Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1970 AHSME Problems/Problem 33"

m (→‎Solution)
m (→‎Solution 3)
 
(10 intermediate revisions by 2 users not shown)
Line 10: Line 10:
  
 
== Solution ==
 
== Solution ==
−
'''Solution 1'''
+
===Solution 1===
 
We can find the sum using the following method. We break it down into cases. The first case is the numbers <math>1</math> to <math>9</math>. The second case is the numbers <math>10</math> to <math>99</math>. The third case is the numbers <math>100</math> to <math>999</math>. The fourth case is the numbers <math>1,000</math> to <math>9,999</math>. And lastly, the sum of the digits in <math>10,000</math>. The first case is just the sum of the numbers <math>1</math> to <math>9</math> which is, using <math>\frac{n(n+1)}{2}</math>, <math>45</math>. In the second case, every number <math>1</math> to <math>9</math> is used <math>19</math> times. <math>10</math> times in the tens place, and <math>9</math> times in the ones place. So the sum is just <math>19(45)</math>. Similarly, in the third case, every number <math>1</math> to <math>9</math> is used  <math>100</math> times in the hundreds place, <math>90</math> times in the tens place, and <math>90</math> times in the ones place, for a total sum of <math>280(45)</math>. By the same method, every number <math>1</math> to <math>9</math> is used <math>1,000</math> times in the thousands place, <math>900</math> times in the hundreds place, <math>900</math> times in the tens place, and <math>900</math> times in the ones place, for a total of <math>3700(45)</math>. Thus, our final sum is <math>45+19(45)+280(45)+3700(45)+1=4000(45)+1=\boxed{\text{A)}180001}.</math>
 
We can find the sum using the following method. We break it down into cases. The first case is the numbers <math>1</math> to <math>9</math>. The second case is the numbers <math>10</math> to <math>99</math>. The third case is the numbers <math>100</math> to <math>999</math>. The fourth case is the numbers <math>1,000</math> to <math>9,999</math>. And lastly, the sum of the digits in <math>10,000</math>. The first case is just the sum of the numbers <math>1</math> to <math>9</math> which is, using <math>\frac{n(n+1)}{2}</math>, <math>45</math>. In the second case, every number <math>1</math> to <math>9</math> is used <math>19</math> times. <math>10</math> times in the tens place, and <math>9</math> times in the ones place. So the sum is just <math>19(45)</math>. Similarly, in the third case, every number <math>1</math> to <math>9</math> is used  <math>100</math> times in the hundreds place, <math>90</math> times in the tens place, and <math>90</math> times in the ones place, for a total sum of <math>280(45)</math>. By the same method, every number <math>1</math> to <math>9</math> is used <math>1,000</math> times in the thousands place, <math>900</math> times in the hundreds place, <math>900</math> times in the tens place, and <math>900</math> times in the ones place, for a total of <math>3700(45)</math>. Thus, our final sum is <math>45+19(45)+280(45)+3700(45)+1=4000(45)+1=\boxed{\text{A)}180001}.</math>
−
'''Solution 2'''
+
===Solution 2===
−
Consider the numbers from <math>0000-9999</math>. We have <math>40000</math> digits and each has equal probability of being <math>0,1,2....9</math>
+
Consider the numbers from <math>0000-9999</math>. We have <math>40000</math> digits and each has equal an probability of being <math>0,1,2....9</math>.
−
Our requested sum then is <math>4000(45)+1=\boxed{\text{A)}180001}.</math>
+
Thus, our desired sum is <math>4000\left( \frac{9 \cdot 10}{2} \right)+1=4000(45)+1=\boxed{\text{A)}180001}.</math>
−
Credit: Math1331Math
+
 
 +
''Credit: Math1331Math''
 +
===Solution 3===
 +
As in Solution 2, we consider the four digit numbers from <math>0000-9999.</math> We see that we have <math>10000\times4=40000</math> digits with each digit  appearing equally.
 +
 
 +
Thus, the digit sum will be the average of the digits multiplied by <math>40000.</math> The digit average comes out to be <math>\frac{0+9}{2},</math> since all digits are consecutive.
 +
 
 +
So, our answer will be <math>\frac{9}{2} \times 40000 = 180000.</math> However, since we purposely did not include <math>10000,</math> we add one to get our final answer as <math>\boxed{\textbf{(A)}180001}.</math>
 +
 
 +
Solution by davidaops.
 +
 
 +
''Credit to Math1331Math''
  
 
== See also ==
 
== See also ==

Latest revision as of 14:54, 15 December 2018

Problem

Find the sum of digits of all the numbers in the sequence $1,2,3,4,\cdots ,10000$.

$\text{(A) } 180001\quad \text{(B) } 154756\quad \text{(C) } 45001\quad \text{(D) } 154755\quad \text{(E) } 270001$

Solution

Solution 1

We can find the sum using the following method. We break it down into cases. The first case is the numbers $1$ to $9$. The second case is the numbers $10$ to $99$. The third case is the numbers $100$ to $999$. The fourth case is the numbers $1,000$ to $9,999$. And lastly, the sum of the digits in $10,000$. The first case is just the sum of the numbers $1$ to $9$ which is, using $\frac{n(n+1)}{2}$, $45$. In the second case, every number $1$ to $9$ is used $19$ times. $10$ times in the tens place, and $9$ times in the ones place. So the sum is just $19(45)$. Similarly, in the third case, every number $1$ to $9$ is used $100$ times in the hundreds place, $90$ times in the tens place, and $90$ times in the ones place, for a total sum of $280(45)$. By the same method, every number $1$ to $9$ is used $1,000$ times in the thousands place, $900$ times in the hundreds place, $900$ times in the tens place, and $900$ times in the ones place, for a total of $3700(45)$. Thus, our final sum is $45+19(45)+280(45)+3700(45)+1=4000(45)+1=\boxed{\text{A)}180001}.$

Solution 2

Consider the numbers from $0000-9999$. We have $40000$ digits and each has equal an probability of being $0,1,2....9$. Thus, our desired sum is $4000\left( \frac{9 \cdot 10}{2} \right)+1=4000(45)+1=\boxed{\text{A)}180001}.$

Credit: Math1331Math

Solution 3

As in Solution 2, we consider the four digit numbers from $0000-9999.$ We see that we have $10000\times4=40000$ digits with each digit appearing equally.

Thus, the digit sum will be the average of the digits multiplied by $40000.$ The digit average comes out to be $\frac{0+9}{2},$ since all digits are consecutive.

So, our answer will be $\frac{9}{2} \times 40000 = 180000.$ However, since we purposely did not include $10000,$ we add one to get our final answer as $\boxed{\textbf{(A)}180001}.$

Solution by davidaops.

Credit to Math1331Math

See also

1970 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 32 		Followed by
Problem 34
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1970_AHSME_Problems/Problem_33&oldid=99455"