How to find the solution to a rational equation with LCD - GRE Quantitative Reasoning

The definition of an arithmetic mean of a set of values is given as the sum of all the values divided by the total count of values:

Where represents the value in a set, and is the number of values in the set.

Quantity B can thus be defined as follows:

Which simplifies to:

or, simplifying:

We are told that the mean of , , , and is 14, which can be written as:

and then as

Plugging this value into our definition of Quantity B, we can find its numerical value:

So

The mathematical approach to this problem would be to set up a system of equations; one which represents the current relationship of Satoshi's and Reginald's number of bottle caps:

And one which represents the relationship 6 bottle caps later, wherein Satoshi would have three times as many bottlecaps as Reginald:

Subtracting the first equation from the second equation provides:

Which simplifies to

Solving this yields the value for Reginald's number of bottle caps, which is 17, which can be plugged back into the very first equation to find Satoshi's number of bottle caps, which is 63.

There is, however, another way to find this solution, which is by first looking at the answer choices.

Since we know that Satoshi's number of bottlecaps plus six must be divisible by 3 (since it is three times Reginald's new number of caps), we can quickly eliminate the choices 86, 143, and 52. Furthermore, since Satoshi has 46 more bottlecaps than Reginald, it'd be impossible for him to have 21 bottlecaps. This leaves only the 63 option.

When taking standardized math tests, where time management is key, it can be helpful to look for quicker ways to find a solution, such as back-solving, rather than just solving the problem normally.

The key to solving this problem is deciphering the language and translating it into a numerical representation. The first part can be written as an equaltiy as follows:

Rearranging terms allows us to solve for this mystery number:

From there we can address the problem's question:

To solve this problem, begin by calculating the amount of time it would take to complete the 120 mile journey at an average of 60 miles per hour:

Next, calculate the time it would take to complete half of the journey, 60 miles, at a speed of 40 miles per hour.

In order to average 60 miles per hour, the rest of the journey would need to be completed in the remaining .05 hours. Thus the speed for the final leg of the journey is determined as follows:

Well over the speed limit.

This problem has three unknowns, , , and, ; however, it also has three separate equations which, if they are independent, enable us to solve for these variables.

One way to begin is by solving for using the first two equations:

Subtracting the bottom equation from the top gives:

The next step will then be to try to combine equations in a way that will eliminate , so that we're only working with our variable of interest, , and , which is now known. To do this, let's compare the second and third equations:

In order to eliminate the , we will want it to have the same coefficient in both equations. We can do this by multiplying the top equation by 4 to get a new system of equations:

Subtracting the top from the bottom yields:

Plugging in the value for then gives:

The definition of an arithmetic mean of a set of values is given as the sum of all the values divided by the total count of values:

Where represents the value in a set, and is the number of values in the set.

Quantity B can thus be defined as follows:

Which simplifies to:

or, simplifying:

We are told that the mean of , , , and is 14, which can be written as:

and then as

Plugging this value into our definition of Quantity B, we can find its numerical value:

So

The mathematical approach to this problem would be to set up a system of equations; one which represents the current relationship of Satoshi's and Reginald's number of bottle caps:

And one which represents the relationship 6 bottle caps later, wherein Satoshi would have three times as many bottlecaps as Reginald:

Subtracting the first equation from the second equation provides:

Which simplifies to

Solving this yields the value for Reginald's number of bottle caps, which is 17, which can be plugged back into the very first equation to find Satoshi's number of bottle caps, which is 63.

There is, however, another way to find this solution, which is by first looking at the answer choices.

Since we know that Satoshi's number of bottlecaps plus six must be divisible by 3 (since it is three times Reginald's new number of caps), we can quickly eliminate the choices 86, 143, and 52. Furthermore, since Satoshi has 46 more bottlecaps than Reginald, it'd be impossible for him to have 21 bottlecaps. This leaves only the 63 option.

When taking standardized math tests, where time management is key, it can be helpful to look for quicker ways to find a solution, such as back-solving, rather than just solving the problem normally.

The key to solving this problem is deciphering the language and translating it into a numerical representation. The first part can be written as an equaltiy as follows:

Rearranging terms allows us to solve for this mystery number:

From there we can address the problem's question:

To solve this problem, begin by calculating the amount of time it would take to complete the 120 mile journey at an average of 60 miles per hour:

Next, calculate the time it would take to complete half of the journey, 60 miles, at a speed of 40 miles per hour.

In order to average 60 miles per hour, the rest of the journey would need to be completed in the remaining .05 hours. Thus the speed for the final leg of the journey is determined as follows:

Well over the speed limit.

This problem has three unknowns, , , and, ; however, it also has three separate equations which, if they are independent, enable us to solve for these variables.

One way to begin is by solving for using the first two equations:

Subtracting the bottom equation from the top gives:

The next step will then be to try to combine equations in a way that will eliminate , so that we're only working with our variable of interest, , and , which is now known. To do this, let's compare the second and third equations:

In order to eliminate the , we will want it to have the same coefficient in both equations. We can do this by multiplying the top equation by 4 to get a new system of equations:

Subtracting the top from the bottom yields:

Plugging in the value for then gives:

The definition of an arithmetic mean of a set of values is given as the sum of all the values divided by the total count of values:

Where represents the value in a set, and is the number of values in the set.

Quantity B can thus be defined as follows:

Which simplifies to:

or, simplifying:

We are told that the mean of , , , and is 14, which can be written as:

and then as

Plugging this value into our definition of Quantity B, we can find its numerical value:

So