4.7 Solve equations with fractions (Page 3/5)

Prealgebra

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\frac{3}{4} x = 24

The coefficient of $x$ is $\frac{3}{4} .$ To solve for $x,$ we need its coefficient to be $1 .$ Since the product of a number and its reciprocal is $1,$ our strategy here will be to isolate $x$ by multiplying by the reciprocal of $\frac{3}{4} .$ We will do this in [link] .

Solve: $\frac{3}{4} x = 24 .$

Solution


Multiply both sides by the reciprocal of the coefficient.
Simplify.
Multiply.
Check:
Substitute $x = 32$ .
Rewrite $32$ as a fraction.
Multiply. The equation is true.

Notice that in the equation $\frac{3}{4} x = 24,$ we could have divided both sides by $\frac{3}{4}$ to get $x$ by itself. Dividing is the same as multiplying by the reciprocal, so we would get the same result. But most people agree that multiplying by the reciprocal is easier.

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Solve: $\frac{2}{5} n = 14 .$

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Solve: $\frac{5}{6} y = 15 .$

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Solve: $- \frac{3}{8} w = 72 .$

Solution

The coefficient is a negative fraction. Remember that a number and its reciprocal have the same sign, so the reciprocal of the coefficient must also be negative.


Multiply both sides by the reciprocal of $− \frac{3}{8}$ .
Simplify; reciprocals multiply to one.
Multiply.
Check:
Let $w = −192$ .
Multiply. It checks.

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Solve: $- \frac{4}{7} a = 52 .$

−91

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Solve: $- \frac{7}{9} w = 84 .$

−108

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Translate sentences to equations and solve

Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.

Subtraction Property of Equality: For any real numbers $a, b,$ and $c,$ if $a = b,$ then $a - c = b - c .$	Addition Property of Equality: For any real numbers $a, b,$ and $c,$ if $a = b,$ then $a + c = b + c .$
Division Property of Equality: For any numbers $a, b,$ and $c,$ where $c \neq 0$ if $a = b,$ then $\frac{a}{c} = \frac{b}{c}$	Multiplication Property of Equality: For any real numbers $a, b,$ and $c$ if $a = b,$ then $a c = b c$

When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.

In the next few examples, we’ll translate sentences into equations and then solve the equations. It might be helpful to review the translation table in Multiply Whole Numbers .

Translate and solve: $n$ divided by $6$ is $−24 .$

Solution

Translate.
Multiply both sides by $6$ .
Simplify.
Check:	Is $−144$ divided by $6$ equal to $−24$ ?
Translate.
Simplify. It checks.

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Translate and solve: $n$ divided by $7$ is equal to $−21 .$

$\frac{n}{7} = −21; n = −147$

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Translate and solve: $n$ divided by $8$ is equal to $−56 .$

$\frac{n}{8} = −56; n = −448$

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Translate and solve: The quotient of $q$ and $−5$ is $70 .$

Solution

Translate.
Multiply both sides by $−5$ .
Simplify.
Check:	Is the quotient of $−350$ and $−5$ equal to $70$ ?
Translate.
Simplify. It checks.

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Translate and solve: The quotient of $q$ and $−8$ is $72 .$

$\frac{q}{−8} = 72; q = −576$

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Translate and solve: The quotient of $p$ and $−9$ is $81 .$

$\frac{p}{−9} = 81; p = −729$

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Translate and solve: Two-thirds of $f$ is $18 .$

Solution

Translate.
Multiply both sides by $\frac{3}{2}$ .
Simplify.
Check:	Is two-thirds of $27$ equal to $18$ ?
Translate.
Simplify. It checks.

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Translate and solve: Two-fifths of $f$ is $16 .$

$\frac{2}{5} f = 16; f = 40$

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Translate and solve: Three-fourths of $f$ is $21 .$

$\frac{3}{4} f = 21; f = 28$

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Translate and solve: The quotient of $m$ and $\frac{5}{6}$ is $\frac{3}{4} .$

Solution

	The quotient of $m$ and $\frac{5}{6}$ is $\frac{3}{4}$ .
Translate.	$\frac{m}{\frac{5}{6}} = \frac{3}{4}$
Multiply both sides by $\frac{5}{6}$ to isolate $m$ .	$\frac{5}{6} (\frac{m}{\frac{5}{6}}) = \frac{5}{6} (\frac{3}{4})$
Simplify.	$m = \frac{5 \cdot 3}{6 \cdot 4}$
Remove common factors and multiply.	$m = \frac{5}{8}$
Check:
Is the quotient of $\frac{5}{8}$ and $\frac{5}{6}$ equal to $\frac{3}{4}$ ?	$\frac{\frac{5}{8}}{\frac{5}{6}} \overset{?}{=} \frac{3}{4}$
Rewrite as division.	$\frac{5}{8} \div \frac{5}{6} \overset{?}{=} \frac{3}{4}$
Multiply the first fraction by the reciprocal of the second.	$\frac{5}{8} \cdot \frac{6}{5} \overset{?}{=} \frac{3}{4}$
Simplify.	$\frac{3}{4} = \frac{3}{4} ✓$

Our solution checks.

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Questions & Answers

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the transfer of energy by a force that causes an object to be displaced; the product of the component of the force in the direction of the displacement and the magnitude of the displacement

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difference between model and theory

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The full note of modern physics

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introduction to applications of nuclear physics

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Show that the equal masses particles emarge from collision at right angle by making explicit used of fact that momentum is a vector quantity

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A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.

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A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike

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Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.

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the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon

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nuclei having the same Z and different N s

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Source: OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9

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