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

Revision as of 16:10, 28 November 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 $40000.$ This is easy. 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 $\boxed{\text{(A)}180001}.$

@@ Line 18: / Line 18: @@
 ''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, and each digit can appear equally.
+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> This is easy. 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{\text{A}180001}.</math>
+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{\text{(A)}180001}.</math>
 == 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
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