Difference between revisions of "2023 AMC 10B Problems/Problem 5"
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| + | ==Solution 2== | ||
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| + | Let <math>x_1,x_2,x_3,...,x_n</math> where <math>x_n</math> represents the <math>n</math>th number written on the board. Lara's multiplied each number by <math>3</math>, so her sum will be <math>3x_1+3x_2+3x_3+...+3x_n</math>. This is the same as <math>3\cdot (x_1+x_2+x_3+...+x_n)</math>. We are given this quantity is equal to <math>45</math>, so the original numbers add to <math>\frac{45}{3}=15</math>. Maddy adds <math>3</math> to each of the <math>n</math> terms which yields, <math>x_1+3+x_2+3+x_3+3+...+x_n+3</math>. This is the same as the sum of the original series plus <math>3 \cdot n</math>. Setting this equal to <math>45</math>, <math>15+3n=45 \Rightarrow n = 10</math>. <math>\boxed{\textbf{(E) }10}.</math> | ||
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| + | ~vsinghminhas | ||
Revision as of 18:04, 15 November 2023
Problem
Maddy and Lara see a list of numbers written on a blackboard. Maddy adds 
to each number in the list and finds that the sum of her new numbers is 
. Lara multiplies each number in the list by and finds that the sum of her new numbers is also . How many numbers are written on the blackboard?
Solution
Let there be 
numbers in the list of numbers, and let their sum be 
. Then we have the following
From the second equation, 
, so 
. Solving, we find 
~Mintylemon66
Solution 2
Let 
where 
represents the 
th number written on the board. Lara's multiplied each number by , so her sum will be 
. This is the same as 
. We are given this quantity is equal to , so the original numbers add to 
. Maddy adds to each of the terms which yields, 
. This is the same as the sum of the original series plus 
. Setting this equal to , 
. 
~vsinghminhas
