# Number Properties

Read the following SAT test question and then select the correct answer.

Read the question carefully so that you are sure you understand what you are being asked.  You must identify the bottom line that you will solve for; note it at the top of your scratch work.  Assess your options to find the most efficient way to solve the problem, and then attack the problem.  Be sure to write out your scratch work clearly so that you do not make careless mistakes.  Your last step is to loop back to make sure that your answer matches the bottom line.  You cannot get a question right if you solved for an answer that you were not asked to find.

For how many positive two-digit integers is the ones digit greater than twice the tens digit?

You must find out how many numbers fit the given requirements.  There is no formula to find numbers in which one digit is more than twice the other, so you must think about this question logically.  Your only option is to methodically check positive integers to see which numbers will work.

You know that you need a positive two digit integer, so your first digit, at the very least, must be a 1.  Your second digit must be more than twice the first.  The number 1 multiplied by 2 is 2, so the number 12 will not fit the requirement of having a ones digit greater than twice the tens digit.   However, any number that begins with a 1 and has a second digit larger than 2 will work.  List all the numbers that begin with 1 and fit the requirements of this problem:

13, 14, 15, 16, 17, 18, 19

After 19, you must start each number with a 2, so find out what the second digit must be.  Again, it must be bigger than twice the first digit, so it must be larger than 4.  24 will not work, so start with 25 and list all of the numbers that fit the requirements of this problem:

25, 26, 27, 28, 29

Follow this procedure for numbers beginning with 3. 3 times 2 is 6, so only numbers larger than 36 will work.

37, 38, 39

Move on to numbers that start with a 4.  4 times 2 is 8, so the second digit must be greater than 8.  This time there is only one number that fits the requirements:

49

Now you have reached numbers beginning with the digit of 5.  5 times 2 is 10.  You cannot have a value as your second digit that is more 10, so any number larger than 49 will not work.

Count up all of the numbers that you have found that fit the requirements of this problem.  That number will satisfy your bottom line.

(A) 16
(B) 20
(C) 28
(D) 32
(E) 36