Difference between revisions of "1993 UNCO Math Contest II Problems/Problem 5"

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A collection of <math>25</math> consecutive positive integers adds to <math>1000.</math> What are the smallest and largest integers in this collection?
 
A collection of <math>25</math> consecutive positive integers adds to <math>1000.</math> What are the smallest and largest integers in this collection?
 
  
 
== Solution ==
 
== Solution ==
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=== Solution 1 ===
 
The thirteenth integer is the average, which is <math>\frac{1000}{25}=40</math>. So, the largest integer is 12 larger, which is <math>40+12=\boxed{52}</math>, and the smallest integer is 12 less, which is <math>40-12=\boxed{28}</math>.
 
The thirteenth integer is the average, which is <math>\frac{1000}{25}=40</math>. So, the largest integer is 12 larger, which is <math>40+12=\boxed{52}</math>, and the smallest integer is 12 less, which is <math>40-12=\boxed{28}</math>.
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=== Solution 2 ===
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By the summation formula, the sum of 25 consecutive numbers (where <math>x</math> is the smallest number in the list) is
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<cmath>\frac{25(2x+24)}{2}</cmath>
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<cmath>25(x+12)</cmath>
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Letting the value of the equation be <math>1000</math>, we have
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<cmath>25(x+12)=1000</cmath>
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<cmath>x+12=40</cmath>
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<cmath>x=28</cmath>
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Thus the smallest value of the list is <math>\boxed{28}</math>, and the largest is <math>28+24=\boxed{52}</math>
  
 
== See also ==
 
== See also ==

Latest revision as of 00:05, 20 January 2023

Problem

A collection of $25$ consecutive positive integers adds to $1000.$ What are the smallest and largest integers in this collection?

Solution

Solution 1

The thirteenth integer is the average, which is $\frac{1000}{25}=40$. So, the largest integer is 12 larger, which is $40+12=\boxed{52}$, and the smallest integer is 12 less, which is $40-12=\boxed{28}$.

Solution 2

By the summation formula, the sum of 25 consecutive numbers (where $x$ is the smallest number in the list) is \[\frac{25(2x+24)}{2}\] \[25(x+12)\] Letting the value of the equation be $1000$, we have \[25(x+12)=1000\] \[x+12=40\] \[x=28\] Thus the smallest value of the list is $\boxed{28}$, and the largest is $28+24=\boxed{52}$

See also

1993 UNCO Math Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
1 2 3 4 5 6 7 8 9 10
All UNCO Math Contest Problems and Solutions