## Saturday, May 26, 2012

### Math Cartoons from Calvin and Hobbes by Bill Waterson

by Bill Watterson

The math questions are on the above link. Please give me feedback on the comment.
Thanks a lot.

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 3There 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.

Calvin and Hobbes Cartoon 4
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?

#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:   $\frac{2}{\frac{1}{40} + \frac{1}{60}}=\frac{2}{\frac{3 + 2}{120}} = 48$ 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. $\frac{D}{30}+ \frac{D}{45} = 1$ $\rightarrow$ 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.

#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.
$\frac{D}{4}+ \frac{1}{4} =\frac{D}{3}-\frac{1}{4}$ $\rightarrow$ $\frac{D}{12} = \frac{1}{2}$  (Remember to convert 15 minutes to $\frac{15}{60}$ or $\frac{1}{4}$ hour)
D = 6 miles

## Friday, May 25, 2012

### 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 - ___

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 - 33 - 6            b.  2 - 2√3 - 4        c.   2 6  - 2√ 2       d. 2.5 - 2.53 - 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.

## Thursday, May 24, 2012

### For Young Mathletes 2

May 16th, 2012 homework for young Mathletes.

Please give me feedback on the comment.
I would like to use students' problems soon.

Thanks a lot.

## Wednesday, May 23, 2012

### Mathcounts: Problem Solving Strategies: for the Beginning Level

Discounts, percentage off questions:
Consider the following:

There is a melon on sale for $100 dollars. The vendor offers to sell it to you at a 20% discount. How much would you have to pay for the melon? In situations such as this, it's better to look backwards at the percentages. A 20% discount means that there is 100-20, or 80%, of the original price left. Therefore, you would have to pay$100 x 80% = $100 x 80/100 = 80$ for the melon.

This method can be used for all percentage problems, including more complicated ones, as follows:

The melon vendor has switched to selling strawberries. Each box of strawberries will cost you only $480! After a period of financial devastation, the vendor lowers the price to 40% off. He then offers you an additional 10% off, because you purchased so many melons from him earlier.How much would you have to pay for the box of strawberries? The 40% off means that only 60% [100% -40%] is left. The additional 10% off means that only 90% [100% - 10%] of the 60% is left. You would have to pay ($480 x 60% x 90%), or $480 x 3/5 x 9/10 =$259.20.

How Consumers Miscaculate Sales Prices : from Consumeraffairs.com

Profits, marked-up, percentage increase questions:
To make 40% profit, the vendor charges his customer 84 dollars, what is the wholesale price?

Let the wholesale price be x.
(100% + 40%) * x = 84;     1.4x = 84;      x = 60 dollars

Most students forgot to use 100%, which is equivalent to the original price, on top of 40% and get a real weird answer. Make sure you understand why and how.

This month Mary's monthly income has increased from $2400 to$3000, what is her salary increase?

Two ways to approach this problem:
Solution I: |2400 - 3000| over 2400 = 1/4 = 25% increase
Percentage increase/decrease = |positive difference of the two prices| over the original price

Solution II: (3000/2400) -1 = 25% You find what multiple of the new price versus the original and
then minus 1 (the original 100%). If it's negative, it's percentage decrease.

More questions to ponder (answers and solutions below):

#1 :You’re walking by a store window and you see a sign that says, “20% off the original price plus an additional 25% off the already reduced sale price.What percentage of the original price do you need to pay? How much is the discount?

# 2: Would you be more likely to buy a product that is 45 % off, or the same product in a store wide 25 % off sale with an additional 25 % off that product?

# 3: A store is offering a promotion; every week, another 50% is taken off current sales prices. Your friend calls you excitedly and says: "Hey, I'll wait for two weeks, since the item will then be free." What's wrong with that statement?

#4: Which is a better deal? 20% off an item that costs 13500 or 10% off and another 10% off of the same item? How much will you save for the cheaper option.

#5: A computer is on sale for $1260, which is 25% discount off the regular price. What is the regular price? #6: A computer's retail price is$1260, which is a 25% markup of the wholesale price. What is the wholesale price?

#7: To make 30% profit, you charge your customer $195, what is the wholesale price? #8: A store owner offers 20% discounts of a sales item; however, he doesn't want to lose money on the deal. What percentage markup of the wholesale price should he implement so that he's not losing money? #9: To make 20% profit, you charge 132 dollars for certain items you sold; however, your friends get 20% discount off the wholesale price. How much do they need to pay? #10: If A is 130% of B and C is 195% of B, what fraction of C is A? Answer key and Solutions: #1: 0.8 [100% -20%] * 0.75 [100% - 25%] = 0.6 or 60% so you pay 60% of the original price, which is (100% - 60%) or 40% discount. #2: After 45% discount, you pay (100% - 45%) or 55% of the regular price However, using the second pricing method you need to pay (100% - 25%) * (100% - 25%) = 0.75 * 0.75 = 0.5625 = 56.25% of the regular price. The first one is cheaper by 1.25% #3: (100% - 50%) (100% - 50%) = 0.5 times 0.5 = 0.25 = 25% so you still need to pay 25% of the regular price. #4: After 20% off, you pay (100% - 20%) = 80% or 0.8 of the regular price. After 10% off and another 10% off, you pay (100-10%) * (100-10%) = 0.9 * 0.9 = 0.81 or 81% of the regular price. Thus, 20% off is a better deal and you save 1% of$13500, which comes up to 135 dollars.
#5: Let the original price be "x" dollars. (100%-25%) * x = 1260;  x = 1260 divided by 0.75 =
1680 dollars
#6: Let the wholesale price be "x". (100% + 25%) * x = 1260;  x = 1260 divided by 1.25 =                     1008 dollars.
#7: 195 divided by (100% + 30%) = 1.95 divided by 1.3 = 150.
#8: If the store owner offers 20% discount of an item, the customers will pay (100% - 20%) = 80% or 4/5 of the regular price. To not lose any money, let the markup percentage be "x".
4/5 of (100% + x) = 1, x = 25%
#9: 132 divided by 1.2 = 110;    110 * 0.8 = 88 dollars
#10: 130%  over 195% = 130/195 = 2/3

You

### A Skill for the 21st Century: Problem Solving by Richard Rusczyk

A Skill for the 21st Century: Problem Solving by Richard Rusczyk, founder of  "Art of Problem Solving".

## Monday, May 21, 2012

### Base Numbers: Level 1

Base numbers:

Convert other bases to base 10.
Example #1:
1 | 0 |0
22|21 |20                  0*20 + 0*21 + 1*22 = 4

Thus, 1002  = 4 in base 10
The first few base 2 (or binary system) numbers that are equivalent to their base 10 (or decimal) counterpart.
base 2                       base 10
0                               0

1                               1

10                             2

11                             3

100                           4

101                           5

110                           6

111                           7

1000                         8

1001                         9

1010                        10

Example #2:
1|  2|  45
52|51|50                    4*50 + 2*51 + 1*52  = 4 + 10 + 25 = 39

Thus, 1245 = 39 in base 10

Some important points:
* All the digits in the number has to be smaller than the base. (Why?)
* Hexadecimal is base 16. In that system, you can use numbers from "0" to "9", the same as base 10.
However, numbers 10 to 15, you use letters: A for 10, B for 11, C for 12, D for 13, E for 14 and last, F      for 15.

Short cut to some base conversions:
#1: Convert 11011112 to base 8.
Since 8 = 23, you can convert base 2 to base 8 by regrouping every 3 digits.
Thus the original number will become 1578
because if you regroup the original number starting from the right, you have 12-1012-1112 and it's 157 in base 2's term.
#2: Try convert 111100112 to base 4, which is 22 so you regroup by 2.
11 (3 in base 2)-11(3 in base 2)-00 (0 in base 2)-11(3 in base 2) so the answer is 33034.

Convert other bases to base 10.
#: 101112
#23304
#31005
#42103
#5246
#6: Convert 10111 to base 4
#7: Convert 123321to base 16 (or hexadecimal) [student Andrew's problem]
#8 110100112  is what to base 16 ?
#9Convert 1648 to binary (or base 2) [Daniel's question]
#10: What is 100005  minus 15 ? [Remaining :Willie's questions]
#11: What is 0.18 to base 10?
#12: What is 5378  divided by 108 ?
#13: 3716  is what in base 4? what in base 10?
#14: What is 12314  + 22324 ?

#1: 23
#2: 60
#3: 25
#4: 21
#5: 16
#6: 23 4
#7: 6F9 16
#8: D3 16
#9: 1110100 2
#10: 1111 5
#11: 1/ 8
#12: 53.7 8
#13: 313 in base 4 and 55 in base 10
#14: 10123 4

### Mathcounts State Perparations: Counting problems

Check out Mathcounts here, the best competition math program for middle school students.

Problems (solutions below)
#1:How many times does the digit 7 appear in the list of all the integers from 1 to 1000 inclusive?

#2. How many integers from 1 to 1000 inclusive do not contain the digit 7?

#3. How many integers from 1 to 1000 inclusive contain the digit 7?

#4.How many four-digit positive integers have at least one digit that is a 1, 3 or 5?

#5. How many three-digit numbers exist of which the middle digit is the average of the first and the last digits?

Solutions:
#1. Every 100 will give you 20 “7”s – 10 for the unite digit (07, 17, 27…97) and 10 for the tenth
digit (70, 71, 72…79) so from 1 to 1000 there will be 20 times 10 = 200 times the number 7 appear on the tenth and unit digits.
However, there are 100 times the number “7” appear on the hundredth digit. (from 700 to 799), so the answer is 300.

#2. There are 8 single digit numbers that do not contain the digit 7.
There are 8 x 9 = 72 two digit numbers that do not contain the digit 7.
There are 8 x 9 x 9 = 648 three digit numbers that do not contain the digit 7.
Add them together and + 1 (the number 1000 itself) and you get 729.

#3. Do complementary counting. Use the total minus no "7" appeared.
1000 – 729 (from the previous answer) = 271

#4. Do complementary counting. There are 9 x 10 x 10 x 10 = 9000 four-digit positive integers.
Of those, there are 6 (exclude 1, 3, or 5) x 7 (again exclude 1, 3, or 5) x 7 x 7= 2058  integers without 2 3 or 5.
9000 – 6 x 7 x 7 x 7 = 6942

5. For the middle digit to be an integer, the first and last digits must be both odd or both even.
(Why?) There are 5 x 5 = 25 numbers that the hundredth and unit digit are odd and there are 4 (no zero at the hundredth place) x 5 = 20 numbers that the hundredth and unit digit are both even.
The total is 45.

## Sunday, May 20, 2012

### Counting : Beginning Advanced Level

Problems:
# 1: How many 3 digit numbers are there? 2 digit even numbers are there?

Solution I: a. 3 digit starts with 100 and ends in 999 so there are 999 -100 + 1 = 999 - 99 = 900 three digit numbers
b.  2 digit numbers start with 10 and ends in 99 so there are 99 - 9 = 90 two digit numbers.

Solution II: a. ___ ___ ___ You can't have "0" on the hundredth digit so there are 9 numbers (1 to 9) for the hundredth digit, 10 (you can use "0" now) for both the tenth and unit digit.
There are 9 * 10 * 10 = 900 three digit numbers.
b. ___   ___ Again, you can't have "0" on the tenth digit so there are 9 * 10 = 90 two digit numbers.

Quite a few students are confused with questions such as this.
Use easier example to demonstrate why you multiply, not add the numbers.

How many ways to pair A, B, C  with 1, 2, 3.
You have (A, 1), (A, 2), (A, 3), (B, 1), (B, 2), (B, 3), (C, 1), (C, 2) and finally (C, 3), which is
3 * 3 = 9 ways to pair them instead of adding 3 + 3.

# 2: How many 3 digit numbers are there that the hundredth digit is larger than 4, the tenth digit is odd and
the ones (or unit digit) is even?

Solution:  ___ ___ ___  There are 5 possible numbers for the hundredth digit (5, 6, 7, 8, 9) that is larger than 4,  5 digits that are odd (1, 3, 5, 7, 9) and 5 digits that are even (0, 2, 4, 6, 8) so the answer is
5 * 5 * 5 = 125 three digit numbers

# 3: How many 3-digit numbers chosen from 2, 4, 5, 7, 9 can you make if you can repeat the digits, but
the same number can't be right next to each other?

Solution: ___ ___ ___  There are 5 possible digits for the first slot; however, since you can't use the same digit right next to each other, only 4 digits left for the second slot. Again, you can't use the same digit as the
second slot so 4 left for the third slot.
5 * 4 * 4 = 80  three digit numbers.

#1: How many three digit positive odd integers?
#2: How many 3 digit odd numbers can be formed with the digits 1, 2, 3, 4, 5 if you can repeat digits?
What about if you can't repeat digits?
#3: How many 3 digit numbers can be formed with the digits 0,1,2, 3, 4, 5 if repeating digits are allowed? How many are even and how many are odd?
#4: How many 3 digit numbers can be formed with the digits 0,1,2, 3, 4, 5 if no digit is repeated in any number? How many are even and how many are odd? (This one is tricky!!)

#1: 9 * 10 * 5 = 450
#2: 5 * 5 * 3 = 75 (digits can be repeated) ; 4 * 3 * 3 = 36 (digits can not repeated)
#3: 5 * 6 * 6 = 180; There will be 5 * 6 * 3 = 90 even numbers and the same 5 * 6 * 3 = 90 odd numbers
Don't forget that "0" can't be placed on the hundredth digit.
#4: 100 numbers can be formed. 48 odd and 52 even numbers.

### For Young Mathletes 1

May 2nd, 2012 homework for young Mathletes.