Problem:

Solution:
1/(1+1/3) – 1/(1+1/2) (simplify both denominators)
= 1/(4/3) – 1/(3/2) (divide numerator by denominator)
= 3/4 – 2/3 (find common denominator)
= 9/12 – 8/12 = 1/12

Strategy:
When adding and subtracting fractions, the first step is always to get a common denominator.
When multiplying fractions, simplify first, then multiply numerators and denominators.
When dividing fractions, keep the first fraction, change the sign to multiplication, and invert the last fraction. Then, multiply.

Problem:
If n is a prime number greater than 3, what is the remainder when n-squared is divided by 12?

Solution:
The fastest way to solve this problem is to pick any prime number larger than 3 (5 being the easiest choice), squaring it, and dividing it by 12. 5-squared = 25 and the remainder when 25 is divided by 12 is 1.

Strategy:
Because only one answer choice can be correct, picking numbers works great for this problem.

You can also use modular arithmetic to solve this problem. Any prime number larger than 3 must be odd, so the remainder when that number is divided by 12 is 1, 5, 7, or 11 (note it can’t be 3 or 9 because then that number would be divisible by 3 and not prime). When you square the number the remainder is 1, 25, 49, or 121, and the remainder of 25, 49, and 121 when divided by 12 are all 1, thus the remainder when the square of a prime number is divided by 12 is 1.

Problem:
Of the five coordinates associated with points A, B, C, D, and E on the number line above, which has the greatest absolute value?

Solution:
The absolute value is greatest for the point farthest from the origin (0), which is point A.

Strategy:
Absolute value is the distance away from the orgin. Because you can never have a negative distance, the absolute value is always positive. For this problem, choose the coordinate that has the greatest distance from the orgin.

Problem:
Running at the same constant rate, 6 identical machines can produce a total of 270 bottles per minute. At this rate, how many bottles could 10 such machines produce in 4 minutes?

Solution:
6 machines produce 270 bottles per minute.
Therefore, 1 machine produces 270 / 6 = 45 bottles per minute.
Thus 10 machines produce 45 * 10 = 450 bottles in 1 minute, and
450 * 4 = 1,800 bottles in 4 minutes.

Strategy:
When working with rate problems, be sure to keep track of units. Your final answer should be the correct unit, in this case, minutes.

Problem:
The ratio of 2 to ⅓ is equal to the ratio of what?

Solution:
Multiply each side of the ratio 2 to ⅓ by 3, yielding 6 to 1.

Strategy:
You can also choose to write the ratio 2 to ⅓ as a fraction: 2 ÷ ⅓ which yields 6.