Four staff members at a certain company worked on a project. The amounts of time that the four staff members worked on the project were in the ratio 2 to 3 to 5 to 6. If one of the four staff members worked on the project for 30 hours, which of the following CANNOT be the total number of hours that the four staff members worked on the project?
(A) 80
(B) 96
(C) 160
(D) 192
(E) 240
It is given that the time contributions of four project members were in the ratio 2:3:5:6. By introducing a multiplier ( x ), these contributions can be expressed as ( 2x ), ( 3x ), ( 5x ), and ( 6x ), where ( x ) represents a common unit of time.
Summing these contributions, the total time spent by the group is ( 2x + 3x + 5x + 6x = 16x ).
Given that one member worked 30 hours, we derive four scenarios, depending on which member this was:
- Option 1: If Person 1 worked 30 hours, then ( 2x = 30 ) leading to ( x = 15 ).
- Option 2: If Person 2 worked 30 hours, then ( 3x = 30 ) leading to ( x = 10 ).
- Option 3: If Person 3 worked 30 hours, then ( 5x = 30 ) leading to ( x = 6 ).
- Option 4: If Person 4 worked 30 hours, then ( 6x = 30 ) leading to ( x = 5 ).
Using these ( x ) values, the total working hours for the group are calculated as follows:
- Option 1: Total = ( 16 \times 15 = 240 ) hours
- Option 2: Total = ( 16 \times 10 = 160 ) hours
- Option 3: Total = ( 16 \times 6 = 96 ) hours
- Option 4: Total = ( 16 \times 5 = 80 ) hours
These results outline the total hours the group could have worked based on each scenario. Each ( x ) value provides a different total, highlighting the impact of the assumed work distribution.
answer D
Ratio and proportion are fundamental mathematical concepts that often come up in various contexts, from solving problems in algebra to making recipes in the kitchen.
Ratio
A ratio is a way to compare two quantities by division, expressing how many times one quantity is as large as another. It is typically written in three forms: as a fraction (e.g., 1/2), with a colon (e.g., 1:2), or with the word “to” (e.g., 1 to 2). Ratios are used to express the relative sizes of two or more things. For example, if you have 2 apples and 3 oranges, you can describe the ratio of apples to oranges as 2:3.
Proportion
A proportion, on the other hand, is an equation that states that two ratios are equal. It involves four terms, with the first term divided by the second term being equal to the third term divided by the fourth term; for example, 1/2 = 3/6. This indicates that the two fractions or ratios are equivalent. Proportions are used to solve problems where you need to find an unknown member of one of the ratios, maintaining the equality of the two ratios.
Together, these concepts are used extensively in problems involving scaling, resizing, and distributing quantities in a balanced way.
Rule of Three for Direct Proportionality GMAT
The Rule of Three for Direct Proportionality is a mathematical method used to solve problems where three values are known and a fourth value is to be found, and the quantities involved are directly proportional. This method is particularly useful when dealing with problems where two corresponding sets of quantities increase or decrease at the same rate.
Here’s how it works:
- Establish the proportion: Recognize that two quantities or sets of quantities are in direct proportion. This means if one quantity increases, the other increases in such a way that the ratio between them remains constant.
- Set up the known ratios: Arrange the three known values and the unknown value into two ratios. The known three values include two values that correspond to one set of conditions, and one value that corresponds to another set of conditions. The unknown is the value you need to find that corresponds to the second condition.
- Apply the cross-multiplication method: Create an equation such that the product of the means equals the product of the extremes (i.e., in the ratio a/b = c/d, ad = bc). This method helps to find the unknown value that will keep the proportion consistent.
For example, if you know that 5 apples cost $10 and you want to find out how much 8 apples cost, assuming the cost is directly proportional to the number of apples, you would set up the proportion:
- 5 apples / $10 = 8 apples / x dollars
- Solving for x using cross-multiplication gives you (5)(x) = (10)(8), leading to x = $16.
This demonstrates that the rule of three is a straightforward way to resolve problems in direct proportion by maintaining a consistent ratio across different scenarios. GMAT PREP.
