Weird but Delicious Math Questions

Check out Mathcounts here--the best competition math for middle school mathletes.

Problem: (Solution below)
#1: 1993 Mathcounts National Team Round #4 :The teacher whispers positive integer A to Anna, B to Brett, and C to Chris. The students don't know one another's numbers but they do know that the sum of their numbers is 14. Anna says: "I know that Brett and Chris have different numbers." Then Brett says: "I already knew that all three of our numbers were different." Finally, Chris announces: "Now I know all three of our numbers." What is the product ABC?

#2: 2000 AMC10 #22: One morning each member of Angela’s family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?

Solution: 

#1: For Anna to know that Brett and Chris have different numbers, she must have an odd number
because 14 - odd = odd and you can only get odd sum of two numbers if they are different, one odd
one even.

For Brett to say that he already know all three have different numbers, he not only must have an odd number, but also his number has to be larger or equal to 7 and is not the same as what A have.

Otherwise, A-B-C could be 1-1-12; 3-3-8; or 5-5-4.
It would exceed 14 if you have 7-7-__. (All numbers are positive) so Brett"s and Anna's numbers must
be different.

If Brett has 7, then the numbers could be A-B-C = 1-7-6 ; 3-7-4 or 5-7-2.

If Brett has 9, then the numbers could be A-B-C = 1-9-4; or 3-9-2.

If Brett has 11, then then numbers could be A-B-C = 1-11-2.

From the above possibilities you know Chris has to have 6 for him to be sure he knows all the numbers. 

So A-B-C = 1-7-6 and the product of ABC = 1 x 7 x 6 = 42

#2: Let there be m cups of mild, c cups of coffee. and x people in Angela's family.
According to the given, you can set up the following equation: 
  m + c = = 8. Two ways to solve this equation.

Solution I:

m + c =   Both sides times 12 x

Both m and c need to be positive so the only x that works is when x = 5.

Solution II: 

m + c  = 8, Both sides times 12 and you have 

3m + 2c = 96; m + c is a multiple of 8.

30      3

28      6

26      9

.

.

.

2       45;  from  (30 + 3 ) = 33 to (2 +45) = 47 only 40 is a multiple of 8

and 40 divided by 8 = 5 so 5 is the answer.

 

Mental Math Tricks I

Multiples of 11:

11 x 14 = 1 ___ 4 , the middle number is (1 + 4) so the answer is 154

Let's try a few calculations mentally.

11 x 33 = 3 __ 3, the middle number is ( 3 + 3 ) so the answer is 363

11 x 52 = 5 __ 2, the middle number is ( 5+2 ) so the answer is 572

11 x 27 = 2 __ 7 , the middle number is ( 2 + 7 ) so the answer is 297

What about if the sum of the middle number is > 9? Then you carry over the "1" to the digit on its left
as you are doing the addition.

Examples:

11 x 67 = 6 __ 7, the middle number is (6 + 7) = 13 so the answer is 737

11 x 89 = 8 __ 9, the middle number is (8 + 9) = 17 so the answer is 979

11 x 47 = 4 __ 7, the middle number is (4 + 7) = 11 so the answer is 517

Harder trick:

11 x 223 = ?  2 _ _3, Write the two digits on the far left and right. Now add two numbers together,
starting with the unit digit. When the sum is  larger than 9,carry over to the next digit to the left.
2 _5(3+2) 3;  2 4 (2 + 2) 53 so the answer is 2453.

11 x 40532 = 4_ _ _ _ 2; 4_ _ _ 5 (2 +3) 2;  4_ _ 8(5+3)52;  4_ 5(0+5)852;  44(4 +0)5852,
so the answer is 445852  (This is fun!!)

Other interesting pattern:

11 x 11 =  121
111 x 111 = 12321
1111 x 1111 = 1234321
etc… till  111111111 x 111111111 = 12345678987654321

Divisibility rules for 11: 

The difference  of the sum of alternative digits is a multiple of 11, including “0”                                               (0 x 11 = 0, a multiple of 11.)

Example:

61985   (6 + 9 + 5) – (1 + 8) = 11   The number is divisible by 11.

7469     (7 + 6) – (4 + 9) = 0   The number is divisible by 11.

Try these mentally:(Answers below.)

#1: 11 x 23 =

#2: 11 x 72 =

#3: 11 x 97 =

#4: 11 x 55 =

#5: 11 x 76 =

#6: 11 x 60 =

#7: Sum of the first multiples of 11 smaller than 150.

#8:11 x 3421 =

#9: 11 x 452360 =

#10:  11 x 204673=

#11: 11111 x 11111=

#12: 1111111 x 1111111=

#13: What is the sum of the digits of 11111111 x 11111111?

#14: What is n if 45732n is divisible by 11?

#15: How many solutions for distinct numbers A and B if 4A8B is divisible by 11?

#16: How many solutions for distinct numbers A and B if A7B2 is divisible by 11?  What is their sum?

#17: How many solutions for distinct numbers A and B if A3B41 is divisible by 11?  

Answer key:
#1: 253
#2: 792
#3: 1067
#4: 605
#5: 836
#6: 660
#7 1001 The easiest way to do this is to see that this is an arithmetic sequence, starting with 11, 22, 33...
143 (11 x 13). There are 13 terms and the median is 77 so 13 x 77 or 13 x 7 x 11 = 1001
To do Mathcounts well, you need to know 7 x 11 x 13 =1001 by heart.
#8: 37631
#9: 4975960
#10: 2251403
#11:  123454321
#12:  1234567654321
#13:  The number is 123456787654321 so the sum of the digit is (4 x 7) x 2 + 8 = 64
#14:  n = 5
#15:  A + B = 12 (9, 3); (8, 4); (7, 5); skip (6, 6) because A and B are distinct (5, 7); (4, 8), (3, 9)
#16:  90, 81, 72, 63, 54, 45, 36, 27, 18;  A can’t be “0” so there are 9 pairs and the numbers are equally spaced, an arithmetic sequence. Thus the sum is median times how many numbers.                                      54 x 9 = 486
#17: A + B + 1 = 7 so A + B = 6 ; There are (6, 0);(5, 1); (4, 2); (2, 4); (1, 5) 
A + B + 1 = 7 + 11 = 18; A + B = 17; There are (9, 8) and (8, 9) so total 7 pairs

Mathcounts strategies: Some sums

The following sequences are all arithmetic sequences and for any arithmetic sequences, the sum is 

always average times the terms (how many numbers). 

To find the average, you can use 

a. sum divided by how many numbers.
b. average of the first and the last term, the second first and the second last term, or the third first and the third last term, etc...

Sum of the first consecutive natural numbers:

1 + 2 + 3 + 4 + 5 + ... + ( n -1 ) + n = \( \dfrac {n\left( n+1\right) } {2}\) 

Examples : 

#1: 1 + 2 + 3 + ... + 100 = \(\dfrac {100\left ( 101\right) } {2}=5050\)

#2: 1 + 2 + 3 + ... + 27 = \( \dfrac {27\left( 28\right) } {2}=378\)

#3: 4 + 8 + 12 + ... + 80 = 4 (1 + 2 + 3 + ... + 20) = \(\dfrac {4\times 20\times 21} {2}=840\)

Sum of the first consecutive natural even numbers: Proof without words 

Have you noticed in this sequence, every number is double the numbers in the first example,

so you don't need to divide by 2.  

\(2+4+6+\ldots +2n=n\left( n+1\right) \)

Examples: 

#1: 2 + 4 + 6 + ... 100 = 50 * 51 = 2550

#2: 2 + 4 + 6 + 8 + 10 + 12 = 6 x 7 = 42 

In this case, you can also find the midpoint, which is 7 and then 7 x 6 = 42

#3: 2 + 4 + 6 + ...420 = 210 * 211 = 44,310

Sum of the first consecutive odd numbers: Proof without words

In this special case, the mean is the same as how many numbers.

So it's easier if you find the mean by averaging the first and the last term and then square the mean.

1 + 3 + 5 + ... + ( 2n -1) = \(n^{2}\)

Examples: 

#1: 1 + 3 + 5 + ... + 39 = \(\left[ \dfrac {\left( 39+1\right) } {2}\right] ^{2}\) = (20^{2}\)

#2: 1 + 3 + 5 + ... + 89 = \(\left[ \dfrac {89+1} {2}\right] ^{2}=45^{2}=2025\)

#3: 1 + 3 + 5 + ... + 221 = \(\left[ \dfrac {221+1} {2}\right] ^{2}= 111^{2}=12321 \)

Applicable problems:

#1: What is the sum of the first 40 consecutive positive integers? 

#2: What is the sum of the first 40 consecutive positive even integers?

#3: What is the sum of the first 40 consecutive positive odd integers? 

#4: How many times does a 12-hour clock strikes in one day if it strikes once on one o'clock, twice on two o'clock, etc...?

#5: At a game show,you win $100 for the first correct answer and $200 for the second correct answer, etc. How much do you win if you answer 8 questions in a row correctly? 

Answer key:

#1: 820

#2: 1640

#3: 1600

#4: 156

#5: 100 + 200 + 300 ... + 800 = 100 (1 + 2 + 3 + ...8) = 100* \(\dfrac {8\times 9} {2}\) =  3600 

Painted Cube Problems: Beginning Level

Please refrain yourself from checking the answers too soon. Use Lego blocks, unit cubes, or Rubik's cubes to help you think. If you are really stuck, check out this link on painted cube problems.

Please comment and help me make my blog more user friendly. Thanks a lot!! 

Painted Cube Problem, competition math, problem solving by Mrs. Lin

Answer Key to the Painted Cube Problem

Polygon Part II: Interior/Exterior angles, Central Angles and Diagonals and Practice Problems

Learn the Basics here: Polygon Part I

Some important notes:

#1: Sum of all the interior angle of an n-sided polygon is \((n-2) * 180\)

#2: To find one interior angle of an n-sided regular polygon, you use :

\(\frac{(n-2)*180}{n}\)

or \({180 -\frac{360}{n}}\)", the latter will always give you one exterior angle of a regular n-gon.

#3:  Interior and its exterior angles are supplementary to each other. 

Interior angle A + exterior angle A = 180 degrees. 

#4: In every convex polygon, the exterior angles always add up to 360 degree.

Proof without words

#5: The central angle of a regular n-sided polygon is : \({\frac{360}{n}}\), same method as finding the exterior angle of a regular n-gon. Get more details on central angle here.

#6: Since \({\frac{360}{n}}\) will give you one exterior angle of a regular n-gon , 360 divided by one exterior angle of a regular n-gon will give you how many sides of that polygon. 

#7: To find how many diagonals an n-sided polygon has, you use: 

\(nC2 - n\) (Any two vertices except the sides will render one diagonal; however, order doesn't matter, thus choose 2.)

b. \( \frac{n(n-3)}{2}\) Any vertex, except its neighboring vertices and itself, can connect with other vertex to form a diagonal and there are n vertices; however, since order doesn't matter, AC is the same as CA so you divide the number by 2.

Questions to ponder: (Answer and solutions below.)

#1: The sum of the diagonals of two regular polygons is 44 and the sum of each of their interior angles is 264, what is the sum of their sides?

#2: If an exterior angle of a regular n-gon is 72, what is the measure of its interior angle? How many diagonals does that n-gon have? 

#3: What is one interior, exterior angle as well as how many diagonals are there for a 20 sided regular polygon?

#4: If each of the exterior angle of a regular polygon is 30, how many sides does that polygon have? 

#5: If the sum of all the interior angles of a polygon is 1440, how many sides does the polygon have? 

#6: How many degrees are there in the sum of a pentagon + a heptagon + a nonagon?

#7: The sum of the interior angles of a regular polygon is 720, what is the measure of one interior angle of that polygon? 

#8: 2000 Mathcounts State Target #8: (Check out Mathcounts here) : The total number of degrees in the sum of the interior angles of two regular polygons is 1980. The sum of the number of diagonals in the two polygons is 34. What is the positive difference between the numbers of sides of the two polygons?

                                                                               

# 9: Both pentagon and hexagon are regular. What is angle ABF?

#10:   B is the center of this regular pentagon. 

What is angle A, B and C? 

Answer key and solutions:

#1:  16: Learn the common polygon property by heart and check what the question is asked for.

In this case, let's see a few common polygons:

pentagon   5 sides     diagonals    5     interior angle  108 degrees

hexagon    6 sides     diagonals    9     interior angle  120  degrees

octagon     8 sides     diagonals   29    interior angle  135  degrees

nonagon    9 sides     diagonals   27    interior angle  140 degrees

decagon    10 sides   diagonals   35    interior angle  144 degrees

so the two polygons asked are hexagon and decagon, the sum of their sides are 6 + 10 = 16

#2: Interior and exterior angles are supplementary so 180 - 72 = 108 degrees  for its interior angle.

It's a pentagon and there are 5 diagonals in a pentagon.

#3: \(\frac{360}{20}\) = 18 degrees for the exterior angle

180 - 18 = 172 degrees for the interior angle of a 20-sided polygon.

#4: \(\frac{360}{30}\) = 12 sides 

#5:\(\frac{1440}{180}\) = 8 ; 8 + 2 = 10. It's a decagon (10 sides) -- when you find the sum of the interior angles you use (n - 2) * 180 so now you do the reverse.

#6: [ (5-2) + (7-2) + (9-2)] x 180 = (3 + 5 + 7) x 180 = 5 x 3 x 180 = 2700 degrees.

#7:\(\frac{720}{180}\)= 4 and 4 + 2 = 6 so this is a regular hexagon and one of its interior is \("\frac{720}{6}\) = 120 degrees. 

Or you can also do \(\frac{720 + 360}{180}\)

= 6 because all the interior angles and their exterior angles are supplementary and the sum of any exterior angle of a convex polygon is 360 degrees.

Using this method, you get how many sides instantly.

#8: This one is tricky because there are actually two different polygon pairs that the total number of degrees add up to 1980 degrees. One is a regular hexagon and a nonagon.  (720 + 1260)

Another is a heptagon and an octagon (900 + 1080). Looking at the sum of their diagonals, the latter is the right pair.  (14 + 20 = 34) so the answer is 8 - 7 = 1

#9: 360 - 108 - 120 = 132 degrees. 

#10: To find the central angle B, you do  \(\frac{360}{5}\)

= 72 degrees. BA and BC are both radius so the angle is

congruent. \(\frac{180-72}{2}\) = 54 degrees

Prime Factorization : Part II

   Learn basic facts here.











Another fun trick is to find the total number of factors any number has. To do this, take the prime factorization of the number, add 1 to each of the exponents of the prime factor, and multiply these new numbers together.

Example:

#1: 12 = 2² x 3¹
(2+1) x (1+1) = 3 x 2 = 6
Therefore, 12 has 6 factors. {1, 2, 3, 4, 6, 12}

#2: 8= 2³ , (3+1) = 4 Therefore, 8 has 4 factors. {1, 2, 4, 8}

Problems to ponder: answers and solutions below.
1. What is the smallest number with 5 factors?
2. What is the smallest number with 6 factors?
3. What is the smallest number with 12 factors?
4. How many factors does any prime number have?
5. How many even factors does 36 have? (tricky question)
6. How many perfect square factors does 144 have? ( don’t just list them, think how you get the answers)









Answers:
1. 16  Since 5 itself is a prime, the smallest number that has 5 factors would look like this: n⁴  put in the smallest prime number and the answer is  2⁴ = 16 

2. 12  The number 6 can be factor as 6 x 1 or 3 x 2 so you can have either n⁵ or x² * y¹
so either 2⁵ or 2² * 3 so the answer is 12. 

3. 60  12 = 12 x 1 = 6 x 2 = 4 x 3 = 3 x 2 x 2  The last would give you the smallest number which is  
2² * 3¹ * 5¹ = 60

4. 2 Any prime number has two factors, which are 1 and itself. The smallest prime number is 2, which is the oddest prime -- the only even number that is a prime. (Why??)

5. 6  Any even number multiples another integer will give you another even number. To get only even factors, you  need to always leave the smallest even number, which is 2, with all the other factors. 
36 = 2² * 3²
= 2 (2¹ *3²)  There are(1 + 1) ( 2 + 1 ) = 6 even factors

6. 6  The number 144 = 2⁴ * 3²
To get only square factors, you need to keep the smallest square number of all the prime numbers, leave out the others that are not square and find how many factors that new arrangement has. 
( 2²)² * (3²)¹  There are (2 + 1) ( 1 + 1)= 6 square factors.

See what they are on the left.

Polygons: Part I: Interior and Exterior angles, Sum and Diagonals

Some helpful links: Looking at Polygons by Kadie Armstrong, a mathematician.

Year 7 Interactive Maths: Polygons

Notes below. It takes a few seconds to download the pages. Please be patient.
Send me feedback or notify me of errors. Answer key is below for self-study.Polygons, Interior Angles, Exterior Angles, Sum













Answer Key to Polygons, Interior Angles, Exterior Angles and Sum

Math Cartoons from Calvin and Hobbes by Bill Waterson

 by Bill Watterson 

Solutions:
Calvin and Hobbes Cartoon 2You and Mr. Jones are both traveling towards each other. Mr. Jones is traveling at 35 mph, and you are traveling at 40 mph. In other words, every hour, the distance between you to lessens by 35 + 40 = 75 miles.

You can therefore think of this problem as one car covering 50 miles while going at 75 mph. The time this will take is distance/rate, or 50/75 = 2/3 hours.

2/3 hours times 60 minutes/hour = 40 minutes.

Add this to the starting time of 5:00, and you will pass Mr. Jones at 5:40.

Calvin and Hobbes Cartoon 

There are actually two answers ( 10 or 10/3 ) to this problem, because there are two different ways the points can be lined up.

If the points are in the order A-B-C and BC = 5 inches (given) , which means AB must also be 5 inches. This way, AC will be 10 inches, which satisfies the condition that AC = 2AB.

If the points are in the order B-A-C and BC = 5 inches, which means that AB + AC = 5 inches (make sure you see why – the parts of a whole add up to the whole) and, since AC = 2AB, AB = 5/3 and AC = 10/3 inches.

Once again, Jack and Joe are driving towards each other. Since Jack is driving at 60 mph, and Joe is driving at 30 mph, the distance between them is decreasing by 90 mph. Therefore, the combined result is one car driving at 90 mph.

Distance = rate x time, so distance = 90 mph x 10/60 hours (10 mins is 10/60 or 1/6 of an hour) The answer is 15 miles.

Here are some other problems that you can try. Have fun. (Answer key and solutions below.)

#1: Mary drove 50 miles at an average speed of 40 miles per hour to her friend's house, and drove back home on the same route at an average speed of 60 miles per hour. What was her average speed for the entire trip, in miles per hour? 

#2: John traveled 60 miles an hour and Edwards traveled 45 miles an hour. Both departed at the same spot in the same direction. After 45 minutes of driving, how far were they apart?

#3: A car drives in a circular path with a diameter of 800 feet. If the car completes exactly 2 laps each minute, what is the car's speed, in feet per second? Express your answer to the nearest whole number.

#4: If it takes you an hour total to go to work and your driving speed is 30 miles per hour to work and 
45 miles per hour back home, how far away is your home to the office where you work? 

#5: Driving to work, if your speed is 20 miles per hour, you'll be 10 minutes late; however, if your driving speed is 30 miles per hour, you'll be 10 minutes earlier. What speed will get you to work on time?

#6: If you walk from your house to the school at the rate of 4 km/hour, you'll reach the school 15 minutes earlier than the scheduled time. If you walk at the rate of 3 km/hour, you'll reach the school 15 minutes late. What is the distance of the school from your house? 















 Answer key and solutions: 
#1. Solution I:
Average speed, or rate, is equal to total distance divided by total time: R = D/T.
We know the total distance: Mary drives 50 miles to her friend’s house, and 50 miles back, for a total distance of 100 miles.

Finding the total time is a little trickier. Here, the equation has to be split up into two parts. Just as R = D/T, RT = D, and T = D/R (you can find these equations simply by rearranging the variables). Therefore, the time it takes for Mary to reach her friend’s house can be found as follows:

T = D/R
T = 50 miles / 40 mph
T = 5/4 hours

And similarly, the time it takes for Mary to return home can be found as follows:

T = D/R
T = 50 miles / 60 mph
T = 5/6 hours

Combining these two results, we find that her total time is 5/4 hours + 5/6 hours, or 25/12 hours.
Now we have the total distance and the total time, so her average speed for the entire trip is 100 miles / (25/12 hours), or 48 mph.

Solution II: 

Two segments of the same length (to work and back home).
Use harmonic mean:   mph

#2. 11.25 , very straight-forward questions, just remember that the time given is in minutes, not hours, so you need to convert minutes to hours.

#3. The first step in any rate problem is to see which of the three variables – rate, distance, and time – we have.

Since we’re looking for the car’s speed, or rate, that’s going to be our unknown. The question tells us that the car completes 2 laps each minute, which gives us distance and time. Notice that the problem wants the rate in feet per second, so we’ll need to do a few conversions at the end.

First, to find distance, we need to convert the phrase ‘2 laps’ into units of feet. The car is on a circular path with diameter 800 feet. In other words, 1 lap is equal to the circumference of a circle with diameter 800 feet, which is equal to 800 π feet.

Therefore, two laps is equal to 1600π  feet.

Now we’ve got total distance. We know that this distance is covered in 1 minute, or 60 seconds. Therefore, the car’s speed is equal to 1600 π feet / 60 seconds, or approximately 84 feet/second.

# 4: Let D be the distance. Times 90 (LCM) on both sides.
3D + 2 D = 90 ; 5D = 90; D = 18 miles

#5: Use harmonic mean [See solution II on question #1], since both time late and early is the same. You can just ignore it. 
The answer is 24 mpr. 

#6: D = RT; We can set up the equation and make the time on both sides = to you'll walk to the school on time. 
    (Remember to convert 15 minutes to or hour)
D = 6 miles

Special Right Triangles: 30-60-90 and 45-45-90 Degrees Right Triangles

Please give me feedback on the comment.

Thanks a lot!!  Mrs. Lin







These are the two most common right triangles.

For 45-45-90 degrees, the ratio is  1 - 1 - √ 2

For 30-60-90 degrees, the ratio is  1 - √ 3 - 2

Other concepts to remember are that in any triangle a. larger angle corresponds to longer side and b.same 
angles have the same side length.

Level 0 skills check practices: 

Find the missing side length (answers below)

I: 45-45-90 degrees right triangle

a. 3 - ___- ___            b. 5 - ___- ___         c.  √ 2 - ___ - ___      d. tricky : ___ - ___ - √3

e. ___ - ___ - 4           f. ___ - ___ - 6

II. 30-60-90 degrees right triangle

a. 3 - ___- ___            b.  ___- ___  - 4        c.  √ 2 - ___ - ___      d. ___ - ___ - 5

e.  ___ - 6 - ___           f. ___ - 3 - ___

Answer key and some notes:

I: 45-45-90 degrees right triangle

a. 3 - 3- 3 √ 2              b. 5 - 5 - 5 √ 2         c.  √ 2 - √ 2  - 2       d.√ 6 /2 - √ 6 /2 - √3

e. 2√ 2 - 2√ 2 - 4         f. 3√ 2 - 3√ 2  - 6

Notes:
a. Given the side length to 45 degrees, the easiest way to get the 90 degree side length is to time that number by √ 2 .

b. Given the side length to 90 degrees, the easiest way to get the 45 degree side length is to divide that number by 2 and then times √ 2 .

II. 30-60-90 degrees right triangle

a. 3 - 3√3 - 6            b.  2 - 2√3 - 4        c.  √ 2 - √ 6  - 2√ 2       d. 2.5 - 2.5√3 - 5

e.  2√3 - 6 - 4√3         f.  √3 - 3 -  2√3

Notes:
a. Given the side length to 30 degrees, the easiest way to get 90 degrees side length is to times 2 to the 30 degree side length. To get the side length to 60 degrees, times √3 to the side length to 30 degrees.

 b. Given the side length to 90 degrees, divide 90 degree side length by 2 to get the side length to 30 degrees.Times √3 to the side length of 30 degrees to get the side length of 60 degrees.

c. Given the side length to 60  degrees, divide that number by 3 and then times √3  to get the side length of 30 degrees. Times 2 to get the side length of 90 degrees.

How Many Numbers (Terms)? Space, Inclusive and Exclusive Notes

Please write a comment and give me feedback. Thanks a lot!!

Quite a lot of my students have problems figuring out this type of problems so here are the notes.

#1: How many consecutive numbers from 1 to 5 inclusive?

1 _ 2 _ 3 _ 4 _ 5   There are 5 numbers if you just list them and count them out; however, what if
the question is:

#2: How many consecutive numbers from 34 to 200 (SAT type problem)?

Most students would think it's 200 - 34 = 166, but it's not.

Using #1 case, if you do 5 - 1 = 4, you are only getting how many spaces between those consecutive numbers.

Thus for question #2, the correct answer is 200 - 34 + 1 or 200 - 33 = 167

#3: What about how many consecutive numbers from 5 to 100 exclusive?

Inclusive means including the first and the last numbers; exclusive means not including the first and the last numbers, so for this question, you do 100 - 5 - 1 = 94.

Use # 1 case to help you figure out and really understand the concepts involved.

Here are other questions to help you practice the skills.

Word problems:  Answers below.

#1: How many numbers from 45 to 100 inclusive? 

#2: How many numbers from 17 to 127 inclusive? 

#3: How many numbers from 12 to 34 exclusive? 

#4: How many multiples of 9 from 1 to 200 inclusive?

#5: The dimension of the square on the left is 20 feet by 20 feet. If you  place a post every four feet, starting at one corner, how many posts will be placed? 

#6:  The distance from exit 13 to 21 is 216 miles. How many miles is the distance between two exits if all exits are equally spaced? 

#7: How many multiples of 5 from 120 to 218 exclusive? 

#8: Who is right? The teacher or the student?  Try this question.

#9: How many numbers from -12, -11, -10.........56 inclusive? 

What is their sum?

#10: How many numbers are in the list: 17.25, 18.25, 19, 25...111.25? 

Solutions: To excel at Mathcounts state/national, you need to practice all these questions mentally.

#1: 100 - 45 + 1 = 100 - 44 = 56
#2: 127-17 + 1 = 127 - 16 = 111
#3: Exclusive: 34 -12 -1 = 34 - 13 = 21
#4: Multiples of 9 from 1 to 200 starts with 9 and ends in 198.

Solution I: 9 , 18, 27...198  = 9 (1, 2, 3, ...22) The answer is 22.

Solution II:   \(\frac{(198 - 9)}{9} + 1 = 22"\)

#5:  Just observe one side first. Exclude the 4 corners, the other posts
are similar to those exclusive type problems.
There are posts on each side so 4 * 4 + 4 (corner posts)
= 20



 #6: There are 21 - 13 = 8 space so miles.  The answer is 27 miles.
#7:  Multiples of 5 from 120 to 218 start with 120 and end in 215.
Since it's asking exclusive, 120, 125, ...215 = 5(24, 25, ...43)
43 - 24 - 1 = 43 - 25 = 18
#8: You only need two cuts to get 3 pieces so 2 * 10 = 20 minutes. The student is right.
#9: 56 - ( -12) + 1 = 56 + 13 = 69
The sum is from 13 to 56 since up to 12 it got cancelled with the negative equivalent numbers.
Use average * the term you got  . The sum is 1518.
#10:111.25 - 117.25 + 1 = 111.25 - 116.25 = 95

Problem Solving Strategy: Probability, Counting, Grid

Check out Mathcounts here, the best competition math program for middle school students.
Download this year's Mathcounts handbook here. (It's free.)


#5 1993 Mathcounts National Target : Find the probability that four randomly selected points on the geoboard below will be the vertices of a square? Express your answer as a common fraction.

#5 2004 AMC 10A: A set of three points is chosen randomly from the grid shown. Each three-point set has the same probability of being chosen. What is the probability that the points lie on the same straight line?

Solution:

#5 National Target: There are 16C4 = (16 x 15 x 14 x 13)/ 4 x 3 x 2 = 1820 ways to select 4 points on the geoboard.

There are 3 x 3 = 9  one by one squares and 2 x 2 = 4 two by two squares and 1 x 1 = 1 three by three squares. (Do you see the pattern?)

                                                         

There are 4 other squares that have side length of √ 2
and 2 other larger squares that have side length of √ 5.

9 + 4 + 1 + 4 + 2 = 20 and 20/1820 = 1/91

#5 AMC-10A: There are 9C3 = (9 x 8 x 7) / 3 x 2 x 1 = 84 ways to chose the three dots and 8 of the lines connecting the three dots will form straight lines. (Three verticals, three horizontals and two diagonals.) so 

8/84 = 2/21

Problem Solving Strategy: Counting Coins

Q: How many ways can you make 25 cents if you can use quarters, dimes, nickels or pennies?
Start with the largest value:

1 Quarter      1 way

2 Dimes, 1 Nickel   You can stop here since it implies 2 ways.
2 Dimes, 0 Nickel ( which implies 5 pennies)

1 Dime, 3 Nickels , which implies 4 ways.
1 Dime, 2 Nickels ( 5 pennies)
1 Dime, 1 Nickel   (10 pennies)
1 Dime, 0 Nickel ( 15 pennies)

0 Dime, 5 N, which implies 6 ways.
: , 4 N  (5 pennies)
: , 3 N  (10 pennies)
: , 2 N  (15 pennies)
: , 1 N  (20 pennies)
: , 0 N (25 pennies)

So altogether 13 ways.

Practice questions: (answers below)

Q 1 : How many ways can you make a. 15cents, b. 20cents, c. 30 cents if you can use quarters, dimes, nickels or pennies?

Q 2 : How many different combination of coins could a person have if she has exactly 21 cents?

Q 3 :Using nickels, dimes, quarters and/or half-dollars, how many ways can you make 75 cents?

Q 4: 20 coins of quarters and nickels add up to 4 dollars. How many nickels are there? 

Q5:  What is the least number of US coins to make changes possible from 1 to  99 cents inclusive? (half dollar is allowed)










Answers: 

#1: a. 6 ways ; b. 9 ways; c 18 ways ;
#2: 9 ways
#3: 22  ways
#4: 5 nickels and 15 quarters   
#5: 9 coins (1 half dollar, 1 quarter, 2 dimes, 1 nickel and 4 pennies)

Prime Numbers: Mathcounts Beginning Level

This year's Mathcounts' handbook can be downloaded free here .

Please take a look at what that program is all about. It's team work, problem solving, fun, friendship building and lots and lots more. The majority of students we met at the Mathcounts Nationals all went to the most selected colleges and are thriving there.

Useful Definitions:

Prime – a number which cannot be divided by any numbers other than 1 and itself.

Factors – all whole numbers which can evenly divide a given number

Factoring – the breakdown of any number into its prime components

Greatest Common Factor (GCF)– the greatest number which is a factor of two or more given numbers

Least Common Multiple/Denominator (LCM)– the smallest number which is a multiple of two or more given numbers

Relatively Prime – two numbers with a GCF of 1

A prime, as stated in the list of useful definitions, is a number which cannot be divided by any numbers other than 1 and itself. The smallest prime is 2. [Or, as some people claim, the oddest prime.]

Whole numbers which are not primes are called composite. The smallest composite number is 4.

1 is the exception: it is considered to be neither a prime nor a composite number.

Given a chart of the whole numbers 2-100, the primes can be easily recognized:

The easiest thing to do is to the look at the smallest primes – namely, 2, 3, 5, 7 – and cross all multiples of them from the chart. 

The numbers most commonly mistaken for primes are 51, 57 and 91. The first two (51 and 57) as can be shown by adding up the digits, is divisible by 3, while 91 is equal to 7x13. 

To decide whether or not a number is a prime, take its square root and try dividing the original number by all primes less than the square root. If it is not divisible by any of them, the number is a prime. 

Questions: (beginning level) 

#1: List all the two digit prime numbers that end in a. unit digit 1. b. unit digit 3. c. unit digit 7. d. unit digit 9.

#2: What is the smallest prime number that is the sum of two prime square numbers? 

#3: An emirp (prime spelled backwards) is a prime that gives you a different prime when its digits are reversed. What is the smallest emirp? What are all of the emirps between 1 and 100 inclusive?

#4: The number "p" has three distinct prime factors. How many factors does the number "p" have?

#5: What is the smallest number that has 5 factors? 7 factors? 11 factors? Any pattern?

#6: What is the smallest number that has 6 factors? 10 factors? 12 factors? 20 factors?

#7: How many positive factors does the number 24 have?

#8: Find the sum of all the positive factors of 24?

#9: The GCF (greatest common factor) of x and 21 is "3". If x is smaller than 200, how many possible x are there?

#10: How many even factors does the number 180 have?

Solutions:

#1: See the prime number chart.

#2:  2² + 3² = 13

#3:  13 ( 31 is the reverse prime). The other emirps below 100 are 17, 31, 37, 71, 73, 79, and 97.

#4:  To fine how many factors a number has, you prime factorize that number and add one to each of the
exponents of those prime and multiply them together.
Let x, y, z be the three prime numbers that make up the number p.
p = x * y * z  (1 + 1) (1 + 1) (1 + 1) = 8 -- each prime number has exponent 1.
The 8 factors of p are 1, x, y, z, xy, xz, yz, xyz (or p)

#5: The smallest number that has 5 factors is 2⁴ or 16. The factors are 1, 2, 4, 8, 16.
Since the exponent is 4, there are (4 + 1) = 5 factors.
The smallest number that has 7 factors is 2⁶ or 64.
The smallest number that has 11 factors is 2¹⁰ = 1024
5, 7, 11 are all prime numbers. 

#6: This one is harder than the previous question. 6 =1*6 or 3 * 2
2⁵ = 32  ;   2² * 3 = 12  (both numbers have 6 factors but the latter is much smaller).
The answer is 12.

10 = 1 x 10 = 5 x 2  ;  2⁴ * 3 = 48

12 = 1 x 12 = 6 x 2 = 4 x 3 = 3 x 2 x 2
2²* 3 * 5 = 60

20 = 1 x 20 = 2 x 10 = 5 x 2 x 2
2⁴* 3 * 5 = 240

#7: 24 = 3¹*2³ (1 + 1) (3+1) =8; Those factors are:

#8: Continue to the previous number 24, the easiest way to find the sum is to use the
following method:
(1 + 2 + 4 + 8) (1 + 3) = 60

#9: 

#10:  180 = 2²* 3 ²* 5
To get just the even factors, you need to keep a 2 to guarantee the factors stay even.
2 ( 2 * 3²* 5)  There are (1+1)(2+1)(1+1) = 12 factors.