let's just describe QR as the variable x.
QR x RS x^2
------- = --- = (3/5)^2
OP x OQ (4-x)^2
Use cross product:
x 3
---- = --
4-x 5
5x = 3(4-x)
5x = 12-3x
Therefore, 12 = 8x, so x = 12/8.
So the final answer is x = 1.5
Sorry, the fractions are all messed up.
ReplyDelete38) Oops, I forgot to find out the value of b. So in order to find b (which is the y coordinate of P), you have to continue by solving OQ. OQ would equal 4 - 1.5.
ReplyDeleteThat equals: 2.5. Since OQ equals to OP, OP is also 2.5. Therefore, the coordinate for P is (0, 2.5), but you only need the y coordinate so b is 2.5 .
#12. A collection of nickels, dimes, and pennies has an average value of $0.07 per
ReplyDeletecoin. If a nickel were replaced by 5 pennies, the average would drop to $0.06
per coin. What is the number of dimes in the collection?
A=total # of coins
Sum/A=0.07
Sum/A+4=0.06
Sum=0.07A
S=0.06(A+4)
0.07A=0.06(A+4)
0.07A=0.06A+0.24
0.1A=0.24
A=24
24 times 0.07=1.68
Sum=1.68
168 cents
10D+5N+P=168
16 dimes, 1 nickel, 3 pennies
160+5+3=168.
$1.68
Answer: 16 dimes.
*there can also be other answers.
#40. . Pipe A will fill a tank in 6 hours. Pipe B will fill the same tank in 4 hours.
ReplyDeletePipe C will fill the tank in teh same number of hours that it will take pipes A and
B working together to flill the tank. What fraction of th tank will be filled if all
three pipes work together for one hour?.
Tank A=T/6
Tank B= T/4
Tank C= T/6+T/4\
T(1/6+1/4+1/6+1/4)
T(20/24)
5/6 Tank
Answer: 5/6
#26. I can give it in class on Friday.
ReplyDeleteI am posting for Shivam. Here are his answers.
ReplyDeleteSolutions to Problems #5, #19, #33, and #47
#5. Question: For how many days in a century is the date increasing sequence of consecutive digits?
The date is in the format: mm/dd/yy. This problem requires some casework.
We can start with the month being a two digit number (Case 1). The only two digit number that would work (out of 10, 11, 12) is 12.
Next , for the dd, we can substitute either 3, or 34, because days can be 1 digit, or 2 digits. However, 34 cannot work according to our calendar. For the year, the only option left is 45. Thus the date becomes 12/3/45.
Now, for Case 2, we can take the month as one of 9 1-digit numbers. However, after the 6th month, the date doesn’t line up with our calendar. So, we have only 6 months to choose from.
For the first month, “1” we can have the day as 2, or 23. (1/2/yy or 1/23/yy). However, for the other 5 numbers, we only have 1 case. This makes 5 + 2 + 1 dates. So, there are 8 dates.
#19. Question: What is the first value that exceeds 5000?
The starting term: 1
A term in the sequence is the sum of the elements of the series preceding it.
Therefore, the series becomes: 1, 1, 2, 4, 8, 16, 32 …
We can see that starting after the 2nd 1, all of the integers are powers of 2.
(i.e. 2^1=2, 2^2=4, 2^3=8…)
We know that 2^10 is 1024, and so, by a little arithmetic, we can find that 2^12 is 4096.
Therefore, the smallest term in this series after 5000 is 2^13, or, 8192.
#33. Question: What is the measure of the angle of the sector that was cut away?
The radius of the cone formed is 12 cm, and the volume is 432 π cubic cm. By manipulating the Volume formula for a cone, we can find the height to be 9 cm. With the Pythagorean Theorem, we can find that the slant height is 15 cm.
So, the area of the circle (entire circle) is 225 π square cm. Next we can find the surface area of the cone, not including the base. This, by using the formula, comes to be 180 π square cm. By subtracting the area of the circle from this, we find the area of the sector.
Now, we can use the sector area formula to find the measure of the angle inside the sector. By some siple calculations, we find that the angle, which I denoted as theta, to be 72 degrees.
#47. Question: What is product of the two smallest factors of (2^1024) – 1?
We can start with the fact that this is a difference of 2 squares [of the form (a + b)(a – b)]
Also, 1024 is simply 2^10. So, there will be 11 factors. The greatest will be: (2^512 -1).
The next one will be (2^256 -1), and so on, until you reach (2^0 + 1)(2^0 – 1).
This is simply (1 + 1)(1 – 1). So, the product of the two smallest factors is 0.
I am posting for Adiyan. Here are his answers.
ReplyDelete2. On the refrigerator, MATHCOUNTS is spelled out with 10 magnets, one letter
per magnet. Two vowels and three consonants fall off and are put away in a
bag. If the Ts are indistinguishable, how many distinct possible collections of
letters could be in the bag?
Ans- If there are two Ts in the bag, the third consonant can be any of the other 5.
If there are not two Ts, the consonants in the bag can be any three of six possibilities. So the total number of consonant combinations is
5 + 6C3 = 5 + [6! / (3! 3!)] = 5 + 20 = 25
and the total number of possible collections of letters in the bag is the product of the vowel and consonant combinations:
3 * 25 = 75
16. I have no idea
30. Among all triangles with integer side lengths and perimeter 20 units, what is
the area of the triangle with the largest area? Express your answer in simplest
radical form.
Ans-
In geometry, Heron's (or Hero's) formula states that the area A of a triangle whose sides have lengths a, b, and c is
+++A+=+sqrt%28s%28s-a%29%28s-b%29%28s-c%29%29+
Using this formula and plugging all the values in after simplifying the expression you end up with 100/9 * the square root of three
Which is 19.245009
I don't know how to put in simplest radical form
44. The student council sold 661 T-shirts, some at $10 and some at $12. When
recording the amount of T-shirts they had sold at each of the two prices, they
reversed the amounts. They thought they made $378 more than they really did.
How many T-shirts actually were sold at $10 per shirt?
Ans- The equation I used was
12x+ 10 (661-x)= 10x+ 12 (661-x)+378
12x-10x+6610= 10x-12x+7932+378
4x=1700
x=425
4. Three-fourths of the students in Mr. Shearer's class have brown hair and xis sevenths of his students are right-handed. If Mr. Shearer's class has 28 students, what is the smallest possible number of students that could be both right-handed and have brown hair?
ReplyDeleteFor this problem, let's say x is the right handed and brown haired. I first calculated 3/4 and 6/7 of 28. It came out as 21 and 24. If it takes 21 (3/4) are brown haired, the left including x will be 7. If it takes 24 (6/7) are right handed, the left including x will be 4. Then it means that x is 29-7-4 which is 17.
Smallest possible number of students that could be be both right handed and have brown hand is 17.
18) What is the sum of the seven smallest distinct positive integer multiples of 9?
i used the formula for sums of arithmetic equation.
s=n[(A1+An)/2]
but I needed An which is An=A1+(n-1)d
An came out as 63 and S came out as 252.
32) The cube shown has edges of length 2cm. What is the area of triangle BCD? Express your answer in simplest radical form.
If I look each at a one side, each side has length of 2√2 due to the hypotenuse theorem. If I look at the triangle and divide in half, one half of the triangle get a base of √2 which makes the height √6. Since we found the height and base, the answer in radical form comes out in 2√3.
46) jan is thinking of a positive integer. Her integer has exactly 16 positive factors, two of which are 12 and 15. What is Jan's number?
I first knew that it had to be 60, 120, 180, 240, .... because of the greatest common factor of 12 and 15.
i tried 60 but it didn't have 16 positive factors.
For 120, I listed out factors:
12,10 15,8 1,120 2,60 3,40 4,30 24,5 6,20
Number 120 had 16 factors and so Jan's number is 120.
Since a multiple of 100 must have 2 factors of 2 and 2 factors of 5, we can count the pairs by focusing on the factors of 5. For one thing, 50 can be paired with any number that has one factor of 2, since $50=2 \cdot 5^2$ takes care of all the other primes. So, 50 can be paired with 2, 4, 10, 12, and 20, for 5 pairs. Then, 20 can be paired with (excluding 50 which we already counted) 15 and 10, both of which have the necessary factor of 5, giving us 2 more pairs. There are no remaining pairs of numbers 15 and smaller that are multiples of 100, because the only pair with two factors of 5, $\{10, 15 \}$, lacks a factor of 2. So, there are $5+2 = 7$ pairs. And in total, there are ${7 \choose 2 } =21$ possible pairs, giving us a probability of $\frac{7}{21} = \boxed{\frac{1}{3}}$.
ReplyDelete