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Complex Fractions

Definition of a fraction 

A fraction has two parts: a numerator and a denominator; the numerator is the number above the line, and the denominator is the number below the line. In a fraction, the line or slash that divides the numerator and denominator denotes division. It is being used to show how many parts we have compared to the actual quantity.

Examples

  • ½ 
  • $3\frac{3}{7}$
  • $\frac{y}{7}$ 
  • 1 + ½
  • $\frac{1/2}{3/4}$ 
  • $\frac{x+2}{6+7/y}$

Complex fractions

Definition 

Whenever a “normal” fraction has fractions there in numerator, denominator, or even both, it is referred to as a complicated fraction. A compound fraction is another name for this sort of fraction. Any fraction with a fraction is on the top, bottom, or even both sides.

Interlocking dolls are analogous to complex fractions. They look like conventional fractions until we realize there was another fraction from inside them.

A fraction with fractions in the numerator, denominator, or both is referred to as a complicated fraction. In plenty of other words, within the overall percent, there is at least one minor fraction. A complex rational expression is one in which the complex fraction comprises a variable.

Complex fractions include the following:

Examples 

          Example 1:

  • 5/(3/2) is a complex fraction with a numerator of 3 as well as a denominator of 1/2.
  • (8/7)/11 is likewise a complex fraction, with the numerator and denominator being 8/7 and 11, correspondingly.
  • One complex fraction is (1/24)/(19/10), which has 1/24 as that of the numerator & 19/10 as the denominator.
  • $\frac{\frac{2-x}{x}}{5-x}$ is complex fraction with variables 
  • $\frac{\frac{2x-1}{5x}}{5x}$
  • $2-\frac{\frac{\frac{1}{c}}{5}}{c}$
  • $\frac{\frac{\frac{y}{7}}{4xy}}{5}$
  • $\frac{x+2}{3/x}$

The challenge asks us to use the FOIL (first outer inner last) method (binomial multiplication) as well as a basic trinomial factorization. It may appear intimidating initially, but if we concentrate attention to the details, then assure that that it is not that difficult.

As you can see, the complex fraction has already one fractional sign, which is exactly what we want. That means we’ll have had to work on the complex numerator a little more. The following step is to transform the complicated numerator to either a “simple” or singular fraction.

Example: 

Here we have another example regarding complex fraction:

$\frac{9/10}{3/10}$

  • =  (9/10)/(3/10) here the 10 and 3 is the denominator and 9 and 10 is nominator  
  • We will Simplify the question in complex fractions:
  • Hence we solve it :-
  • =  (3 x 10) / (10 x 3) complex fraction without variable .
  • =  (3 x 1) / (1 x 1) as per solution 
  • =  3 / 1 
  • = 3

Hints 

  • Most fractions that are difficult can be represented as division problems.
  • This main fraction bar should be read as “divided by.”
  • Complex fractions should indeed be reduced to their simplest form. Leaving a response in this convoluted format is frequently too confusing.
  • Keep in mind which the fraction bar denotes “divided by.”
  • To simplify complex fractions, you can utilize one of two ways. To reduce a complex fraction, remove the fraction from the numerator and/or denominator of the expression.
  • Method 1: Divide complex fractions to make them easier to understand (from the fraction bar).
  • Method 2: Multiply complicated fractions by a common denominator to simplify them.

Math Rules Which Will Assist us 

  • There have been a number of rules in arithmetic, and while they can be frustrating at times, they can be quite useful because we’re trying to simplify complex fractions:
  • Denominator-Numerator Same Quantity Rule: A fraction’s denominator and numerator can always be multiplied by the same amount without modifying the fraction’s value. One could, for instance, multiply 1/2 by 2/2 to get 2/4, that is the same as 1/2.
  • The Multiplicative Property of Equality states that every integer can be multiplied by one without changing its value. We can multiply an integer by almost anything we choose because each number (excluding 0) divided by itself equals 1.

Simplifying a Complex Fraction (Method 1)

  • Phase 1: Modify the numerator to make them each a separate fraction as necessary.

In all the other words, we’ll combine all of the numerator’s parts to make one fraction but all of the denominator’s parts to make other fraction. We really need a refresher in rational expressions addition and subtraction.

  • Step 2: Multiply the numerator by the reciprocal of the denominator to divide the numerator even by denominator.
  • Step 3: Shorten the rational phrase as necessary.

Example 

=     $\frac{1/6}{2/7}$

  • Whenever dividing fractions, remember to invert (turn over, and take the reciprocal of) the second fraction before multiplying.
  • If necessary, reduce its final answer (or reduce as you multiply).

1/6 ÷ 2/7 

When we convert division to multiplication the fraction placing next to division symbol is interchanged by numerator and denominator.  

So        =  1/6 x 7/2

= 7/12

This is the simplified solution 

Example 

= $\frac{5/7}{3}$

Before simplifying, remember to convert a mixed integer to an improper fraction (heavier on top).

= 5/7 ÷ 3/1

=5/7 x 1/3

= $\frac{5}{21}$ it I is the simplified answer of given complex fraction.

Example 1

= $\frac{2}{\frac{1}{5}}$

= $2 ÷ \frac{1}{5}$

= $2 X \frac{5}{1}$

= $\frac{10}{1}$

= 10 answer 

Example 2 

= $\frac{12}{\frac{5}{3}} X \frac{1}{2}$

=  $\frac{12}{1} ÷ \frac{5}{3} X \frac{1}{2}$

= $\frac{12}{1} X \frac{3}{5} X \frac{1}{2}$

= 6 X (3/5)

= 18/5 answer 

  • Hence, we summarize Firstly, in both the numerator and denominator; transform the mixed fraction to an improper fraction.
  • 2. Next, decrease the numerator and denominator fractions to the smallest possible fractions.
  • 3. We can simply transform this complex fraction into a fraction division issue because the fraction implies to divide.
  • 4. It implies to multiply the numerator by the denominator after inverting the denominator.
  • 5. Now it’s time to solve the problem.

Just use inverse multiplying approach whenever possible. To be explicit, you can simplify almost any complex fraction by reducing the numerator and denominator to single fractions and multiplying the numerator by the inverse of the denominator. Inverse multiplication is difficult and time-consuming to utilize in complex fractions with variables. The more complicated the variable expressions in the complex fraction are, the more difficult and time-consuming it is to use it. Inverse multiplication is a decent choice for “simple” complex fractions including variables, but complex fractions with numerous variable components in the numerator and the denominator may be easier to simplify using the alternative approach outlined below.

Example 

= $\frac{\frac{y^2}{16} – 1}{\frac{y}{8} – \frac{1}{2}}$

Create a common denominator and transform the numerator and denominator to single fractions. The numerator’s terms have a least common denominator of 16. The terms in this denominator have a least common denominator of 8. Found that two common denominators used to make individual fractions may or not be the very same. Most of the time, they do not have the same value.)

$\frac{y^2}{16} – 2 ÷ \frac{y}{8} – \frac{1}{2}$

=   $\frac{y^2-16}{16} ÷ \frac{y-4}{8}$

Ensure to multiply the second fraction inverted.

=  $\frac{y^2-16}{16} x \frac{8}{y-4}$

Understand that factoring may be required to lower a fraction.

(y2– 16) = (y + 4) (y – 4 ) 

Make sure our final conclusion is in the simplest simplified possible manner.

When the numerator of the bottom single fraction is reversed, it becomes the denominator, therefore it cannot reach zero.

=$\frac{(y+4){2}$ Answer 

Example 2

$\frac{\frac{x+2}{3}}{\frac{4}{3}}$

= $\frac{x+2}{3} X \frac{3}{4}$

= 3 cancel with 3 by cross multiply 

Hence 

= $\frac{x+2}{4}$

Example 3

$\frac{(x+4)/4}{(2x+4)/2}$

= $\frac{x+4}{4} ÷ \frac{2x+4}{2}$

= $\frac{x+4}{4} X \frac{2}{2x+4}$ 

= (x + 4)/4 ×1/(x + 2) 

= ( x + 4) /(4(x+2))   it is the required simplified answer.

Expression-Involved Complex Fractions

If we have a complex fraction using rational expressions as that of the numerator and denominator, we go through the same stages, and that therefore factoring is essential.

  • Stage 1: Take into account it all.
  • Stage 2 To use the greatest power of each element, compute the overall least common denominator
  • Stage 3: Multiply the LCD by all terms.
  • Stage 4: Merge terms that are similar.
  • Step 5: Cancel and Factor.

Example 

= $\frac{x-2- \frac{1}{x-2}}{3}$

= $\frac{(x-2)^2  – 1}{x-2}     X \frac{1}{3}$

= $\frac{(x-2)^2- 1}{3 (x-2)}$

Then evaluate the square of numerator and will get the final answer.

Method 2 for Simplifying a Complex Fraction

Step 1: Multiply the complex fraction’s numerator and denominator by the LCD of the fractions in the calculation.

Step 2: Keep it simple.

When a term is a fraction, divide LCD( least common factor)  by the denominator then multiply LCD by the numerator to apply the LCD.

If somehow the word is not a fraction, simply multiply it by LCD.

Example 

=  $\frac{\frac{1}{3}}{\frac{2}{9}}$

= $\frac{9 x \frac{1}{3}}{9 x \frac{2}{9}}$

For the full problem, the least common denominator is 8.

Multiply the upper part by 8 to get the total.

=$\frac{3}{2}$

=  $\frac{\frac{1}{2}}{\frac{3}{1}}$

=  $\frac{2 x \frac{1}{2}}{2 x 3}$  

=  $\frac{1}{6}$ ANSWER 

Example 

$\frac{(3 X \frac{1}{3})}{(\frac{3}{4}+\frac{5}{2})}$

= $\frac{1}{(\frac{3+10}{4})}$

= $\frac{1}{\frac{13}{4}}$

= $1 x \frac{4}{13}$

= $\frac{4}{13}$ 

This is the required simplified answer. 

Solving complex fraction with calculators 

The following is using the complex fractions calculator:

  • Part 1: Fill in the input field with the complex fraction.
  • Part 2: To have the simplest form, click the “Simplify” option.
  • Part 3: Lastly, the output field will show the result of converting a complex fraction to the simplest fraction.

Unit rates and complex fractions 

Because the word unit literally means “one,” a unit rate is made up of one unit. Individuals use rates on a daily basis, including when they work 40 hours per week or earn interest at a banking. Unit rates are defined as rates represented as a quantity of one, such as 2 ft / sec (that is, per 1 second) or 5 miles an hour (that is, per 1 hour).

We utilize unit rates to figure out how much of anything goes into one object. Whenever we talk about unit rates, we use terms like ‘per’ and ‘in.

For instance, if someone went on a bicycle journey and cycled 6 miles ‘per’ hour to reach their destination, their unit rate would be 1 hour. They can cover 6 miles in one hour.

Why are Unit Rates Calculated?

When you calculate a unit rate, you’re struggling to sort out how many of something there have been in one unit. To fix the problem, we employ fractional ratios. When solving for unit rates, keep in mind that the word ‘per’ will appear frequently.

Income per hour, breaths per second, miles a gallon, miles per hour, and mins per hour are just some few possibilities.

The unit rate is calculated in three steps.

  • 1st, establish the ratio
  • 2 Simplify
  • 3 Identify with appropriate units

Question

Sara completed 3/5 of a wall in half a day. How long would it take Sara to finish painting a wall?

First step is to the Write the fraction 

 = $\frac{\frac{1}{2}}{\frac{3}{5}}$

= $\frac{1}{2} x \frac{5}{3}$           simplify 

= 5/4 

Now decide a suitable unit 

= 5/4 days per wall 

If a worker works 4/5 hours a day and get $200 then find his/her per hour earning.

So, this per day income is also called unit rate.

Conclusion 

  • A fraction with fractions in the denominator and numerator, or both, is called a complicated fraction. A complex rational expression is a complex fraction that contains variables.
  • For explain complex fractions, there are two techniques.
  • In both the denominator and the numerator, create a single fraction.
  • Multiply the top of the fraction even by reciprocal of a bottom using the dividing rule.
  • Reduce the fraction to its simplest form.
  • We go through the same processes if we have a complex fraction with rational expressions for the numerator and denominator, so factoring is necessary.
  • Please remember that the word ‘per’ will arise frequently when solving for unit rates.
  • Only a few examples include earnings per hour, breaths per second, miles per gallon, miles per hour, and minutes per hour.

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