Two integers are in the ratio of 1 to 3. If 10 is added to the smaller number, the ratio becomes 2 to 1. Find the larger integer. Choose the correct option:  A) 6  B) 9  C) 10  D) 12  E) 15
One option is to set up the equations and solve:   If the ratio of two integers x and y is 1 to 3, then 3x=y, where x is the smaller integer.   If adding 10 to the smaller integer makes the ratio 2 to 1, then x+10=2y.   Substituting y=3x into the second equation yields x+10=6x.   So, x=2 (smaller integer) and, y=3x=6 (larger integer) so A is the correct answer.   Another option is to TEST the answer choices.   A) the larger number is 6   Given: the two integers are in the ratio of 1 to 3   So, the smaller number must be 2   If 10 is added to the smaller number, the ratio becomes 2 to 1   If we add 10 to the smaller number (2), we get 12.   So, the new ratio is 12 to 6 which is the same as 2 to 1.   Therefore choice A is correct.   B) the larger number is 9   Given: the two integers are in the ratio of 1 to 3   So, the smaller number must be 3   If 10 is added to the smaller number, the ratio becomes 2 to 1   If we add 10 to the smaller number (3), we get 13   So, the new ratio is 13 to 9.   No good. We want a resulting ratio of 2 to 1   ELIMINATE B   C) the larger number is 10   Given: the two integers are in the ratio of 1 to 3   So, the smaller number must be 3.33 (which is not an integer)   Since both numbers are integers, the larger number cannot be 10.   ELIMINATE C   D) the larger number is 12   Given: the two integers are in the ratio of 1 to 3   So, the smaller number must be 4   If 10 is added to the smaller number, the ratio becomes 2 to 1   If we add 10 to the smaller number (4), we get 14   So, the new ratio is 14 to 12 which is 7 to 6.   No good. We want a resulting ratio of 2 to 1   ELIMINATE D   E) the larger number is 15   Given: the two integers are in the ratio of 1 to 3   So, the smaller number must be 5   If 10 is added to the smaller number, the ratio becomes 2 to 1   If we add 10 to the smaller number (5), we get 15   So, the new ratio is 15 to 15. This is the same as the ratio of 1 to 1   ELIMIN6TE E   Therefore 6 is the only correct choice.
<result>A</result>
Two integers are in the ratio of 1 to 3. If 10 is added to the smaller number, the ratio becomes 2 to 1. Find the larger integer.