Algebraic Fractions - ACT Math

Card 0 of 909

Question

Simplify the following fraction:

$\frac{24}{256}$

Answer

Find the largest number that divides into both and

$\frac{24}{8}=3$

$\frac{256}{8}=32$

$\frac{24}{256}=\frac{3}{32}$

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Question

Which of the following is the least common denominator for the expression below?

$\frac{1}{d\times e^2\times f}-\frac{1}{d\times f^2}-\frac{1}{d^2\times f}$

Answer

Finding the least common denominator in rational expressions follows the same procedue as finding the least common denominator in fractions.

The least common denominator for this rational exresspion will use all terms with the highest exponents of each.

The first fraction has as the highest term, the second fraction has as the highest term, and the third fraction has as the highest term. Now we combine these and get the least common denominator to be:

$d^2 \times e^2 \times f^2$

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Question

Simplify the following fraction until the numerator and denominator share no factors.

$\frac{616}{210}$

Answer

To simplify a fraction you need to find all the factors that the numerator and denominator have in common. You can see that both share 2 so when you divide both by 2 you get

$\frac{308}{105}$

this is close to the answer but it asks for no common factors, it is hard to see but both of these share 7 as a common factor. When you divide both by 7 you get

$\frac{44}{15}$

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Question

Simplify the following fraction:
$\frac{96}{108}$

Answer

To simplify a fraction, find the gcf (gcd) of the numerator and denominator and divide both by the gcf (gcd). the gcf of and is so:

$\frac{96/12}{108/12}=\frac{8}{9}$

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Question

Simplify the following fraction:

$\small \frac{420}{5390}$

Answer

The first and easiest simplification for this fraction is to divide numerator and denominator by $\small 10$ . This gives you:

$\small \frac{42}{539}$

Next, notice that the two fractions do not both contain factors of $\small 3$ . This is because the denominator's digits, when added up come to $\small 17$ , which is not divisible by $\small 3$ . This means $\small 539$ is not divisible by $\small 3$ . Now, $\small 42$ is $\small 237$ . There is no shared $\small 2$ between these numbers. However, if you try, you will see that they are both divisible by $\small 7$ , which gives you:

$\small \frac{6}{77}$

This is simplest form.

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Question

Simplify the following fraction:

$\small \frac{315}{825}$

Answer

First, begin by noticing that both numerator and denominator contain a $\small 5$ . Dividing this out gives you:

$\small \small \frac{63}{165}$

Now, $\small 1+6+5=12$ and $\small 6+3=9$ . Since each of these are true, we know that both numerator and denominator contain a $\small 3$ . Dividing this out, you get:

$\small \small \small \frac{21}{55}$

This is the simplest possible form of the fraction.

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Question

Maria owns an art studio and spent in supplies. She sells her paintings for each. How many paintings does Maria need to sell until she makes a profit?

Answer

Divide the total money spent by the cost of each painting.

$\frac{$1850}{$150}=12.33333$

Therefore, to make a profit, she needs to sell more than this amount. Since she can't sell a portion of a painting, the answer has to be the next nearest whole number ().

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Question

What is the average of $\frac{2}{3}$ and $\frac{5}{6}$ ?

Answer

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.

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Question

Simplify the following expression:

$\frac{4x^6}{8x^3}$

Answer

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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Question

Simplify the given fraction:

$\frac{125}{2000}$

Answer

125 goes into 2000 evenly 16 times. 1/16 is the fraction in its simplest form.

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Question

For this question, the following trigonometric identities apply:

$\csc x=\frac{1}{\sin x}$ ,

$\sec x=\frac{1}{\cos x}$

$\cot x=\frac{1}{\tan x}$

Simplify: $\sin ^{2}x\csc x\cot x$

Answer

To begin a problem like this, you must first convert everything to $\sin x$ and $\cos x$ alone. This way, you can begin to cancel and combine to its most simplified form.

Since $\csc x=\frac{1}{\sin x}$ and $\cot x=\frac{\cos x}{\sin x}$ , we insert those identities into the equation as follows.

$\sin ^{2}x\cdot \frac{1}{\sin x}\cdot \frac{\cos x}{\sin x}$

From here we combine the numerator and denominators of each fraction together to easily see what we can combine and cancel.

$\frac{\sin^{2}x\cdot \cos x}{\sin x\cdot \sin x}$

Since there is a $\sin ^{2}x$ in the numerator and the denominator, we can cancel them as they divide to equal 1. All we have left is $\cos x$ , the answer.

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Question

Solve

Answer

The common denominator of the left side is x(x–1). Multiplying the top and bottom of 1/x by (x–1) yields

Since this statement is true, there are infinitely many solutions.

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Question

Evaluate when x=11. Round to the nearest tenth.

Answer

Wherever there is an x, plug in 11 and perform the given operations. The numerator will be equal to 83 and the denominator will be equal to 46. 83 divided by 46 is equal to 1.804… and since the second decimal place is not greater than or equal to 5, the first decimal place stays the same when rounding so the final answer is 1.8.

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Question

Simplify: $(\frac{x-4}{0.5})\div (\frac{1}{x+4})$

Answer

Recall that dividing is equivalent multiplying by the reciprocal. Therefore, ((x - 4) / (1 / 2)) / (1 / (x + 4)) = ((x - 4) * 2) * (x + 4) / 1.

Let's simplify this further:

(2x – 8) * (x + 4) = 2x2 – 8x + 8x – 32 = 2x2 – 32

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Question

If 3x = 12, y/4 = 10, and 4z = 9, what is the value of (10xyz)/xy?

Answer

Solve for the variables, the plug into formula.

x = 12/3 = 4

y = 10 * 4 = 40

z= 9/4 = 2 1/4

10xyz = 3600

Xy = 160

3600/160 = 22 1/2

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Question

Mary walked to school at an average speed of 2 miles per hour and jogged back along the same route at an average speed of 6 miles per hour. If her total traveling time was 1 hour, what was the total number of miles in the round trip?

Answer

Since Mary traveled 3 times as quickly coming from school as she did going to school (6 miles per hour compared to 2 miles per hour), we know that Mary spent only a third of the time coming from school as she did going. If x represents the number of hours it took to get to school, then x/3 represents the number of hours it took her to return.

Knowing that the total trip took 1 hour, we have:

x + x/3 = 1

3x/3 + 1x/3 = 1

4_x_/3 = 1

x = 3/4

So we know it took Mary 3/4 of an hour to travel to school (and the remaining 1/4 of an hour to get back).

Remembering that distance = rate * time, the distance Mary traveled on her way to school was (2 miles per hour) * (3/4 of an hour) = 3/2 miles. Furthermore, since she took the same route coming back, she must have traveled 3/2 of a mile to return as well.

Therefore, the the total number of miles in Mary's round trip is 3/2 miles + 3/2 miles = 6/2 miles = 3 miles.

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Question

According the pie chart, the degree measure of the sector representing the number of workers spending 5 to 9 years in the same role is how much greater in the construction industry chart than in the financial industry chart?

Answer

Since the values in the pie charts are currently in terms of percentages (/100), we must convert them to degrees (/360, since within a circle) to solve the question. The "5 to 9 years" portion for the financial and construction industries are 18 and 25 percent, respectively. As such, we can cross-multiply both:

18/100 = x/360

x = 65 degrees

25/100 = y/360

y = 90 degrees

Subtract: 90 – 65 = 25 degrees

Alternatively, we could first subtract the percentages (25 – 18 = 7), then convert the 7% to degree form via the same method of cross-multiplication.

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Question

6 contestants have an equal chance of winning a game. One contestant is disqualified, so now the 5 remaining contestants again have an equal chance of winning. How much more likely is a contestant to win after the disqualification?

Answer

When there are 6 people playing, each contestant has a 1/6 chance of winning. After the disqualification, the remaining contestants have a 1/5 chance of winning.

1/5 – 1/6 = 6/30 – 5/30 = 1/30.

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Question

If $w=\frac{1}{8}$ then which of the following is equal to $w^\frac{2}{3}$ ?

Answer

To raise $\frac{1}{8}$ to the exponent $\frac{2}{3}$ , square $\frac{1}{8}$ and then take the cube root.

$w^\frac{2}{3}=\sqrt[3]{w^2}$

$(\frac{1}{8})^\frac{2}{3}=\sqrt[3]{(\frac{1}{8})^2}=\sqrt[3]{\frac{1}{64}}$

$\sqrt[3]{\frac{1}{64}}=\frac{1}{4}$

$(\frac{1}{8})^{\frac{2}{3}}=\frac{1}{4}$

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Question

Simplify the given fraction:

$\frac{120}{6000}$

Answer

120 goes into 6000 evenly 50 times, so we get 1/50 as our simplified fraction.

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