Solutions

1. 100

   First set up a proportion and then cross-multiply:

   

   Isolate x and use the calculator to solve.

   
2. 45
   

   First substitute 3x for y.

   

   Then solve for x :  x = 3 × 15 = 45.

3. 11

   Write a proportion to solve this problem:

   

   Cross-multiply to solve:

   
4. 43

   First, establish the starting number of men and women with a proportion, and simplify:

   

   Cross-multiply:

   5x = 7 × 35

   Isolate x and use the calculator to solve:

   

   If 6 women leave the room, there are 49 − 6 = 43 women left.

5. 108

   If the ratio of cats to dogs is 5 : 6, then there are 5x cats and 6x dogs (using the Unknown Multiplier). Express the fact that there are 18 fewer cats than dogs with an equation:

   

   Therefore, there are 6(18) = 108 dogs.

6. 33 hours

   Use an equation with the Unknown Multiplier to represent the total hours put in by the three people:

   

   Therefore, the person who worked the most hours put in 5(11) = 55 hours, and the person who worked the least hours put in 2(11) = 22 hours. This represents a difference of 55 − 22 = 33 hours.

7. (A)

   You can use an Unknown Multiplier x to help express the number of students and teachers. In light of the given ratio, there would be x teachers and 8x students, and the total number of people on the field trip would therefore be x + 8x = 9x. Note that x in this case must be a positive integer, because you cannot have fractional people.

   The total number of people must therefore be a multiple of 9. The only multiple of 9 between 60 and 70 is 63. Therefore, x must be which equals 7. Rewrite the quantities:

   Quantity A	Quantity B
   The number of teachers on the field trip = 7	6

   Therefore, Quantity A is greater.

8. (D)

   While you know the ratio of men to women, you do not know the actual number of men and women. The following Before and After charts illustrate two of many possibilities:

   Case 1	Men	Women
   Before	3	4
   After	4	3

   Case 2	Men	Women
   Before	9	12
   After	10	11

   These charts illustrate that the number of men may or may not be greater than the number of women after the move. Therefore, the relationship cannot be determined from the information given.

9. (B)

   This Multiple Ratio problem is complicated by the fact that the number of rubies is not consistent between the two given ratios, appearing as 2 in one and 3 in the other. You can use the least common multiple of 2 and 3 to make the number of rubies the same in both ratios:

   E:R:S		E:R:S
   1:2	multiply by 3	3:6
   3:5	multiply by 2	6:10

   Combining the two ratios into a single ratio yields:

   E : R : S: Total = 3 : 6: 10 : 19

   The smallest possible total number of gemstones is 19. Therefore, Quantity B is greater.