Difference between revisions of "1985 AJHSME Problem 2"

Latest revision as of 08:40, 22 March 2025

Problem

$90+91+92+93+94+95+96+97+98+99=$

$\text{(A)}\ 845 \qquad \text{(B)}\ 945 \qquad \text{(C)}\ 1005 \qquad \text{(D)}\ 1025 \qquad \text{(E)}\ 1045$

Solution 1

We can add as follows: $90+91+92+93+94+95+96+97+98+99= 10(90) +1+2+3+4+5+6+7+8+9 = 900 + 45 = \boxed{945}$ The answer is $\boxed{\textbf{(B)}\ 945}$ .

Solution 2

Pair the numbers like so: $(90+99)+(91+98)+(92+97)+(93+96)+(94+95)$ The sum of each pair is $189$ and there are $5$ pairs, so the sum is $945$ and the answer is $\boxed{\textbf{(B)}\ 945}$ .

Solution 3 (Cheap and Quick)

We know that $10(90) = 900$ and $10(100) = 1000.$ Quick estimation reveals that this sum is in between these two numbers, so the only answer available is $\boxed{\textbf{(B)}\ 945}$ .

Solution $3\frac{1}{2}$

You see that $90+91+92+93+94+95+96+97+98+99$ is equal to $0+1+3+4+5+6+7+8+9+900$ . You can use the formula $\frac{n(n+1)}{2}$ to get $45+900=\boxed{\textbf{(B)}\ 945}$ .

Video Solution

https://youtu.be/1NtsgKc6mXs

~savannahsolver

@@ Line 8: / Line 8: @@
 We can add as follows:
 <cmath>90+91+92+93+94+95+96+97+98+99= 10(90) +1+2+3+4+5+6+7+8+9 = 900 + 45 = \boxed{945}</cmath>
-The answer is <math>\text{(B)}.</math>
+The answer is <math>\boxed{\textbf{(B)}\ 945}</math>.
 ==Solution 2==
 Pair the numbers like so:
 <cmath>(90+99)+(91+98)+(92+97)+(93+96)+(94+95)</cmath>
-The sum of each pair is <math>189</math> and there are <math>5</math> pairs, so the sum is <math>945</math> and the answer is <math>\text{(B)}.</math>
+The sum of each pair is <math>189</math> and there are <math>5</math> pairs, so the sum is <math>945</math> and the answer is <math>\boxed{\textbf{(B)}\ 945}</math>.
-==Cheap Solution==
+==Solution 3 (Cheap and Quick)==
-We know that <math>10(90) = 900</math> and <math>10(100) = 1000.</math> Quick estimation reveals that this sum is in between these two numbers, so the only answer available is <math>\text{(B)}.</math>
+We know that <math>10(90) = 900</math> and <math>10(100) = 1000.</math> Quick estimation reveals that this sum is in between these two numbers, so the only answer available is <math>\boxed{\textbf{(B)}\ 945}</math>.
-==Solution 3==
-You see that 90+91+92+...
+==Solution <math>3\frac{1}{2}</math>==
+You see that <math>90+91+92+93+94+95+96+97+98+99</math> is equal to <math>0+1+3+4+5+6+7+8+9+900</math>. You can use the formula <math>\frac{n(n+1)}{2}</math> to get <math>45+900=\boxed{\textbf{(B)}\ 945}</math>.
 ==Video Solution==

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