## Saturday, June 9, 2012

### Unit digit, Tenth digit and Digit Sum

Word problems on unit digit, tenth digit or digit sum.

#1: How many digits are there in the positive integers 1 to 99 inclusive?
Solution I:  From 1 to 9, there are 9 digits.
From 10 to 99, there are 99 - 10 + 1 or 99 - 9 = 90 two digit numbers. 90 x 2 = 180

Solution II:  ___ There are 9  one digit numbers (from 1 to 9).
___ ___ There are 9 * 10 = 90 two digit numbers (You can't use "0" on the tenth digit but you
can use "0" on the unit digit.) 90 * 2 + 9 = 189

# 2: A book has 145 pages. How many digits are there if you start counting from page 1?
There are 189 digits from page 1 to 99. (See #1, solution I)
From 100 to 145, there are 145 - 100 + 1 or 145 - 99 = 46 three digit numbers.
189 + 46*3 = 327 digits.

#3: "A book has N pages, number the usual way, from 1 to N. The total number of digits in the page number
is 930. How many pages does the book have"?  Similar to one Google interview question.
Read the questions and others here from the Wall Street Journal.
Solution I:
930 - 189 (digits of the first 99 pages) =741
741 divided by 3 = 247. Careful since you are counting the three digit numbers from 100 if the book has N
pages N - 100 + 1 or N - 99 = 247. N = 346 pages.

Solution II:
There are N pages, so there will be N unit digits, N - 9 (the number of unit digits) ten digits, and N - 99 hundred digits.
N + N - 9 + N - 99 = 930; 3 N = 930 + 108 ; N = 346 pages

#4: If you write consecutive numbers starting with 1, what is the 50th digit you write?
Solution I:
50 - 9 = 41, leave the odd 1 out for now. 40 divided by 2 = 20 and N - 10 + 1 or N - 9 = 20
N = 29 so the last digit you write is the next one, which is "3".

Solution II:  Let there be n numbers: n + (n - 9) = 50, 2n = 59, n = 29.5 which mean the last digit
you write is the first digit of the next number, which is "30", so 3 is the answer.

#5: What is the sum if you add up all the digits from 1 to 100 inclusive?
00  10  20  30  40  50  60  70  80  90
01  11  21  31  41  51  61  71  81  91
02  12  22  32  42  52  62  72  82  92
03  13  23  33  43  53  63  73  83  93
04  14  24  34  44  54  64  74  84  94
05  15  25  35  45  55  65  75  85  95
06  16  26  36  46  56  66  76  86  96
07  17  27  37  47  57  67  77  87  97
08  18  28  38  48  58  68  78  88  98
09  19  29  39  49  59  69  79  89  99

Solution I: Do you see the pattern?  From 00 to 99 if you just look at the unit digits.
There are 10 sets of ( 1+ 2 + 3 ... + 9) , which gives you the sum of 10 * 45 = 450
How about the tenth digits? There are another 10 sets of (1 + 2 + 3 + ...9) so another 450
Add them up and you have 450 * 2 = 900 digits from 1 to 99 inclusive.
900 + 1 ( for the "1" in the extra number 100) = 901

Solution II: If you add the digits on each column, you have an arithmetic sequence, which is
45 + 55 + 65 ... + 135  To find the sum, you use average * the terms (how many numbers)
${\frac{(45 + 135)}{2}}*{[\frac{(135-45)}{10}+ 1]}$ = 900
900 + 1 = 901

#1: A book has 213 pages, how many digits are there?
#2: A book has 1012 pages, how many digits are there?
#3: If you write down all the digits starting with 1 and in the end there are a: 100, b: 501 and c: 1196 digits, what is the last digit you write down for each question?
#4: What is the sum of all the digits counting from 1 to 123?

#1: 531 digits.
#2: 2941 digits.
#3: a. 5, b. 3, c. 3
#4: 1038

## Thursday, June 7, 2012

### For Young Mathletes 4

For Young Mathletes 4

This Week's Problems
May 30th, 2012 for Young Mathletes Answer Key to May 30th, 2012 for 4th and 5th Graders

## Tuesday, June 5, 2012

### 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.
#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:

a.  $nC2 - n$ (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?

#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        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. $\frac{20*19}{2}-20$ = 170 diagonals
#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