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? | ||
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== Solution == | == Solution == | ||
+ | === 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>. | ||
+ | |||
+ | === Solution 2 === | ||
+ | By the summation formula, the sum of 25 consecutive numbers (where <math>x</math> is the smallest number in the list) is | ||
+ | <cmath>\frac{25(2x+24)}{2}</cmath> | ||
+ | <cmath>25(x+12)</cmath> | ||
+ | Letting the value of the equation be <math>1000</math>, we have | ||
+ | <cmath>25(x+12)=1000</cmath> | ||
+ | <cmath>x+12=40</cmath> | ||
+ | <cmath>x=28</cmath> | ||
+ | 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 consecutive positive integers adds to What are the smallest and largest integers in this collection?
Solution
Solution 1
The thirteenth integer is the average, which is . So, the largest integer is 12 larger, which is , and the smallest integer is 12 less, which is .
Solution 2
By the summation formula, the sum of 25 consecutive numbers (where is the smallest number in the list) is Letting the value of the equation be , we have Thus the smallest value of the list is , and the largest is
See also
1993 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
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 |