If the question can be answered by one of the statement alone

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Jake and Colin are painting a house together. How long would it take Colin to paint the house alone?

(1) Jake and Colin can paint the house together in 6 hours, working at the same rate.

(2) Jake can paint the house alone in 12 hours

Possible Answers:

Each statement alone is sufficient.

Both statements taken together are sufficient.

Statement (2) alone is sufficient.

Statements (1) and (2) are insufficient.

Statement (1) alone is sufficient.

Correct answer:

Statement (1) alone is sufficient.

Explanation:

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

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.

Celine and Jack are painting a house together at their respective rates. After working 1 hour together, Jack starts to feel tired and stops. How long does it take Celine to finish the work?

(1) Celine could have painted the house alone in 3 hours.

(2) It would have taken Jack 5 hours to paint the house alone.

Possible Answers:

Both statements taken together are sufficient.

Statement (1) alone is sufficient.

Statements (1) and (2) taken together are not sufficient.

Each statement alone is sufficient.

Statement (2) alone is sufficient

Correct answer:

Both statements taken together are sufficient.

Explanation:

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.

Machine X and machine Y both produce screwdrives at their respective rates. What is the rate of the machines working together?

(1) Machine X can produce 200 screwdrivers in an hour and machine Y can produce as many in 2 more hours.

(2) Machine Y can produce 200 screwdrivers in 3 hours. Machine X is three times as fast as machine Y.

Possible Answers:

Statements (1) and (2) are not sufficient.

Statement (1) alone is sufficient.

Each statement alone is sufficient.

Statement (2) alone is sufficient.

Both statements taken together are sufficient.

Correct answer:

Each statement alone is sufficient.

Explanation:

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:

Each statement alone is therefore sufficient.

Machine O working together with machine P produces telephones at a rate of 1056 per hour. How many telephones can machine O produce in an hour?

(1) It takes machine P four hours longer than machine O to produce 880 telephones.

(2) Machine P produces telephones at a rate of 176 telephones per hour.

Possible Answers:

Statements 1 and 2 together are not sufficient.

Statement (2) alone is sufficient.

Each statement alone is sufficient.

Statement (1) alone is sufficient.

Both statements together are sufficient.

Correct answer:

Each statement alone is sufficient.

Explanation:

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.

Machines A,B and C working together take 12 minutes to complete a watch.

What is the rate of both machines B and C working together?

(1) Machines A and B together can complete a watch in 20 minutes.

(2) Machines A and C together can complete a watch in 15 minutes.

Possible Answers:

Each statement alone is sufficient.

Statement (1) alone is sufficient.

Statements (1) and (2) taken together are not sufficient.

Both statements taken together are sufficient.

Statement (2) alone is sufficient.

Correct answer:

Both statements taken together are sufficient.

Explanation:

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

16 very precise machines, working at the same constant rate, can produce a luxury necklace working together. How many machines should we add in order for the necklace to be produced in an hour.

(1) It takes 5 hours for these machines to produce the necklace

(2) A single machine produces a necklace in 80 hours.

Possible Answers:

Statement (1) alone is sufficient.

Both statements together are sufficient.

Statements 1 and 2 together are not sufficient.

Each statement alone is sufficient.

Statement (2) alone is sufficient.

Correct answer:

Each statement alone is sufficient.

Explanation:

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.

Train A and Train B are moving toward one another. How long does it takes for train A to pass by train B?

(1) The distance between train A and train B is 180 miles.

(2) Train A's rate is 45 mph and train B's rate is 60 mph.

Possible Answers:

Statements 1 and 2 taken together are not sufficient.

Statement (1) alone is sufficient.

Statement (2) alone is sufficient.

Each statement alone is sufficient.

Both statements taken together are sufficient.

Correct answer:

Both statements taken together are sufficient.

Explanation:

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.

A train makes roundtrips between two cities at an average speed of 75 mph. What is the distance between the two cities, taking into consideration that the train does not travel at the same speed for both trips?

(1) The train takes 50 minutes to do one way.

(2) The train takes 60 minutes to do the other way, which is uphill.

Possible Answers:

Statement (2) alone is sufficient.

Both statements taken together are sufficient.

Each statement alone is sufficient.

Statement (1) alone is sufficient.

Statements 1 and 2 taken together are not sufficient.

Correct answer:

Both statements taken together are sufficient.

Explanation:

Firstly, we should remember that the average rate is given by the following formula

, where

is the total distance and

is the total time. So to answer this question we should find a value for

can only be found by adding the two times for both trips. By pluging in the values we can find a value for

, therefore we need both statements.

How long does it take train A to reach a town which is 500 miles away, knowing that the entire portion of the rails are damaged?

(1) The train usually goes 500 miles in 675 minutes.

(2) Since the rails are damaged it typically takes train A twice the usual time.

Possible Answers:

Statement 1 alone is sufficient.

Statement 2 alone is sufficient.

Both statements taken together are sufficient.

Statements 1 and 2 taken together are insufficient.

Each statement alone is suffcient.

Correct answer:

Both statements taken together are sufficient.

Explanation:

To solve this problem, we need to find the rate of the train considering the fact that the rails are damaged. The first statement tells us only the usual rate of the train and is therefore not sufficient because we don't know how fast the train will be going on the damaged portions of the railroad.

Statement two only tells us that the train must progress at a rate half as slow as its usual rate.

Using statements 1 and 2 we can easily find the rate which is given by

. Note that this rate is given in miles per minutes but we don't have to calculate it, we just need to know that we can calculate it.

A plane makes a round trip at an average speed of 650 mph.

What was the speed of the plane on the second portion of the flight?

(1) It took the plane twice as long to do the first portion of the flight

(2) The plane flew at a speed of 750mph on the second portion of the flight

Possible Answers:

Both statements together are sufficient.

Statement 1 alone is sufficient.

Statements 1 and 2 together are not sufficient.

Statement 2 alone is sufficient.

Each statement alone is sufficient.

Correct answer:

Each statement alone is sufficient.

Explanation:

Since we are told an average speed for a round trip we should be able to set the following equation

, where

is the distance of a one way and

is the total time of the round trip and

is the average speed of the plane.

Statement 1 tells us that it took the plane twice as long to do the first portion of the round trip, therefore we can figure out

from this statement, indeed,

where

is the time it took to do the second portion of the trip. Therefore,

. Since

is the rate for the second portion of the trip, we can figure it out by pluging in the values

, which is sufficient to answer the problem.

Statement 2 tells us that the plane was flying at a speed of

, this alone answers the question.

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