The rate formula  - or, actually, the equivalent equation  - will help.

Let  be the speed limits along the interstate, the first half of the state route, and the second half of the state route, respectively.

Then Tom can drive I-10 in  hours; the first half of SR 34, ; the second half of SR 34, .

The time  it takes, in hours, for Tom to make the entire trip is the sum of these fractions: 

.

To calculate , we need . Statement 1 only tells us ; Statement 2 only tells us  and . Therefore, both together, but neither alone, are enough to allow us to calculate .

From the first statement, we can calculate the number of cupcakes Cassie can decorate in 150 minutes. From there, we can calculate the rate at which Derek decorates. Therefore, Statement 1 alone is sufficient to answer the question.

The second statement gives the rate at which Derek decorates cupcakes. Therefore, Statement 2 alone is also sufficient to answer the question.

Statement 1 alone does not tell us enough about the speed of train B to answer the question. Statement 2 alone does not tell us about the rates of either individual train, so it is also not sufficient by itself to answer the question.

But can we figure out the answer using both pieces of information? Yes, we can! If we know that the trains meet at 6:40, then their combined rate of travel is 60 total miles in 40 minutes. Statement 1 says that Train A travels twice as fast as Train B, so we can determine the distances covered by the two trains in those 40 minutes. From there, we can find rates for both trains and then answer the question. Thus, both statements together are sufficient to answer the question, but neither statement alone is sufficient.

Note: We didn't actually answer the question of Train B's arrival time. For data sufficiency questions, don't waste time trying to find the specific answer. All that is necessary is determining whether or not it is POSSIBLE to answer the question with the information given.

Here we can see that there are two people painting the house, so we should set the following equation : , where  is the rate for Jake and where  is the rate for Colin,  stands for their rate together. Statement one gives us a value for  and even though it doesn't tell us what is  or  it tells us that  which allow us to answer the question. 

For Statement 2, we are left with two other unknowns in our equation and we can clearly see that (2) is not sufficient.

We should start by setting equations  and , these are the two equation we should be able to solve. Where  is the amount of the work they did togetherm  is the amount of work left,  is Celine's rate and  is the rate of Celine and Jack together. So with  we can see that we would need both Jack and Celine's rate. Since they work at a different rate, we shoud only be able to answer this problem taking both statements together.

To be able to solve this problem, we need to figure out  where  is the rate of machine X and  is the rate of machine Y. Statement 1 tells us the rate for machine X, but tells us as well that it takes 2 more hours for machine Y to produce the same amount of screwdrivers. Therefore, we are told that  and , which is sufficient to answer the question. Statement 2 tells us that the rate of machine Y is  and also that that rate of machine X is 3 times as fast as machine Y:  so . 

Each statement alone is therefore sufficient.

To solve this problem we first have to set up an equation for our variables:  where  is the rate of machine O and  is the rate of machine P.

The first statement tells us that  and , where  is the time it takes machine O to produce 880 telephones.

At first it might look insufficient but, by pluging in the values for  and  in our first equation we get :

, this gives us a quadratic equation, in which we can solve and find t, and therefore find the number of telephones O can produce in 1 hour.

The second statement tell us that , therefore, we can plug in this value in our first equation to find the rate for machine O and this will allow us to answer the question.

Firstly we should set up an equation for this problem:  where  are the respective rates of machines A, B and C. We need to find .

Statement 1 tells us that .

This alone is not suffient since we will have no precise information on machine B. Similarly, with statement 2 We can find the value of the rate for machine B but we cannot know what is the rate of machine C, from what we are told. However, taking both statements together, with statement 1 we can find the value for  and with statement 2 we can find the value for  and thereby we can find the value of .

We know that there are 16 machines and we are looking for how many we should add. Let us set an equation representing the number of machines in use , where  is the total number of machines we need and  is the rate of an individual machine. We should therefore find a rate for a single machine to be able to solve this problem, since we can then calculate .

Statement 1 tells us that for 16 machines, it takes 5 hours to produce a necklace. From this we can find the rate of a single machine.

Statement 2 tells us direclty the rate of a single machine, therefore both these statements allow us to answer the problem.

Since the problem doesn't tell us anything about the rates and the distance between the two train, there is not much we can say. Statement one tells us that there is 180 miles between the two trains. This is not sufficient, since we don't know how fast the trains are.

Statement 2 alone tells us the rates of the trains, but we don't know how far away they are, this statement alone doesn't help us answer the question.

If we take both statements together however, we can see that the distance that each train would have made when both trains meet, is a total of 180 miles, since both trains were 180 miles away.

We can create the following equation , where  is the time it takes for both train to meet at a given point and  and  are the trains A and B respective rates. The information we have allows us to solve the equation for  and therefore we can answer the problem.