# SAT Mathematics : SAT Math

## Example Questions

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### Example Question #1 : Divisibility & Number Fluency

Jesse has a large movie collection containing  movies.  of his movies are action movies,  of the remainder are comedies, and the rest are historical movies. How many historical movies does Jesse own?

Explanation:

of the movies are action movies.  of  of the movies are comedies, or , or . Combining the comedies and the action movies ( or ), we get  of the movies being either action or comedy. Thus,  of the movies remain and all of them have to be historical.

### Example Question #2 : Divisibility & Number Fluency

If  and , find the value of .

Explanation:

Substitute the values of  and  into the given expression:

### Example Question #3 : Divisibility & Number Fluency

Let the nth term of a sequence be denoted as  and given by the following equation:

For example, the tenth term of the sequence is .

What is the sum of the first five terms of the sequence?

Explanation:

In order to find the sum of the first five terms, we will need to find the values of each of the first five terms using the equation given above. Essentially, we will let  range from  to  to determine each term.

Remember that anything to the power of zero is equal to 1. Therefore,  raised to the zero power is also .

In general, when  is raised to an even power, the result is . Conversely, if  is raised to an odd power, the result is .

Thus, the first five terms of the sequence are . We must now add these all together. Because we are adding fractions with unlike denominators, we need to find the smallest multiple that  have in common. Because  is a multiple of , we really only need to worry about . If we were to list out the multiple of , e would see that the smallest one they have in common is . Sometimes, the easiest way to find the least common multiple of several numbers is o multiply them together. The products of  is .

We will now convert each fraction to an equivalent form with a denominator of . For example, if we were to convert  to a fraction with a denominator of , we would multiple both the numerator and denominator by , as shown below:

### Example Question #4 : Divisibility & Number Fluency

What is the solution, reduced to its simplest form, for ?

Explanation:

### Example Question #5 : Divisibility & Number Fluency

What is the result of adding  of  to ?

Explanation:

Let us first get our value for the percentage of the first fraction.  of   is found by multiplying  by  (or, simplified, ):

Our addition is therefore . There are no common factors, so the least common denominator will be  or . Multiply the numerator and denominator of  by  and the numerator of  by

This yields:

, which cannot be reduced.

### Example Question #6 : Divisibility & Number Fluency

Explanation:

Find the least common denominator to solve this problem

Multiply 27 with , and multiply  with 3 to obtain common denominators.

Convert the fractions.

Combine the terms as one fraction.

### Example Question #7 : Divisibility & Number Fluency

A record store has stocked up on new supplies so that their current inventory consists of pop, rock, and hip-hop.  of their records are pop records and  of their records are rock records. If the store has a total of  records, how many hip-hop records are stocked in the store?

Explanation:

First we must find what fraction of hip-hop records are there in the store. We find a common denominator for the faction of pop records and rock records by multiplying  by  and  by  to get  records being pop records and  records being rock records.

In order to get the faction of hip-hop records remaining in the store, we must write out:

since the sum total of the fractions must add up to . We solve to get .

This tells us that fraction of hip-hop records must be .

The question is asking how many records are stocked in the store, therefore we need a denominator of . We can write out  and solve for .

Similarly, this question can be solved using deduction. If  of  is  and  of  is , knowing that we need the amount remaining, we can subtract  from  and get .

Answer choice  is wrong because it is finding the total amount of Pop records.

Answer choice  is wrong because it is finding the total amount of rock records.

### Example Question #8 : Divisibility & Number Fluency

Jackson wrote a song with equal parts pre-chorus, chorus, and bridge. The song ends up being  minutes and  seconds with a beat occurring every  seconds. How many beats are there in the chorus?

Explanation:

First, we must convert the song length from minutes and seconds to simply seconds.
.  We add  seconds to  seconds and get  seconds total song length.

If the song is equal parts pre-chorus, chorus, and bridge that means  of the song is composed of pre-chorus,  of the song is composed of chorus, and  of the song is composed of bridges. Therefore, we divide  by  and get  seconds of chorus.
Since a beat occurs every five seconds, we divide  seconds by  seconds to get the amount of beats in that time frame. . Since a beat can not be sectioned, the answer is rounded down to .

Answer choice  is incorrect because it is finding the total number of beats in the song.

Answer choice  is incorrect because it rounds up to the nearest beat.

### Example Question #9 : Divisibility & Number Fluency

Explanation:

The rule for dividing fractions is invert the second fraction and multiply.  We keep the  as it is then change the division sign to a multiplication sign and invert  into .
becomes .

Similarly, you can memorize, “top, bottom, bottom, top” meaning  is on top and it stays on top,  is labeled bottom and it stays on the bottom,  is labeled bottom and it moved to the bottom, then  is labeled top so it moves to the top.

### Example Question #10 : Divisibility & Number Fluency

Find the value of  if