Difference between revisions of "2016 AMC 10A Problems/Problem 14"
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==Solution 2== | ==Solution 2== | ||
| − | You can also see that you can rewrite the word problem into a equation <math>2x</math> + <math>3y</math> = <math>2016</math>. Therefore the question is just how many multiples of 3 subtracted from 16 will be an even number. We can see if <math>y</math> = 0, <math>x</math> = 1008. All the way to <math>x</math> = 0, and <math>y</math> = <math>672</math>.Therefore, between <math>0</math> and <math>672</math>, the number of multiples of 2 is (C) 337. | + | You can also see that you can rewrite the word problem into a equation <math>2x</math> + <math>3y</math> = <math>2016</math>. Therefore the question is just how many multiples of 3 subtracted from 16 will be an even number. We can see if <math>y</math> = 0, <math>x</math> = 1008. All the way to <math>x</math> = 0, and <math>y</math> = <math>672</math>.Therefore, between <math>0</math> and <math>672</math>, the number of multiples of 2 is \boxed{\textbf{(C)}\$ 337}$. |
Revision as of 18:46, 3 February 2016
Problem
How many ways are there to write 
as the sum of twos and threes, ignoring order? (For example, 
and 
are two such ways.)
Solution 1
The amount of twos in our sum ranges from 
to 
, with differences of 
because 
.
The possible amount of twos is 
Solution 2
You can also see that you can rewrite the word problem into a equation 
+ 
= . Therefore the question is just how many multiples of 3 subtracted from 16 will be an even number. We can see if 
= 0, 
= 1008. All the way to = 0, and = 
.Therefore, between and , the number of multiples of 2 is \boxed{\textbf{(C)}$ 337}$.
