SAT Math : Algebraic Fractions

Study concepts, example questions & explanations for SAT Math

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Example Questions

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Example Question #1 : Algebraic Fractions

The expression (\frac{a^{2}}{b^{3}})(\frac{a^{-2}}{b^{-3}}) = ?

Possible Answers:

\frac{a^{-4}}{b^{-9}}

0

1

\frac{b^{9}}{a^{4}}

b^{-9}

Correct answer:

1

Explanation:

A negative exponent in the numerator of a fraction can be rewritten with a positive exponent in the denominator. The same is true for a negative exponent in the denominator. Thus, \frac{a^{-2}}{b^{-3}} =\frac{b^{3}}{a^{2}}.

When \frac{a^{2}}{b^{3}} is multiplied by \frac{b^{3}}{a^{2}}, the numerators and denominators cancel out, and you are left with 1.

Example Question #2 : How To Simplify A Fraction

Two two-digit numbers, and , sum to produce a three-digit number in which the second digit is equal to . The addition is represented below. (Note that the variables are used to represent individual digits; no multiplication is taking place).

What is ?

Possible Answers:

Correct answer:

Explanation:

Another way to represent this question is:

In the one's column, and add to produce a number with a two in the one's place. In the ten's column, we can see that a one must carry in order to get a digit in the hundred's place. Together, we can combine these deductions to see that the sum of and must be twelve (a one in the ten's place and a two in the one's place).

In the one's column:

The one carries to the ten's column.

In the ten's column:

The three goes into the answer and the one carries to the hundred's place. The final answer is 132. From this, we can see that because .

Using this information, we can solve for .

You can check your answer by returning to the original addition and plugging in the values of and .

Example Question #3 : How To Simplify A Fraction

Let  \frac{\frac{1}{4}+\frac{1}{3}-\frac{1}{5}}{\frac{1}{2}-\frac{1}{6}+1}  = \frac{a}{b}, where  and  are both positive integers whose greatest common factor is one. What is the value of ?

Possible Answers:

115

44

34

73

103

Correct answer:

103

Explanation:

First we want to simplify the expression: \frac{\frac{1}{4}+\frac{1}{3}-\frac{1}{5}}{\frac{1}{2}-\frac{1}{6}+1}.

One way to simplify this complex fraction is to find the least common multiple of all the denominators, i.e. the least common denominator (LCD). If we find this, then we can multiply every fraction by the LCD and thereby be left with only whole numbers. This will make more sense in a little bit.

The denominators we are dealing with are 2, 3, 4, 5, and 6. We want to find the smallest multiple that these numbers have in common. First, it will help us to notice that 6 is a multiple of both 2 and 3. Thus, if we find the least common multiple of 4, 5, and 6, it will automatically be a multiple of both 2 and 3. Let's list out the first several multiples of 4, 5, and 6.

4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60

5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60

6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60

The smallest multiple that 4, 5, and 6 have in common is 60. Thus, the least common multiple of 4, 5, and 6 is 60. This also means that the least common multiple of 2, 3, 4, 5, and 6 is 60. Therefore, the LCD of all the fractions is 60.

Let's think of the expression we want to simplify as one big fraction. The numerator contains the fractions 1/4, 1/3, and –1/5. The denominator of the fraction is 1/2, –1/6 and 1. Remember that if we have a fraction, we can multiply the numerator and denominator by the same number without changing the value of the fraction. In other words, x/y = (xz)/(yz). This will help us because we can multiply the numerator (which consists of 1/4, 1/3, and –1/5) by 60, and then mutiply the denominator (which consists of 1/2, –1/6, and 1) by 60, thereby ridding us of fractions in the numerator and denominator. This process is shown below:

\frac{\frac{1}{4}+\frac{1}{3}-\frac{1}{5}}{\frac{1}{2}-\frac{1}{6}+1}\frac{60}{60}\cdot\frac{\frac{1}{4}+\frac{1}{3}-\frac{1}{5}}{\frac{1}{2}-\frac{1}{6}+1} = \frac{60\cdot \frac{1}{4}+60\cdot \frac{1}{3}-60\cdot \frac{1}{5}}{60\cdot \frac{1}{2}-60\cdot \frac{1}{6}+60\cdot 1}

= \frac{15+20-12}{30-10+60}=\frac{23}{80}

This means that a/b = 23/80. We are told that a and b are both positive and that their greatest common factor is 1. In other words, a/b must be the simplified form of 23/80. When a fraction is in simplest form, the greatest common factor of the numerator and denominator equals one. Since 23/80 is simplified, a = 23, and b = 80. The sum of a and b is thus 23 + 80 = 103.

The answer is 103.

Example Question #1 : Mixed / Improper Fractions

Change to a mixed number 

Possible Answers:

Correct answer:

Explanation:

To convert from a fraction to a mixed number we must find out how many times the denominator goes into the numerator using division and the remainder becomes the new fraction. 

 

Example Question #1 : Algebraic Fractions

What is the average of and ?

Possible Answers:

Correct answer:

Explanation:

To average, we have to add the values and divide by two. To do this we need to find a common denomenator of 6. We then add and divide by 2, yielding 4.5/6. This reduces to 3/4.

Example Question #1 : Algebraic Fractions

Which of the following is equivalent to  ?

Possible Answers:

None of the answers are correct

 

Correct answer:

Explanation:

This problem is solved the same way ½ + 1/3 is solved.  For example,  ½ + 1/3 = 3/6 + 2/6 = 5/6.  Find a common denominator then convert each fraction into an equivalent fraction using that common denominator.  The final step is to add the two new fractions and simplify.

Example Question #1 : Algebraic Fractions

A train travels at a constant rate of  meters per second. How many kilometers does it travel in  minutes?

 

Possible Answers:

Correct answer:

Explanation:

Set up the conversions as fractions and solve:

\dpi{100} \small \frac{20m}{1sec}\times \frac{60sec^}{1min}\times \frac{1km}{1000m}\times \frac{10min}{1}

Example Question #1 : How To Simplify A Fraction

Simplify.  \frac{4x^{4}z^{3}}{2xz^{2}}

Possible Answers:

Can't be simplified

2x^{4}z^{3}

\frac{2x^{4}z^{3}}{xz}

2x^{3}z

4x^{3}z

Correct answer:

2x^{3}z

Explanation:

To simplify exponents which are being divided, subtract the exponents on the bottom from exponents on the top.  Remember that only exponents with the same bases can be simplified

Example Question #2621 : Sat Mathematics

Simplify:

 

 

Possible Answers:

Correct answer:

Explanation:

x2 – y2 can be also expressed as (x + y)(x – y).

Therefore, the fraction now can be re-written as (x + y)(x – y)/(x + y).

This simplifies to (x – y).

Example Question #1 : Algebraic Fractions

Simplify:

Possible Answers:

Correct answer:

Explanation:

Notice that the term appears frequently. Let's try to factor that out:

Now multiply both the numerator and denominator by the conjugate of the denominator:

 

 

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