What is the ones digit of 347^105? a) 1 b) 3 c) 5 d) 7 e) 9
The correct answer is (d), or 7. When you get a problem like this -- where the number is so big your calculator can't display it properly -- you'll need to find a pattern to how the number changes when you take different powers of it. For example, 47^1 = 47, which has a ones digit of 7. 47^2 = 2209, which has a ones digit of 9. 347^3 has a ones digit of 3. 347^4 has a ones digit of 1, and 47^5 has a ones digit of 7. If you take the next few powers, you'll see the pattern repeat itself. This means that every fourth power of 47 has a ones digit of 1. So 47^104 has a ones digit of 1. That means that 47^105 must have a ones digit of 7 -- the next digit in the pattern we found before. If you didn't have enough time to work out the answer, you could still make a good guess by getting rid of (c) before you guessed. Remember -- the only numbers with a ones digit of 5 are multiples of 5, and since 47 isn't a multiple of 5, none of its powers will be either.
If 65 percent of the people who check out a certain library book are Spanish majors, what is the ratio of the number of Spanish majors who check out the book to non-Spanish majors who check out the book? a) 6 to 3 b) 13 to 5 c) 13 to 7 d) 26 to 10 e) 65 to 35
The answer is (c), or 13 to 7. An easy way to answer questions like this is to assume that a certain number of people checked out the book, and see how many were Spanish majors. For example, we could assume that 100 people checked it out -- 100 is an easy number to use with percentages. That would mean 65 Spanish majors checked it out, and 100-65 = 35 non-Spanish majors checked it out. So you know that, if 65 Spanish majors were to check out the book, 35 non-Spanish majors would check it out. The ratio of 65 to 35 simplifies to a ratio of 13 to 7. (You'll get the same ratio for any number you start with, not just 100, since ratios are consistent for any starting value.)
If triangle ABC has one side (A) of length 8, and one side (B) of length 10, which of the following could NOT be the length of side C? a) 4 b) 8 c) 10 d) 18
The answer is (d), or 18. Any two sides of a triangle must be longer, combined, than the third side of that triangle. (Try to draw a triangle where this isn't true, and you'll quickly see why!) Since side C can't be as long as sides A and B combined, it must be less than 8+10, or less than 18.