what to do if you have a fraction in the denominator

Download Article

Download Article

Complex fractions are fractions in which either the numerator, denominator, or both incorporate fractions themselves. For this reason, circuitous fractions are sometimes referred to equally "stacked fractions". Simplifying complex fractions is a procedure that can range from easy to difficult based on how many terms are nowadays in the numerator and denominator, whether any of the terms are variables, and, if then, the complexity of the variable terms. Meet Step one beneath to get started!

1. ane

   If necessary, simplify the numerator and denominator into unmarried fractions. Complex fractions aren't necessarily hard to solve. In fact, complex fractions in which the numerator and denominator both incorporate a single fraction are unremarkably fairly like shooting fish in a barrel to solve. And then, if the numerator or denominator of your complex fraction (or both) contain multiple fractions or fractions and whole numbers, simplify as needed to obtain a single fraction in both the numerator and denominator. This may require finding the least common denominator (LCM) of ii or more than fractions.

   • For example, allow'southward say we desire to simplify the complex fraction (iii/5 + two/fifteen)/(5/7 - 3/ten). Commencement, we would simplify both the numerator and denominator of our complex fraction to single fractions.
     • To simplify the numerator, nosotros will use a LCM of 15 by multiplying 3/5 by iii/3. Our numerator becomes 9/fifteen + 2/fifteen, which equals 11/15.
     • To simplify the denominator, we will use a LCM of 70 by multiplying 5/vii by 10/10 and three/10 by 7/seven. Our denominator becomes 50/70 - 21/70, which equals 29/70.
     • Thus, our new complex fraction is (11/15)/(29/lxx).

2. ii

   Flip the denominator to find its inverse. By definition, dividing one number by another is the same as multiplying the get-go number by the inverse of the second. Now that we have obtained a complex fraction with a single fraction in both the numerator and the denominator, nosotros can use this belongings of segmentation to simplify our complex fraction! First, find the inverse of the fraction on the lesser of the complex fraction. Do this by "flipping" the fraction - setting its numerator in the place of the denominator and vice versa.

   • In our example, the fraction in the denominator of the complex fraction (11/15)/(29/70) is 29/70. To find its inverse, we only "flip" information technology to get lxx/29.
     • Notation that, if your circuitous fraction has a whole number in its denominator, you can care for it equally a fraction and find its changed all the same. For instance, if our complex fraction was (11/xv)/(29), we can define the denominator as 29/ane, which makes its inverse 1/29.

   Advertisement

3. iii

   Multiply the numerator of the complex fraction by the changed of the denominator. Now that you've obtained the inverse of your complex fraction's denominator, multiply it past the numerator to obtain a single simple fraction! Think that to multiply ii fractions, nosotros merely multiply across - the numerator of the new fraction is the product of the numerators of the two erstwhile ones, and similarly with the denominator.

   • In our instance, we would multiply xi/15 × lxx/29. 70 × 11 = 770 and xv × 29 = 435. And so, our new elementary fraction is 770/435.

4. 4

   Simplify the new fraction by finding the greatest common gene. We now take a single, simple fraction, so all that remains is to render it in the simplest terms possible. Find the greatest common factor (GCF) of the numerator and denominator and separate both by this number to simplify.

   • Ane common gene of 770 and 435 is 5. So, if we divide the numerator and denominator of our fraction past 5, we obtain 154/87. 154 and 87 don't accept whatever common factors, and so we know nosotros've plant our final reply!

   Advertizing

1. i

   When possible, employ the inverse multiplication method above. To exist clear, nearly whatsoever circuitous fraction can exist simplified by reducing its numerator and denominator to single fractions and multiplying the numerator past the inverse of the denominator. Complex fractions containing variables are no exception, though, the more complicated the variable expressions in the complex fraction are, the more than hard and time-consuming information technology is to use inverse multiplication. For "like shooting fish in a barrel" complex fractions containing variables, inverse multiplication is a good pick, just complex fractions with multiple variable terms in the numerator and denominator may exist easier to simplify with the alternate method described below.

   • For example, (1/x)/(x/half-dozen) is piece of cake to simplify with changed multiplication. 1/10 × 6/x = 6/ten² . Here, at that place is no need to use an alternating method.
   • However, (((1)/(10+iii)) + ten - 10)/(ten +four +((1)/(x - 5))) is more hard to simplify with changed multiplication. Reducing the numerator and denominator of this complex fraction to single fractions, inverse multiplying, and reducing the result to simplest terms is probable to exist a complicated process. In this case, the alternating method below may be easier.

2. 2

   If changed multiplication is impractical, kickoff by finding the lowest common denominator of the fractional terms in the circuitous fraction. The first footstep in this alternate method of simplification is to notice the LCD of all the fractional terms in the complex fraction - both in its numerator and in its denominator. Usually, if one or more of the fractional terms have variables in their denominators, their LCD is just the product of their denominators.

   • This is easier to understand with an example. Let's try to simplify the complex fraction nosotros mentioned to a higher place, (((1)/(ten+iii)) + 10 - 10)/(x +4 +((1)/(x - 5))). The fractional terms in this circuitous fraction are (1)/(x+3) and (one)/(10-5). The common denominator of these two fractions is the product of their denominators: (ten+iii)(x-5).

3. 3

   Multiply the numerator of the complex fraction by the LCD you lot just found. Adjacent, nosotros'll need to multiply the terms in our complex fraction by the LCD of its fractional terms. In other words, we'll multiply the unabridged circuitous fraction by (LCD)/(LCD). We can exercise this freely because (LCD)/(LCD) is equal to one. First, multiply the numerator on its own.

   • In our example, we would multiply our complex fraction, (((ane)/(x+3)) + x - 10)/(10 +4 +((1)/(x - 5))), past ((10+3)(ten-5))/((10+3)(x-5)). We'll have to multiply through the numerator and denominator of the complex fraction, multiplying each term by (x+iii)(x-v).
     • First, allow'south multiply the numerator: (((ane)/(x+iii)) + 10 - 10) × (x+3)(x-5)
       • = (((ten+3)(x-5)/(x+3)) + ten((x+3)(x-v)) - 10((10+3)(ten-5))
       • = (x-v) + (x(xⁱⁱ - 2x - 15)) - (ten(x² - 2x - fifteen))
       • = (x-5) + (xⁱⁱⁱ - 2x² - 15x) - (10xⁱⁱ - 20x - 150)
       • = (x-five) + tenⁱⁱⁱ - 12x² + 5x + 150
       • = 10³ - 12xⁱⁱ + 6x + 145

4. four

   Multiply the denominator of the complex fraction past the LCD as you did with the numerator. Continue multiplying the complex fraction past the LCD y'all plant past proceeding to the denominator. Multiply through, multiplying every term by the LCD.

   • The denominator of our complex fraction, (((1)/(x+three)) + x - 10)/(x +4 +((1)/(x - five))), is x +iv +((1)/(x-5)). We'll multiply this by the LCD we found, (x+three)(x-5).
     • (x +4 +((ane)/(x - 5))) × (x+3)(10-five)
     • = x((x+3)(x-5)) + 4((x+3)(x-5)) + (ane/(10-5))(ten+iii)(x-five).
     • = x(x² - 2x - fifteen) + 4(x² - 2x - 15) + ((x+3)(x-5))/(x-5)
     • = 10³ - 2x² - 15x + 4x^two - 8x - threescore + (x+3)
     • = x³ + 2x² - 23x - sixty + (10+three)
     • = x³ + 2x² - 22x - 57

5. v

   Form a new, simplified fraction from the numerator and denominator y'all only found. After multiplying your fraction by your (LCD)/(LCD) expression and simplifying by combining like terms, you should be left with a uncomplicated fraction containing no fractional terms. As you may take noticed, by multiplying through by the LCD of the fractional terms in the original circuitous fraction, the denominators of these fractions abolish out, leaving variable terms and whole numbers in the numerator and denominator of your answer, but no fractions.

   • Using the numerator and denominator nosotros found to a higher place, nosotros can construct a fraction that'south equal to our initial circuitous fraction but which contains no fractional terms. The numerator we obtained was x³ - 12x² + 6x + 145 and the denominator was 10³ + 2x² - 22x - 57, so our new fraction is (10^three - 12x² + 6x + 145)/(x³ + 2x² - 22x - 57)

   Advertisement

Add together New Question

• Question

  How do I solve 5/6 divided by ane 1/4?

  

  That'due south 5/6 divided by 5/four, which is solved by multiplying v/6 past 4/v, which is 20/30 or 2/3.

• Question

  How practise I solve four/one/4 - 5/2?

  

  four/1/4 is four divided by ¼. That's equal to (iv)(four) = 16. Then 16 - 5/2 = 16 - 2½ = 13½ or 27/two.

• Question

  How do I solve iii 2/3 = 2 x/3?

  

  First alter the left side to an improper fraction: 3 ii/iii = xi/3. Then change the right side to (2x)/3. Now solve for x: xi/3 = 2x/3. Multiply both sides past 3, and so that 11 = 2x, and ten = five½.

• Question

  How do I solve (i/iii)six + (2/3)6?

  

  Looking at information technology logically, only as one-3rd of a cake plus ii-thirds of the same cake amounts to the full cake, one-tertiary of half-dozen plus two-thirds of 6 equals 6. Doing the multiplication, (1/3)(6) = half dozen/3. (2/3)(6) = 12/three. 6/3 + 12/3 = 18/iii = 6.

• Question

  How do I solve 2/vii divided by fourteen?

  

  Dividing past 14 is the aforementioned equally multiplying by 1/14. (ii/vii) multiplied by (ane/14) equals 2/98 or i/49. Note that dividing a fraction by a whole number is the same as multiplying the denominator by that whole number.

• Question

  How do I solve (ten/3 - two) - ten + 4/2 - x/6?

  

  You can't "solve" this expression, but yous can simplify it: (ten/3) - (2) - (x) + (4/ii) - (10/6) = (2x/6) - ii - (6x/six) + (ii) - (10/six) = [(2x - 6x - x) / 6] + (-2 + 2) = [(-5x) / 6] + 0 = -5x / 6. (This is not a solution. A solution is something like x = 10. Expressions can be "solved" only when they are equations.)

• Question

  How do I solve 3/x - ane?

  

  Since this expression is not an equation, information technology tin can't be "solved." Information technology tin can be simplified slightly: (3/ten) - 1 = (3/x) - (x/x) = (3-x) / x. However, this form is non any more useful than the original expression, so at that place's really no indicate in making the modify.

Ask a Question

200 characters left

Include your electronic mail address to become a message when this question is answered.

Submit

Advertisement

Video

• Show each step of your work. Fractions tin can easily become confusing if yous are trying to motion also rapidly or attempting to practise them in your head.

• Find examples of complex fractions online or in your textbook. Follow along with each step until you get the hang of it.

Thanks for submitting a tip for review!

Advert

Near This Article

Article Summary X

To simplify complex fractions, start by finding the inverse of the denominator, which you lot can do by simply flipping the fraction. Then, multiply this new fraction past the numerator. You lot should now have a unmarried uncomplicated fraction. Finally, simplify the new fraction by finding the greatest common gene between the numerator and the denominator, and dividing both fractions past this number. If you desire to larn how to simplify fractions that have variables in them, keep reading the article!

Did this summary help you?

Thanks to all authors for creating a page that has been read 242,516 times.

Did this commodity aid you lot?

kaufmanlumpose.blogspot.com

Source: https://www.wikihow.com/Simplify-Complex-Fractions

Share this post