Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (2024)

PRODUCTS OF FRACTIONS

The product of two fractions is defined as follows.

The product of two fractions is a fraction whose numerator is the product of the numerators and whose denominator is the product of the denominators of the given fractions.

In symbols,

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (1)

Any common factor occurring in both a numerator and a denominator of either fraction can be divided out either before or after multiplying.

Example 1 Find the product of Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (2)

Solution
Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (3)

The same procedures apply to fractions containing variables.

Example 2 Find the product of Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (4)

Solution First, we divide the numerator and denominator by the common factors to get

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (5)

Now, multiplying the remaining factors of the numerators and denominators yields

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (6)

If a negative sign is attached to any of the factors, it is advisable to proceed as if all the factors were positive and then attach the appropriate sign to the result. A positive sign is attached if there are no negative signs or an even number of negative signs on the factors; a negative sign is attached if there is an odd number of negative signs on the factors.

Example 3
Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (7)

When the fractions contain algebraic expressions, it is necessary to factor wherever possible and divide out common factors before multiplying.

Example 4 Find the product of Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (8).

Solution First, we must factor the numerators and denominators to get

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (9)

Now, dividing out common factors yields

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (10)

We now multiply the remaining factors of the numerators and denominators to obtain

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (11)

Note that when writing fractional answers, we will multiply out the numerator and leave the denominator in factored form. Very often, fractions are more useful in this form.

In algebra, we often rewrite an expression such as Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (12) as an equivalent expression Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (13).Use whichever form is most convenient for a particular problem.

Example 5
Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (14)

Common Errors: Remember that we can only divide out common factors, never common terms! For example,

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (15)

because x is a term and cannot be divided out. Similarly,

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (16)

because 3 is not a factor of the entire numerator 3y + 2.

QUOTIENTS OF FRACTIONS

In dividing one fraction by another, we look for a number that, when multiplied by the divisor, yields the dividend. This is precisely the same notion as that of dividing one integer by another; a ÷ b is a number q, the quotient, such that bq = a.

To find Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (17), we look for a number q such that Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (18). To solve this equation for q, we multiply each member of the equation by Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (19). Thus,

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (20)

In the above example, we call the number Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (21) the reciprocal of the number Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (22). In general, the reciprocal of a fraction Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (23) is the fraction Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (24). That is, we obtain the reciprocal of a fraction by "inverting" the fraction. In general,

The quotient of two fractions equals the product of the dividend and the reciprocal of the divisor.

That is, to divide one fraction by another, we invert the divisor and multiply. In symbols,

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (25)

Example 1
Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (26)

As in multiplication, when fractions in a quotient have signs attached, it is advisable to proceed with the problem as if all the factors were positive and then attach the appropriate sign to the solution.

Example 2
Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (27)

Some quotients occur so frequently that it is helpful to recognize equivalent forms directly. One case is

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (28)

In general,

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (29)

Example 3

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (30)

When the fractions in a quotient involve algebraic expressions, it is necessary to factor wherever possible and divide out common factors before multiplying.

Example 4

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (31)

SUMS AND DIFFERENCES OF FRACTIONS WITH LIKE DENOMINATORS

The sum of two or more arithmetic or algebraic fractions is defined as follows:
The sum of two or more fractions with common denominators is a fraction with the same denominator and a numerator equal to the sum of the numerators of the original fractions.

In general,

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (32)

Example 1

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (33)

When subtraction is involved, it is helpful to change to standard form before adding.

Example 2

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (34)

We must be especially careful with binomial numerators. For example, we should rewrite

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (35)

where the entire numerator is enclosed within parentheses.

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (36)

SUMS OF FRACTIONS WITH UNLIKE DENOMINATORS

In Section 6.3, we added fractions with like denominators. In this section, we will add fractions with unlike denominators.

LEAST COMMON DENOMINATOR

In general, the smallest natural number that is a multiple of each of the denominators of a set of fractions is called the lowest common denominator (LCD) of the set of fractions. Sometimes, we can obtain the LCD by inspection. If the LCD is not immediately evident, we can use a special procedure to find it.

To find the LCD:

  1. Completely factor each denominator, aligning common factors when possible.
  2. Include in the LCD each of these factors the greatest number of times it occurs in any single denominator.

Example 1 Find the lowest common denominator of the fractions Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (37)

Solution The lowest common denominator for Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (38) contains among its factors the factors of 12, 10, and 6.

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (39)

Thus, the LCD is 60. (This number is the smallest natural number that is divisible by 12, 10, and 6.)

The LCD of a set of algebraic fractions is the simplest algebraic expression that is a multiple of each of the denominators in the set. Thus, the LCD of the fractions

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (40)

because this is the simplest expression that is a multiple of each of the denominators.

Example 2 Find the LCD of the fractions

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (41)

Solution Following the method of Example 1, we get

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (42)

Thus, the LCD is x2(x + l)(x - 1).

We can add fractions with unlike denominators by first building the fractions to equivalent fractions with like denominators and then adding.

To add fractions with unlike denominators:

  1. Find the LCD of the set of fractions.
  2. Build each fraction to an equivalent fraction with the LCD as the denominator.
  3. Add the fractions using the property
    Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (43)

Example 3 Write the sums of Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (44) and Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (45) as single terms.

Solution In each case, the LCD is 10. We build each fraction to a fraction with 10 as the denominator. Thus,

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (46)

are equivalent to

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (47)

from which we obtain

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (48)

Sometimes, the fractions have denominators that are binomials.

Example 4 Write the sum of Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (49) as a single term.

Solution The LCD is (x + 2)(x - 1). We build each fraction to a fraction with denominator (x + 2)(x - 1), inserting parentheses as needed, and get

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (50)

Now that we have like denominators, we can add the numerators, simplify, and obtain

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (51)

Example 5 Write the sum of Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (52) as a single term.

Solution First we factor the denominators in order to obtain the LCD.

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (53)

We now build each fraction to fractions with this denominator and get

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (54)

We can now add the numerators, simplify, and obtain

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (55)

Common Errors Note that we can only add fractions with like denominators. Thus,

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (56)

Also, we only add the numerators of fractions with like denominators. Thus,

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (57)

DIFFERENCES OF FRACTIONS WITH UNLIKE DENOMINATORS

We subtract fractions with unlike denominators in a similar way that we add such fractions. However, we first write each fraction in standard form. Thus, any fraction in the form

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (58)

is first written as

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (59)

We can now add the fractions.

Example 1 Write the difference Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (60) as a single term.

Solution We begin by writing Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (61) in standard form as Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (62). The LCD is 12x. We build each fraction to an equivalent fraction with this denominator to get

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (63)

Now, adding numerators yields

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (64)

Again, special care must be taken with binomial numerators.

Example 2 Write the difference of Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (65) as a single term.

Solution Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (66) should first be written as

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (67)

where the entire numerator is enclosed within parentheses. Then, we obtain the LCD 6 and build each fraction to fractions with denominator 6, add numerators, and simplify.

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (68)

The next examples involve binomial denominators.

Example 3 Write the difference of Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (69) as a single term.

Solution We begin by writing Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (70) in standard form as Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (71). The LCD is (x - l)(x + 2) and we build each fraction to an equivalent fraction with this denominator to get

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (72)

Now adding numerators and simplifying yields

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (73)

Example 4 Write the difference of

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (74)

as a single term

Solution We first factor the denominators and write the fractions in standard form to get

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (75)

We find the LCD (x + 7)(x - 3)(x + 3) and build each fraction to an equivalent fraction with this denominator to get

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (76)

Now, adding numerators and simplifying yield

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (77)

COMPLEX FRACTIONS

A fraction that contains one or more fractions in either its numerator or denominator or both is called a complex fraction. For example,

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (78)

are complex fractions. Like simple fractions, complex fractions represent quotients. For example,

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (79)

In cases like Equation (1), in which the numerator and denominator of the complex fraction do not contain sums or differences, we can simply invert the divisor and multiply. That is,

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (80)

In cases like Equation (2), in which the numerator or denominator of the complex fraction contains sums or differences, we cannot simply invert the divisor and multiply. However, we can use the fundamental principle of fractions to simplify complex fractions. In fact, we can also use the fundamental principle to simplify complex fractions of Form (1) above.

Example 1 Simplify Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (81) by using the fundamental principle of fractions.

Solution We multiply the numerator and denominator by the LCD of all fractions in the numerator and denominator; in this case, the LCD is 4. The result is a simple fraction equivalent to the given complex fraction.

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (82)

The simplification of Equation (2) on page 255 appears in the next example.

Example 2 Simplify Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (83)

Solution We multiply the numerator and denominator by the LCD of all fractions in the numerator and denominator; in this case, the LCD is 6. We obtain

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (84)

FRACTIONAL EQUATIONS

To solve an equation containing fractions, it is usually easiest to first find an equivalent equation that is free of fractions. We do this by multiplying each member of an equation by the lowest common denominator of the fractions.

Although we can apply the algebraic properties we have studied in any order, the following steps show the order most helpful in solving an equation when the solution is not obvious. Of course, not all the steps are always necessary.

To solve an equation:

  1. Clear fractions," if there are any, by multiplying each member of the equation by the LCD.
  2. Write any expression that contains parentheses as an expression without parentheses.
  3. Combine any like terms in either member.
  4. Obtain all terms containing the variable in one member and all terms not containing the variable in the other member.
  5. Divide each member by the coefficient of the variable if it is different from 1.
  6. Check the answer if each member of the equation has been multiplied by an expression containing a variable.

Example 1 Solve Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (85).

Solution We multiply each member by the LCD 15to obtain an equivalent equation that does not contain a fraction.

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (86)

The multiplication property of equality (Section 3.4) allows us to multiply each member of an equation by a non zero value in order to obtain an equivalent equation. Thus, to solve the equation

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (87)

we would multiply each member by the LCD 4(x - 5). We note that x cannot equal 5 since 4(x - 5) equals 0 if x = 5. The entire solution is shown in the next example.

Example 2 Solve Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (88).

Solution We multiply each member by the LCD 4(x - 5)to get

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (89)

Applying the distributive property, we obtain

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (90)

Solving for x yields

-21x = -189; x = 9

Note that 4(x - 5) is not equal to zero for a = 9. Thus, a = 9 is a valid solution for the equation.

When equations contain more than one variable, it is sometimes desirable to solve for one variable in terms of the other variable(s).

Example 3 Solve Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (91) for a in terms of a, b, and c.

Solution We multiply each member by the LDC 3xc to get

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (92)

Now, dividing each member by 2x, we obtain

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (93)

APPLICATIONS

The word problems in the following exercises lead to equations involving fractions. At this time, you may want to review the steps suggested to solve word problems and the steps suggested on page 260 to solve equations that contain fractions.

Example 1 If Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (94) of a certain number is added to Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (95) of the number, the result is 11. Find the number.

Solution

Steps 1-2 We first write what we want to find (the number) as a word phrase. Then, we represent the number in terms of a variable.
The number: x

Step 3 A sketch is not applicable.

Step 4 Now we can write an equation. Remember that "of" indicates multiplication.
Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (96)

Step 5 Solving the equation yields
Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (97)

Step 6 The number is 12.

Equations for problems concerned with motion sometimes include fractions. The basic idea of motion problems is that the distance traveled d equals the product of the rate of travel r and the time of travel t. Thus, d = rt. We can solve this formula for r or t to obtain:

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (98)

A table like the one shown in the next example is helpful in solving motion problems.

Example 2 An express train travels 180 miles in the same time that a freight train travels 120 miles. If the express goes 20 miles per hour faster than the freight, find the rate of each.

Solution Steps 1-2 We represent the two unknown quantities that we want to find as word phrases. Then, we represent the word phrases in terms of one variable.

Rate of freight train: r

Rate of express train: r + 20

Step 3 Next, we make a table showing the distances, rates, and times.

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (99)

Step 4 Because the times of both trains are the same, we can equate the expressions for time to obtain
Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (100)

Step 5 We can now solve for r by first multiplying each member by the LCD r(r + 120) and we get

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (101)

Step 6 The freight train’s speed is 40 mph and the express train’s speed is 40 + 20, or 60 mph.

RATIO AND PROPORTION

The quotient of two numbers, a ÷ b or Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (102), is sometimes referred to as a ratio and read "the ratio of a to b." This is a convenient way to compare two numbers.

Example 1 Express as a ratio.

a. 3 in. to 5 in.
b. 8 m to 12 m
c. 6 to 10

Solutions
Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (103)

A statement that two ratios are equal, such as

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (104)

is called a proportion and read "2 is to 3 as 4 is to 6" and "a is to b as c is to d." The numbers a, b, c, and d are called the first, second, third, and fourth terms of the proportion, respectively. The first and fourth terms are called the extremes of the proportion, and the second and third terms are called the means of the proportion.

Example 2 Express as a proportion.

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (105)

If each ratio in the proportion
Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (106)

is multiplied by bd, we have

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (107)

Thus,

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (108)

In any proportion, the product of the extremes is equal to the product of the means.

A proportion is a special type of fractional equation. The above rule to obtain an equivalent equation without denominators is a special case of our general approach.

Example 3 Solve the proportion Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (109).

Solution By applying Property (1) above, we get

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (110)

CONVERSIONS

We can use proportions to convert English units of measure into metric units and vice versa. The following basic relationships will be helpful in setting up appropriate proportions for conversions.

1 meter (m) = 39.37 inches (in.)

1 kilogram (kg) = 2.2 pounds (lb)

1 kilometer (km) = 0.62 miles (mi)

1 liter (1) = 1.06 quarts (qt)

1 pound (lb) = 454 grams (g)

1 inch (in.) = 2.54 centimeters (cm)

When converting units, it is easiest to follow the six steps outlined.

Example 4 Change 8 in. to centimeters.

Solution

Steps 1-2 Represent what is to be found (centimeters) in terms of a word phrase and in terms of a variable.
Centimeters: x

Step 3 Make a table showing the basic relationship between inches and centimeters.

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (111)

Step 4 Using the table from Step 3, write a proportion relating inches to centimeters.

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (112)

Step 5 Solve for x by equating the product of the means to the product of the extremes.

8(2.54) = 1 · x
20.32 = x

Step 6 Eight inches equals 20.32 centimeters.

CHAPTER SUMMARY

  1. The following properties are used to rewrite products and quotients of fractions.

  2. Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (113)

  3. The smallest natural number that is a multiple of each of the denominators of a set of fractions is called the lowest common denominator (LCD) of the fractions. The following properties are used to rewrite sums and differences of fractions.

    Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (114)

  4. A fraction that contains one or more fractions in either numerator or denominator or both is called a complex fraction. We can simplify a complex fraction by multiplying the numerator and denominator by the LCD of all fractions in the numerator and denominator.

  5. We can solve an equation containing fractions by obtaining an equivalent equation in which the solution is evident by inspection. Generally, it is best to obtain an equivalent equation that is free of fractions by multiplying each member of the equation by the LCD of the fractions.

  6. The quotient of two numbers is called a ratio; a statement that two ratios are equal is called a proportion. In the proportion

    Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (115)

    a and d are called the extremes of the proportion and b and c are called the means. In any proportion of this form,

    ad = bc

Reduce Simple or Complex Fractions with Step-by-Step Math Problem Solver (2024)

FAQs

How do you reduce complex fractions? ›

Simplifying a Complex Fraction: Method 1

Simplify the numerator and denominator separately. Step 2. Divide by multiplying the numerator by the reciprocal of the denominator. Step 3 Simplify the resulting fraction, if possible.

How to reduce fractions step by step? ›

Step 1: Find any common factor of the numerator and the denominator. Step 2: Divide the numerator and denominator by the common factor. Step 3: Repeat the same step in the resulting fraction until there are no more common factors other than 1.

How do you simplify expressions with fractions? ›

An algebraic fraction is a fraction where the numerator and denominator are algebraic expressions. To simplify an algebraic fraction, first factorise the expressions in the numerator and denominator and then cancel any common factors in the numerator and denominator.

What is a single fraction in simplest form? ›

We can simplify a fraction by cancelling all common factors of the numerator and the denominator. A fraction which has been simplified it is said to be written in its simplest form or lowest form. An arithmetic fraction is in its simplest form if the numerator and denominator do not have any common integer factors.

What is an example of complex fraction in math? ›

A complex fraction is a fraction that contains one or more fractions in either the numerator or denominator, or both. For example, 2/3 is a simple fraction, but (2/3)/(4/5) is a complex fraction. Complex fractions can be written in various ways, including using horizontal or vertical bars, brackets, or parentheses.

How do you simplify a complex problem? ›

Steps for complex problem-solving
  1. Identify the problem and its cause. ...
  2. Consider the impacts of the problem. ...
  3. Brainstorm solutions. ...
  4. Consider the impacts of solutions. ...
  5. Choose the most optimal solution. ...
  6. Implement a plan. ...
  7. Prepare for changes.

How do you reduce the following fractions in simplest form? ›

Steps to Find Simplest Form
  1. Find the GCF of numerator and denominator of the fraction.
  2. Divide the numerator and denominator by the produced GCF.
  3. Write the simplified fraction of the given fraction.

What is the equation to reduce a fraction? ›

To reduce a fraction, divide both the numerator and denominator by the GCF. (This is also known as "writing a fraction in lowest terms".) This is sometimes shown as "canceling" the common factors. Note that the result is an equivalent fraction in simplest form.

What is a reduced fraction example? ›

For example, the fraction 2/3 is fully reduced. There isn't any whole number, other than 1, that both 2 and 3 can be divided by without having a remainder. Other examples of fully reduced fractions include 7/8, 5/9, and 11/20. An example of a fraction that isn't fully reduced is 2/4.

What is a simple fraction vs complex fraction? ›

Fractions that have only one fraction line are called simple fractions. Fractions with more than one fraction line are called complex fractions.

What is simple fraction with example? ›

Simplified fractions are the fractions in their lowest, reduced and simplest form. The numerator and denominator of the fraction are reduced to the extent that the only common factor between them is 1. For example, 2/3 is the simplified fraction for 4/6, 8/12, 16/24 and so on.

How do you reduce complex numbers? ›

Simplify by combining like terms.

The result of the FOIL rule multiplication should yield two real number terms and two imaginary number terms. Simplify the result by combining like terms together. For the sample 15-9i+10i+6, you can add the 15 and 6 together and add the -9i and the 10i together.

How do you solve complex numbers with fractions? ›

This is done by multiplying the top and bottom of the fraction by the complex conjugate. Once multiplying both the numerator and denominator by the complex conjugate it is time to simplify the answer by combining like terms. Do not forget to substitue the value of which is always -1.

How do you subtract complicated fractions? ›

Subtract Complex Fractions : Example Question #1

Convert the numerators and denominators into single fractions, then simplify. Start by finding the lowest common denominator in both the numerator and denominator of the complex fraction. Add fractions with like denominators. Simplify.

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