ACT Math - How to evaluate a fraction

Evaluate the following equation when and round your answer to the nearest hundredth.

$\frac{x^{3}-2x^{2}-3x+5}{5x+3}$

Explanation

$\frac{x^{3}-2x^{2}-3x+5}{5x+3}$

1. Plug in wherever there is an in the above equation.

$\frac{5^{3}-(2)(5^{2})-(3)(5)+5}{(5)(5)+3}$

2. Perform the above operations.

$\frac{5^{3}-(2)(25)-(3)(5)+5}{(5)(5)+3}$

$\frac{125-50-15+5}{25+3}$

$\frac{65}{28}=2.32$

Solve Actmath_7_113_q10_1

–1

no solution

infinitely many solutions

Explanation

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

Actmath_7_113_q10_2

Actmath_7_113_q10_3

Actmath_7_113_q10_4

Actmath_7_113_q10_5

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

Simplify:

$\frac{\frac{81}{5}+\frac{7}{25}}{\frac{1}{9}-\frac{2}{7}}$

$-\frac{25956}{275}$

$\frac{8414}{41}$

$-\frac{551}{12}$

$-\frac{4532}{1575}$

$\frac{8417}{14}$

Explanation

Begin by simplifying the numerator and the denominator.

Numerator

$\frac{81}{5}+\frac{7}{25}$ has a common denominator of . Therefore, we have:

$\frac{405}{25}+\frac{7}{25} = \frac{412}{25}$

Denominator

$\frac{1}{9}-\frac{2}{7}$ has a common denominator of . Therefore, we have:

$\frac{7}{63}-\frac{18}{63}=-\frac{11}{63}$

Now, reconstructing our fraction, we have:

$\frac{\frac{412}{25}}{-\frac{11}{63}}$

To make this division work, you multiply the numerator by the reciprocal of the denominator:

$\frac{\frac{412}{25}}{-\frac{11}{63}} = \frac{412}{25} \cdot -\frac{63}{11} =-\frac{25956}{275}$

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

4 1/2

360

160

22 1/2

Explanation

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

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

$\cos x$

$\sin x\cos x$

$\sin x$

$\tan x$

$\sec x$

Explanation

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.

If , , and , find the value of $\frac{2x+y}{z}$ .

$\frac{17}{5}$

$\frac{13}{5}$

$\frac{9}{2}$

$\frac{16}{7}$

$\frac{19}{5}$

Explanation

In order to solve $\frac{2x+y}{z}$ , we must first find the values of , , and using the initial equations provided. Starting with :

$x=\frac{55}{5}=11$

Then:

Finally:

With the values of , , and in hand, we can solve the final equation:

$\frac{2x+y}{z}$

$=\frac{2(11)+12}{10}$

$=\frac{22+12}{10}$

$=\frac{34}{10}$

$=\frac{17}{5}$

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

$\frac{3}{2x^2}-16$

$\frac{1}{2}$

None of the other answer choices are correct.

Explanation

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

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

$\frac{1}{4}$

$\frac{1}{2}$

$\frac{1}{16}$

$\frac{1}{32}$

$\frac{1}{64}$

Explanation

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

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

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

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

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

Simplify:

$\frac{\frac{1}{4}+\frac{5}{7}}{\frac{2}{5}}$

$\frac{135}{56}$

$\frac{27}{70}$

$\frac{98}{71}$

$\frac{47}{5}$

$\frac{81}{7}$

Explanation

Begin by simplifying the numerator.

$\frac{1}{4}+\frac{5}{7}$ has a common denominator of . Therefore, we can rewrite it as:

$\frac{7}{28}+\frac{20}{28}=\frac{27}{28}$

Now, in our original problem this is really is:

$\frac{\frac{27}{28}}{\frac{2}{5}}$

When you divide by a fraction, you really multiply by the reciprocal:

$\frac{\frac{27}{28}}{\frac{2}{5}}=\frac{27}{28} \cdot\frac{5}{2}=\frac{135}{56}$

If pizzas cost dollars and sodas cost dollars, what is the cost of pizzas and sodas in terms of and ?

$\frac{3x+5y}{15}$

$5x+\frac{3y}{15}$

Explanation

If 10 pizzas cost x dollars, then each pizza costs x/10. Similarly, each soda costs y/6. We can add the pizzas and sodas together by finding a common denominator:

$2*\frac{x}{10}+2*\frac{y}{6}=\frac{x}{5}+\frac{y}{3}=\frac{3x}{3*5}+\frac{5y}{5*3}=\frac{3x+5y}{15}$

How to evaluate a fraction

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