Difference between revisions of "1985 AJHSME Problems/Problem 2"

(Solution 7 (Brute Force))
 
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<math>\text{(A)}\ 845 \qquad \text{(B)}\ 945 \qquad \text{(C)}\ 1005 \qquad \text{(D)}\ 1025 \qquad \text{(E)}\ 1045</math>
 
<math>\text{(A)}\ 845 \qquad \text{(B)}\ 945 \qquad \text{(C)}\ 1005 \qquad \text{(D)}\ 1025 \qquad \text{(E)}\ 1045</math>
  
==Solution==
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==Solution 1==
 +
To simplify the problem, we can group 90’s together: 90+91+...+98+99=9010+1+2+3+...+8+9.
  
We could just add them all together. But what would the point be of doing that? So we find a slicker way.<br><br>We find a simpler problem in this problem, and simplify -> <math>90 + 91 + ... + 98 + 99 = 90 \times 10 + 1 + 2 + 3 + ... + 8 + 9</math><br>We know <math>90 \times 10</math>, that's easy - <math>900</math>. So how do we find <math>1 + 2 + ... + 8 + 9</math>?<br><br>Well once again (not once again if you didn't read the solution for #1), commutative property comes to the rescue. We rearrange the numbers to make <math>(1 + 9) + (2 + 8) + (3 + 7) + (4 + 6) + 5</math>. You might have noticed that each of the terms i put next to each other add up to 10, which make for easy adding. <math>4 \times 10 + 5 = 45</math>. Adding that on to 900 makes 945.<br><br>945 is (B)
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9010=900, and finding 1+2+...+8+9 has a trick to it.
 +
 
 +
Rearranging the numbers so each pair sums up to 10, we have:
 +
(1+9)+(2+8)+(3+7)+(4+6)+5. 410+5=45, and 900+45=(B) 945.
 +
 
 +
==Solution 2==
 +
We can express each of the terms as a difference from <math>100</math> and then add the negatives using <math>\frac{n(n+1)}{2}=1+2+3+\cdots+(n-1)+n</math> to get the answer.
 +
<cmath>\begin{align*}
 +
(100-10)+(100-9)+\cdots+(100-1) &= 100\cdot10 -(1+2+\cdots+9+10)\
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&= 1000 - 55\
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&= \boxed{\text{(B)}~945}
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\end{align*}</cmath>
 +
 
 +
==Solution 3==
 +
Instead of breaking the sum then rearranging, we can rearrange directly:
 +
<cmath>\begin{align*}
 +
90+91+92+\cdots +98+99 &= (90+99)+(91+98)+(92+97)+(93+96)+(94+95) \
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&= 189+189+189+189+189 \
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&= \boxed{\text{(B)}~945}
 +
\end{align*}</cmath>
 +
 
 +
==Solution 4==
 +
 
 +
The finite arithmetic sequence formula states that the sum in the sequence is equal to <math>\frac{n}{2}\cdot(a_1+a_n)</math> where <math>n</math> is the number of terms in the sequence, <math>a_1</math> is the first term and <math>a_n</math> is the last term.
 +
 
 +
Applying the formula, we have:
 +
<cmath>\frac{10}{2}\cdot(90+99)=\boxed{\text{(B)}~945}</cmath>
 +
 
 +
==Solution 5==
 +
The expression is equal to the sum of integers from <math>1</math> to <math>99</math> minus the sum of integers from <math>1</math> to <math>89</math>, so it is equal to <math>\frac{99(100)}{2} - \frac{89(90)}{2} = 4950 - 4005 = \boxed{\text{(B)}~945}</math>.
 +
 
 +
~ [https://artofproblemsolving.com/wiki/index.php/User:Cxsmi cxsmi]
 +
 
 +
== Solution 6 (Estimate) ==
 +
 
 +
Notice the sum is larger than <math>90 \times 10</math> and less than <math>100 \times 10</math>
 +
 
 +
The only choice that works is 945, thus the answer is <math>\boxed{\textbf{(B)}\ 945}</math>
 +
 
 +
~ lovelearning999
 +
 
 +
== Solution 7 (Brute Force) ==
 +
 
 +
Add all the numbers to get 945, or <math>\boxed{\textbf{(D)}\ 945}</math>
 +
 
 +
~ lovelearning999
 +
 
 +
==Video Solution by BoundlessBrain!==
 +
https://youtu.be/8bVNfa-yEoM
 +
 
 +
==Video Solution==
 +
https://youtu.be/1NtsgKc6mXs
 +
 
 +
~savannahsolver
  
 
==See Also==
 
==See Also==
  
[[1985 AJHSME Problems]]
+
{{AJHSME box|year=1985|num-b=1|num-a=3}}
 +
 
 +
 
 +
{{MAA Notice}}
 +
[[Category:Introductory Algebra Problems]]

Latest revision as of 21:04, 2 October 2024

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

To simplify the problem, we can group 90’s together: 90+91+...+98+99=9010+1+2+3+...+8+9.

9010=900, and finding 1+2+...+8+9 has a trick to it.

Rearranging the numbers so each pair sums up to 10, we have: (1+9)+(2+8)+(3+7)+(4+6)+5. 410+5=45, and 900+45=(B) 945.

Solution 2

We can express each of the terms as a difference from $100$ and then add the negatives using $\frac{n(n+1)}{2}=1+2+3+\cdots+(n-1)+n$ to get the answer. \begin{align*} (100-10)+(100-9)+\cdots+(100-1) &= 100\cdot10 -(1+2+\cdots+9+10)\\ &= 1000 - 55\\ &= \boxed{\text{(B)}~945} \end{align*}

Solution 3

Instead of breaking the sum then rearranging, we can rearrange directly: \begin{align*} 90+91+92+\cdots +98+99 &= (90+99)+(91+98)+(92+97)+(93+96)+(94+95) \\ &= 189+189+189+189+189 \\ &= \boxed{\text{(B)}~945}  \end{align*}

Solution 4

The finite arithmetic sequence formula states that the sum in the sequence is equal to $\frac{n}{2}\cdot(a_1+a_n)$ where $n$ is the number of terms in the sequence, $a_1$ is the first term and $a_n$ is the last term.

Applying the formula, we have: \[\frac{10}{2}\cdot(90+99)=\boxed{\text{(B)}~945}\]

Solution 5

The expression is equal to the sum of integers from $1$ to $99$ minus the sum of integers from $1$ to $89$, so it is equal to $\frac{99(100)}{2} - \frac{89(90)}{2} = 4950 - 4005 = \boxed{\text{(B)}~945}$.

~ cxsmi

Solution 6 (Estimate)

Notice the sum is larger than $90 \times 10$ and less than $100 \times 10$

The only choice that works is 945, thus the answer is $\boxed{\textbf{(B)}\ 945}$

~ lovelearning999

Solution 7 (Brute Force)

Add all the numbers to get 945, or $\boxed{\textbf{(D)}\ 945}$

~ lovelearning999

Video Solution by BoundlessBrain!

https://youtu.be/8bVNfa-yEoM

Video Solution

https://youtu.be/1NtsgKc6mXs

~savannahsolver

See Also

1985 AJHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
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
All AJHSME/AMC 8 Problems and Solutions


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