How to simplify a fraction - PSAT Math
Card 1 of 238
Simplify x/2 – x/5
Simplify x/2 – x/5
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Simplifying this expression is similar to 1/2 – 1/5. The denominators are relatively prime (have no common factors) so the least common denominator (LCD) is 2 * 5 = 10. So the problem becomes 1/2 – 1/5 = 5/10 – 2/10 = 3/10.
Simplifying this expression is similar to 1/2 – 1/5. The denominators are relatively prime (have no common factors) so the least common denominator (LCD) is 2 * 5 = 10. So the problem becomes 1/2 – 1/5 = 5/10 – 2/10 = 3/10.
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If $\frac{p}{6}$ is an integer, which of the following is a possible value of p?
If $\frac{p}{6}$ is an integer, which of the following is a possible value of p?
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$\frac{0}{6}$=0, which is an integer (a number with no fraction or decimal part). All the other choices reduce to non-integers.
$\frac{0}{6}$=0, which is an integer (a number with no fraction or decimal part). All the other choices reduce to non-integers.
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Simplify: $$\frac{4x^{5}$$$y^{3}$$z}{12x^{3}$$y^{6}$$z^{2}$}
Simplify: $$\frac{4x^{5}$$$y^{3}$$z}{12x^{3}$$y^{6}$$z^{2}$}
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$$\frac{4x^{5}$$$y^{3}$$z}{12x^{3}$$y^{6}$$z^{2}$$}=\frac{x^{2}$$$}{3y^{3}$z}
First, let's simplify $\frac{4}{12}$. The greatest common factor of 4 and 12 is 4. 4 divided by 4 is 1 and 12 divided by 4 is 3. Therefore $\frac{4}{12}$=\frac{1}{3}$.
To simply fractions with exponents, subtract the exponent in the numerator from the exponent in the denominator. That leaves us with $$\frac{1}{3}$x^{2}$$y^{-3}$$z^{-1}$ or $$\frac{x^{2}$$$}{3y^{3}$z}
$$\frac{4x^{5}$$$y^{3}$$z}{12x^{3}$$y^{6}$$z^{2}$$}=\frac{x^{2}$$$}{3y^{3}$z}
First, let's simplify $\frac{4}{12}$. The greatest common factor of 4 and 12 is 4. 4 divided by 4 is 1 and 12 divided by 4 is 3. Therefore $\frac{4}{12}$=\frac{1}{3}$.
To simply fractions with exponents, subtract the exponent in the numerator from the exponent in the denominator. That leaves us with $$\frac{1}{3}$x^{2}$$y^{-3}$$z^{-1}$ or $$\frac{x^{2}$$$}{3y^{3}$z}
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Which of the following fractions is not equivalent to $\frac{6}{45}$?
Which of the following fractions is not equivalent to $\frac{6}{45}$?
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Let us simplify $\frac{6}{45}$:
$\frac{6}{45}$=\frac{3times 2}{3times 15}$=\frac{2}{15}$
We can get alternate forms of the same fraction by multiplying the denominator and the numerator by the same number:
$\frac{2times 2}{15times 2}$=\frac{4}{30}$
$\frac{2times 1.5}{15times 1.5}$=\frac{3}{22.5}$
Now let's look at $\frac{12}{89}$:
, but 
.
Therefore, $\frac{12}{89}$ is the correct answer, as it is not equivalent to $\frac{6}{45}$.
Let us simplify $\frac{6}{45}$:
$\frac{6}{45}$=\frac{3times 2}{3times 15}$=\frac{2}{15}$
We can get alternate forms of the same fraction by multiplying the denominator and the numerator by the same number:
$\frac{2times 2}{15times 2}$=\frac{4}{30}$
$\frac{2times 1.5}{15times 1.5}$=\frac{3}{22.5}$
Now let's look at $\frac{12}{89}$:
Therefore, $\frac{12}{89}$ is the correct answer, as it is not equivalent to $\frac{6}{45}$.
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Which of the following is not equal to 32/24?
Which of the following is not equal to 32/24?
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24/32 = 1.33
16/12 =1.33
224/168 =1.33
4/3 = 1.33
96/72 = 1.33
160/96 = 1.67
24/32 = 1.33
16/12 =1.33
224/168 =1.33
4/3 = 1.33
96/72 = 1.33
160/96 = 1.67
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Find the root of
Find the root of
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The root occurs where 
. So we substitute 0 for 
.
This means that the root is at 
.
The root occurs where
This means that the root is at
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Marker Colors Students Blue 13 Pink 10 Orange 5 Brown 5 Green 7
The above chart shows the number of students in a class who chose each of the five marker colors available.
What fraction of the class selected a pink marker OR a brown marker?
| Marker Colors | Students |
|---|---|
| Blue | 13 |
| Pink | 10 |
| Orange | 5 |
| Brown | 5 |
| Green | 7 |
The above chart shows the number of students in a class who chose each of the five marker colors available.
What fraction of the class selected a pink marker OR a brown marker?
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To solve this problem, you must do three things: find out how many students selected a pink OR brown marker, make that a fraction with the total number of students in the class as the denominator, then simplify that fraction. Shown below:
Students who chose pink OR brown:
Pink: 
, Brown: 
Total students:
Total students in the class:
Make it a fraction:
Simplify. To simplify a fraction, you must find the greatest common factor (GCF), then divide both the numerator and the denominator by the GCF. For 
and 
, the GCF is 
, because it is the largest number that goes into both numbers.
The answer is 
.
To solve this problem, you must do three things: find out how many students selected a pink OR brown marker, make that a fraction with the total number of students in the class as the denominator, then simplify that fraction. Shown below:
Students who chose pink OR brown:
Pink:
Total students:
Total students in the class:
Make it a fraction:
Simplify. To simplify a fraction, you must find the greatest common factor (GCF), then divide both the numerator and the denominator by the GCF. For
The answer is
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Simplify the expression,
.
Simplify the expression,
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The numerator of the expression cannot be factored. Therefore, the denominator cannot divide into the numerator, and the expression is in its simplest form.
The numerator of the expression cannot be factored. Therefore, the denominator cannot divide into the numerator, and the expression is in its simplest form.
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Simplify:
Simplify:
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With this problem the first thing to do is cancel out variables. The x2 can all be divided by each other because they are present in each system. The equation will now look like this:
Now we can see that the equation can all be divided by y, leaving the answer to be:
With this problem the first thing to do is cancel out variables. The x2 can all be divided by each other because they are present in each system. The equation will now look like this:
Now we can see that the equation can all be divided by y, leaving the answer to be:
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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 
?
Two two-digit numbers,
What is
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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 
.
Another way to represent this question is:
In the one's column,
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
Using this information, we can solve for
You can check your answer by returning to the original addition and plugging in the values of
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Simplify:
Simplify:
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_x_2 – _y_2 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).
_x_2 – _y_2 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).
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Simplify the following expression:
Simplify the following expression:
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Following this equation, you divide 4 by 8 to get 1/2. When a variable is raised to an exponent, and you are dividing, you subtract the exponents, so 6 – 3 = 3.
Following this equation, you divide 4 by 8 to get 1/2. When a variable is raised to an exponent, and you are dividing, you subtract the exponents, so 6 – 3 = 3.
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Simplify the given fraction:
Simplify the given fraction:
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125 goes into 2000 evenly 16 times. 1/16 is the fraction in its simplest form.
125 goes into 2000 evenly 16 times. 1/16 is the fraction in its simplest form.
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Simplify the given fraction:
Simplify the given fraction:
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120 goes into 6000 evenly 50 times, so we get 1/50 as our simplified fraction.
120 goes into 6000 evenly 50 times, so we get 1/50 as our simplified fraction.
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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 
?
Let $\frac{frac{1}{4}$+\frac{1}{3}$-$\frac{1}{5}$}{$\frac{1}{2}$-$\frac{1}{6}$+1} = $\frac{a}{b}$, where
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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} = $\frac{60cdot frac{1}{4}$+60cdot $\frac{1}{3}$-60cdot $\frac{1}{5}$}{60cdot $\frac{1}{2}$-60cdot $\frac{1}{6}$+60cdot 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.
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} = $\frac{60cdot frac{1}{4}$+60cdot $\frac{1}{3}$-60cdot $\frac{1}{5}$}{60cdot $\frac{1}{2}$-60cdot $\frac{1}{6}$+60cdot 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.
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Simplify:
(2_x_ + 4)/(x + 2)
Simplify:
(2_x_ + 4)/(x + 2)
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(2_x_ + 4)/(x + 2)
To simplify you must first factor the top polynomial to 2(x + 2). You may then eliminate the identical (x + 2) from the top and bottom leaving 2.
(2_x_ + 4)/(x + 2)
To simplify you must first factor the top polynomial to 2(x + 2). You may then eliminate the identical (x + 2) from the top and bottom leaving 2.
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The expression $($\frac{a^{2}$$$}{b^{3}$$})($\frac{a^{-2}$$$}{b^{-3}$}) = ?
The expression $($\frac{a^{2}$$$}{b^{3}$$})($\frac{a^{-2}$$$}{b^{-3}$}) = ?
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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}$$$}{a^{2}$}, the numerators and denominators cancel out, and you are left with 1.
A negative exponent in the numerator of a fraction can be rewritten with a positive exponent in the denominator. The same is true for a negative exponent in the denominator. Thus, $$\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.
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A train travels at a constant rate of 
meters per second. How many kilometers does it travel in 
minutes? 
A train travels at a constant rate of
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Set up the conversions as fractions and solve:
$\frac{20m}{1sec}$times $\frac{60sec^}{1min}$times $\frac{1km}{1000m}$times $\frac{10min}{1}$
Set up the conversions as fractions and solve:
$\frac{20m}{1sec}$times $\frac{60sec^}{1min}$times $\frac{1km}{1000m}$times $\frac{10min}{1}$
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Simplify. $$\frac{4x^{4}$$$z^{3}$$}{2xz^{2}$}
Simplify. $$\frac{4x^{4}$$$z^{3}$$}{2xz^{2}$}
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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
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
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Simplify the following expression:
$$\frac{2x^{4}$$$-32}{2x^{2}$-8}
Simplify the following expression:
$$\frac{2x^{4}$$$-32}{2x^{2}$-8}
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Factor both the numerator and the denominator:
$$\frac{2(x^{2}$$$-4)(x^{2}$$+4)}{2(x^{2}$-4)}
After reducing the fraction, all that remains is:
$(x^{2}$+4)
Factor both the numerator and the denominator:
$$\frac{2(x^{2}$$$-4)(x^{2}$$+4)}{2(x^{2}$-4)}
After reducing the fraction, all that remains is:
$(x^{2}$+4)
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