How Do You Know if Two Fractions Are Equivalent

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Two fractions are equivalent if they have the same value. Knowing how to catechumen a fraction into an equivalent one is an essential math skill that'southward necessary for everything from basic algebra to avant-garde calculus. This article will cover several ways to calculate equivalent fractions from basic multiplication and division to more complex methods for solving equivalent fraction equations.

1. i

   Multiply the numerator and denominator by the same number. Two fractions that are dissimilar merely equivalent have, by definition, numerators and denominators that are multiples of each other. In other words, multiplying the numerator and denominator of a fraction by the same number will produce an equivalent fraction. Though the numbers in the new fraction volition be different, the fractions will take the same value.

   • For case, if we have the fraction 4/eight and multiply both the numerator and denominator by 2, we get (4×2)/(8×2) = 8/16. These 2 fractions are equivalent.
   • (four×2)/(8×two) is essentially the same as four/viii × two/ii Remember that when multiplying ii fractions, we multiply across, pregnant numerator to numerator and denominator to denominator.
   • Notice that 2/2 equals 1 when you carry out the division. Thus, it's like shooting fish in a barrel to see why iv/8 and eight/16 are equivalent since multiplying 4/8 × (ii/2) = 4/8 yet. The aforementioned manner information technology's fair to say that 4/eight = 8/16.
   • Whatsoever given fraction has an infinite number of equivalent fractions. You tin multiply the numerator and denominator by whatever whole number, no matter how big or small to obtain an equivalent fraction.

2. 2

   Separate the numerator and denominator by the aforementioned number. Like multiplication, division tin also be used to find a new fraction that's equivalent to your starting fraction. Only divide the numerator and the denominator of a fraction past the same number to obtain an equivalent fraction. In that location is one caveat to this process--the resulting fraction must have whole numbers in both the numerator and denominator to exist valid.

   • For instance, let'due south look at 4/eight again. If, instead of multiplying, we divide both the numerator and denominator past 2, we get (four ÷ 2)/(8 ÷ 2) = 2/four. 2 and 4 are both whole numbers, so this equivalent fraction is valid.

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1. one

   Observe the number by which the smaller denominator needs to be multiplied to make the larger denominator. Many problems regarding fractions involve determining if ii fractions are equivalent. By calculating this number, you can brainstorm putting the fractions in the same terms to determine equivalency.

   • For case, take the fractions 4/8 and 8/xvi again. The smaller denominator is 8, and nosotros would take to multiply that number x2 in gild to make the larger denominator, which is 16. Therefore, the number in this case is 2.^[1]
   • For more than difficult numbers, you tin can simply separate the larger denominator by the smaller denominator. In this case xvi divided by 8, which still gets us 2.
   • The number may not e'er be a whole number. For instance, if the denominators were 2 and 7, then the number would be three.5.

2. 2

   Multiply the numerator and denominator of the fraction expressed in lower terms past the number from the first step. Two fractions that are unlike merely equivalent have, past definition, numerators and denominators that are multiples of each other. In other words, multiplying the numerator and denominator of a fraction past the aforementioned number volition produce an equivalent fraction. Though the numbers in this new fraction volition be unlike, the fractions will accept the same value.^[2]

   • For instance, if we have the fraction 4/8 from step one and multiply both the numerator and denominator by our previously determined number ii, we become (four×2)/(8×two) = 8/16. Thus proving that these two fractions are equivalent.

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1. 1

   Calculate each fraction as a decimal number. For unproblematic fractions without variables, you can simply express each fraction as a decimal number to determine equivalency. Since every fraction is actually a division problem to begin with, this is the simplest way to determine equivalency.

   • For instance, accept our previously used iv/8. The fraction 4/8 is equivalent to saying 4 divided by viii, which iv/viii = 0.5. Yous can solve for the other example also, which is viii/16 = 0.5. Regardless of the terms of a fraction, they are equivalent if the ii numbers are exactly the aforementioned when expressed equally a decimal.
   • Remember that the decimal expression may go several digits before the lack of equivalence becomes apparent. As a basic case, one/3 = 0.333 repeating while 3/10 = 0.3. Past using more than one digit, we see that these two fractions are non equivalent.

2. 2

   Split up the numerator and denominator of a fraction past the same number to get an equivalent fraction. For more complex fractions, the division method requires additional steps. As with the multiplication method, you can divide the numerator and the denominator of a fraction by the same number to obtain an equivalent fraction. In that location is one caveat to this process. The resulting fraction must accept whole numbers in both the numerator and denominator to be valid.

   • For example, let'south look at 4/8 again. If, instead of multiplying, we divide both the numerator and denominator by 2, we get (iv ÷ 2)/(8 ÷ 2) = two/four. 2 and 4 are both whole numbers, so this equivalent fraction is valid.

3. 3

   Reduce the fractions to their lowest terms. Nigh fractions should typically be expressed in their everyman terms, and you tin convert fractions to their simplest terms by dividing by their greatest common gene (GCF).^[3] This step operates by the aforementioned logic of expressing equivalent fractions past converting them to have the aforementioned denominator, but this method seeks to reduce each fraction to its lowest expressible terms.

   • When a fraction is in its simplest terms, its numerator and denominator are both as minor as they tin can be. Neither can be divided by the whatever whole number to obtain annihilation smaller. To convert a fraction that'due south non in simplest terms to an equivalent form that is, nosotros split the numerator and denominator by their greatest common factor.
   • The greatest common factor (GCF) of the numerator and denominator is the largest number that divides into both to give a whole number result. So, in our iv/8 example, since 4 is the largest number that divides evenly into both 4 and 8, we would divide the numerator and denominator of our fraction by four to get information technology in simplest terms. (4 ÷ 4)/(8 ÷ four) = 1/2. For our other example of 8/16, the GCF is 8, which besides results in ane/2 as the simplest expression of the fraction.

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1. ane

   Set the two fractions equal to 1 another. We utilize cross multiplication for math issues where we know the fractions are equivalent, merely ane of the numbers has been replaced with a variable (typically x) for which we must solve. In cases like this, we know these fractions are equivalent because they're the sole terms on opposite sides of an equal sign, merely information technology's oftentimes not obvious how to solve for the variable. Luckily, with cross multiplication, solving these types of problems is easy.^[4]

2. 2

   Have the two equivalent fractions and multiply across the equals sign in an "X" shape. In other words, you multiply the numerator of i fraction by the denominator of the other and vice versa, then set up these two answers equal to each other and solve.^[5]

   • Accept our 2 examples of 4/8 and 8/16. These two don't contain a variable, but we can testify the concept since nosotros already know they're equivalent. By cross multiplying, we get 4 x 16 = eight x 8, or 64 = 64, which is plainly true. If the two numbers are non the aforementioned, then the fractions are not equivalent.

3. 3

   Introduce a variable. Since cantankerous multiplication is the easiest way to determine equivalent fractions when you lot must solve for a variable, permit's add a variable.

   • For example, let'southward consider the equation 2/ten = ten/13. To cross multiply, nosotros multiply 2 by 13 and ten by x, then set our answers equal to each other:
     • 2 × 13 = 26
     • ten × ten = 10x
     • 10x = 26. From here, getting an answer for our variable is a matter of simple algebra. 10 = 26/10 = two.6, making the initial equivalent fractions ii/ii.6 = 10/13.

4. 4

   Use cantankerous multiplication for equations with multiple variables or variable expressions. One of the best things about cross multiplication is that information technology works in essentially the same manner whether y'all're dealing with two simple fractions (as to a higher place) or with more complex fractions. For instance, if both fractions incorporate variables, you only have to eliminate these variables at the end during the solving procedure. Similarly, if the numerators or denominators of your fractions contain variable expressions (such as x + i), just "multiply through"past using the distributive property and solve as yous normally would.^{[half dozen]}

   • For case, let'southward consider the equation ((10 + iii)/2) = ((ten + 1)/iv). In this case, every bit in a higher place, nosotros'll solve past cross multiplying:
     • (x + 3) × 4 = 4x + 12
     • (x + ane) × 2 = 2x + ii
     • 2x + two = 4x + 12, then we can simplify the equation past subtracting 2x from both sides
     • 2 = 2x + 12, then we should isolate the variable by subtracting 12 from both sides
     • -10 = 2x, and divide by 2 to solve for 10
     • -five = 10

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1. 1

   Cantankerous multiply the ii fractions. For equivalency problems that crave the quadratic formula, we even so begin by using cross multiplication. However, whatsoever cantankerous multiplication that involves multiplying variable terms by other variable terms is likely to result in an expression that tin't hands be solved via algebra. In cases like these, you may need to apply techniques like factoring and/or the Quadratic Formula.^[vii]

   • For example, permit'southward look at the equation ((x +1)/iii) = (4/(2x - 2)). Starting time, let'due south cross multiply:
     • (x + 1) × (2x - 2) = 2x² + 2x -2x - 2 = 2x² - 2
     • four × 3 = 12
     • 2x² - 2 = 12.

2. 2

   Limited the equation as a quadratic equation. At this point, we want to express this equation in quadratic form (ax² + bx + c = 0), which nosotros do by setting the equation equal to naught. In this case, nosotros subtract 12 from both sides to get 2x^two - xiv = 0.

   • Some values may equal 0. Though 2x² - 14 = 0 is the simplest grade of our equation, the true quadratic equation is 2xⁱⁱ + 0x + (-14) = 0. It will probably assistance early on on to mirror the form of the quadratic equation even when some values are 0.

3. 3

   Solve past plugging the numbers from your quadratic equation into the quadratic formula. The quadratic formula (ten = (-b +/- √(b² - 4ac))/2a) will assistance us solve for our value ten at this bespeak.^[8] Don't exist intimidated by the length of the formula. You lot're simply taking the values from your quadratic equation in step two and plugging them into the appropriate spots before solving.

   • x = (-b +/- √(b² - 4ac))/2a. In our equation, 2x² - 14 = 0, a = 2, b = 0, and c = -14.
   • x = (-0 +/- √(0ⁱⁱ - 4(2)(-14)))/2(ii)
   • x = (+/- √( 0 - -112))/2(ii)
   • x = (+/- √(112))/2(2)
   • 10 = (+/- x.58/4)
   • x = +/- ii.64

4. iv

   Bank check your reply by plugging the x value back into your quadratic equation. Past plugging the calculated value of 10 back into your quadratic equation from pace two, you can easily determine if you reached the right answer.^[9] In this case, yous would plug both 2.64 and -two.64 into the original quadratic equation.

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• Question

  How can I use multiplication to observe equivalent fractions?

  

  David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a individual tutoring company based in Los Angeles, California. With over 10 years of educational activity experience, David works with students of all ages and grades in various subjects, as well equally college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. Later attaining a perfect 800 math score and a 690 English language score on the SAT, David was awarded the Dickinson Scholarship from the Academy of Miami, where he graduated with a Bachelor's degree in Concern Assistants. Additionally, David has worked every bit an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Large Ideas Math.

  

  Bookish Tutor

  Expert Respond

  Multiply the top number and bottom number past the aforementioned value to find a fraction that's equivalent to the original.

• Question

  How do I draw a picture to prove that one/three is equivalent to 2/6?

  

  An like shooting fish in a barrel mode is to draw a circumvolve graph. Divide the circle into sixths. Each sixth of the circle would take a central angle of 60°. Then testify (possibly with the utilise of color) that 2 adjacent sixths combine to make one-third of the circle (with a central angle of 120°).

• Question

  What is equivalent to 4 and 2/14?

  

  For easier computation, convert 2/14 to 1/7. So the mixed number is 4 1/vii. If you lot want to catechumen that number to an improper fraction, keep the denominator the same. For the new numerator, multiply the whole number by the denominator and add together the old numerator. In this example, multiply 4 by 7 to get 28, then add i to become a numerator of 29. The denominator is vii.

• Question

  Why do I need to know how to find equivalent fractions?

  

  It is a footstep commonly taken while solving all sorts of math issues.

• Question

  What would 4 be with denominator of ii?

  

  4 = 8/2.

• Question

  What fraction is equal to two fifths?

  

  Multiply both the numerator and the denominator past the aforementioned integer (other than one). That gives you a fraction equal to 2/five. For example, you could multiply both the numerator and the denominator past half-dozen, then that 2/5 is equal to 12/30.

• Question

  How can I observe the equivalent of 1 iii/4?

  

  If you mean an improper fraction: Detect the number of quarters in 1 (4) and add them to the number of quarters in iii/four (three). 4+3 = seven, so the answer is 7/4.

• Question

  What is an equivalent number for five and 7/x?

  

  Danoyachtcapt

  Top Answerer

  Yous split 7 by 10 = .7 and so add 5 = 5.7

• Question

  What's the equivalent fraction of v/half-dozen?

  

  Get an equivalent fraction by multiplying both the numerator and the denominator by the aforementioned number. Two examples: Multiply both v and 6 past iv: 20/24 = 5/6; multiply both v and six by ix: 45/54 = 5/6.

• Question

  What are equivalent fraction of -5/9?

  

  Multiply both the numerator and the denominator by the same integer (any whole number). That will requite you an equivalent fraction. For example, multiply both -five and ix by 3: that forms a new fraction, -xv/27.

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• Converting fractions to equivalent forms is actually a form of multiplying them past 1. In converting 1/2 to 2/4, multiplying the numerator and denominator by 2 is the same as multiplying 1/two by 2/2, which equals 1.

• If desired, convert mixed numbers to improper fractions to make converting easier. Evidently, not every fraction you come across will be as easy to convert as our 4/8 example above. For instance, mixed numbers (e.m. 1 3/four, ii 5/8, v two/3, etc.) tin make the conversion process a footling more than complicated. If you need to convert a mixed number to an equivalent fraction, you tin can do it in two ways: by changing the mixed number to an improper fraction, so converting every bit normal, or by maintaining the mixed number and receiving a mixed number as an reply.

  • To convert to an improper fraction, multiply the whole number component of the mixed number past the denominator of the fractional component and so add together information technology to the numerator. For case, 1 ii/3 = ((1 × three) + 2)/three = 5/iii. And so, if desired, you can catechumen every bit needed. For example, 5/iii × two/2 = 10/6, which is still equivalent to 1 ii/3.
  • However, we don't accept to convert to an improper fraction as above. If we don't, nosotros ignore the whole number component, convert the fractional component lonely, and then add together the whole number component back in unchanged. For example, for 3 four/16, we'll just expect at 4/16. 4/16 ÷ four/4 = i/four. Then, adding our whole number component back in, we get a new mixed number, 3 1/iv.

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• Multiplication and division work for obtaining equivalent fractions because multiplying and dividing past partial forms of the number 1 (ii/2, 3/3, etc.) requite answers that are equivalent to the starting fraction by definition. Improver and subtraction don't allow this possibility.

• Although you multiply the numerators and denominators together when multiplying fractions, you do non add or decrease denominators when adding or subtracting fractions.

  • For instance, above, we establish that four/8 ÷ 4/4 = 1/two . If we instead added by 4/4, we would take gotten a completely unlike answer. iv/eight + 4/iv = iv/8 + 8/eight = 12/8 = one 1/2 or three/2, neither of which are equal to 4/8.

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Article Summary X

To find equivalent fractions, multiply or divide the numerator and the denominator of a fraction past whatever number, as long as it's the same on top and bottom. If you divide the original fraction, the result must have whole numbers in the numerator and denominator to be valid. To bank check the result, utilise cantankerous-multiplication. Multiply the denominator of the first fraction by the numerator of the second fraction, then multiply the first numerator by the 2d denominator. If the 2 answers are equal, the fractions are equivalent. If you demand to acquire how to solve variables in your fraction, keep reading the article!

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