Two fractions are considered equivalent when they have the same value. Knowing how to convert a fraction to an equivalent is an essential math skill used from basic algebra to advanced calculus. This article will cover various ways to calculate equivalent fractions, from basic multiplication and division to more complex methods of solving problems.
Method 1 of 5: Forming Equivalent Fractions
Step 1. Multiply the numerator and denominator by the same number
Two different but equivalent fractions have, by definition, numerators and denominators that are multiples of each. In other words, multiplying the numerator and denominator of a fraction by the same number will produce an equivalent fraction. Although the numbers in the new fraction are different, the fractions will have the same value.
- For example, if we take the fraction 4/8 and multiply both the numerator and denominator by 2, we get (4×2)/(8×2) = 8/16. These two fractions are equivalent.
- (4×2)/(8×2) is essentially equal to 4/8 × 2/2. Remember that when multiplying two fractions, we multiply crosswise, that is, numerator to numerator and denominator to denominator.
- Note that 2/2 equals 1 when the division is performed. So it's easy to see why 4/8 and 8/16 are equivalent, since multiplying 4/8 × (2/2) = 4/8. The same can be said for 4/8 = 8/16.
- Any fraction has an infinite number of equivalent fractions. It is possible to multiply the numerator and denominator by any whole number, no matter how large or small, to get an equivalent fraction.
Step 2. Divide the numerator and denominator by the same number
As with multiplication, division can also be used to find a new fraction equivalent to the initial fraction. Simply divide the numerator and denominator of a fraction by the same number to get an equivalent fraction. There is a point in this process - the resulting fraction must have integers in both the numerator and denominator to be considered valid.
For example, let's look at the 4/8 fraction again. If, instead of multiplying, we divide both the numerator and denominator by 2, we get (4÷2)/(8÷2) = 2/4. Both 2 and 4 are whole numbers, so this equivalent fraction is valid
Method 2 of 5: Using Basic Multiplication to Determine Equivalence
Step 1. Find the number by which the smallest denominator must be multiplied to generate the largest denominator
Many fraction-related problems involve determining whether two fractions are equivalent. When calculating this number, you can start putting both fractions on equal terms to determine equivalence.
- For example, take the 4/8 and 8/16 fractions again. The smallest denominator, 8, and we would have to multiply that number by 2 to make it the largest, which is 16. So the number in this case is 2.
- For more difficult numbers, it is possible to simply divide the largest denominator by the smallest. In this case, 16 will be divided by 8, resulting in 2.
- The number may not always be an integer. For example, if the denominators were 2 and 7, the number in question would be 3, 5.
Step 2. Multiply the numerator and denominator of the fraction expressed in smaller terms by the number in the first step
Two different but equivalent fractions have, by definition, numerators and denominators multiple of each other. In other words, multiplying the numerator and denominator of a fraction by the same number will produce an equivalent fraction. Although the numbers in this new fraction will be different, the fractions will have the same value.
- For example, if we take the 4/8 fraction from the first step and multiply both numerator and denominator by the number 2, determined earlier, we have (4×2)/(8×2) = 8/16 - thus proving that both fractions are equivalent.
Method 3 of 5: Using Basic Division to Determine Equivalence
Step 1. Calculate each fraction as a decimal number
In the case of simple fractions without variables, you can basically express each fraction as a decimal number in order to determine equivalence. Since every fraction is really a division problem from the start, this is the simplest way to determine equivalence.
- For example, take the already used 4/8. The fraction 4/8 is equivalent to calculating 4 divided by 8, ie 4/8 = 0.5. You can also solve the other example, ie 8/16 = 0.5. fraction they are equivalent if both numbers are exactly the same when expressed in decimal form.
- Remember that the decimal expression can go on for several digits before the mismatch becomes apparent. As a basic example, 1/3 = 0, 333, while 3/10 = 0, 3. When using more than one digit, you can see that the two equations are not equivalent.
Step 2. Divide the numerator and denominator of a fraction by the same number to get an equivalent fraction
In the case of more complex fractions, the division method requires additional steps. As with the multiplication method, it is possible to divide the numerator and denominator of a fraction by the same number to obtain an equivalent fraction. There is a secret to this process. The resulting fraction must have whole numbers in both the numerator and denominator to be valid.
- For example, let's look at the 4/8 fraction again. If, instead of multiplying them, we divide the numerator and denominator by 2, we have (4÷2)/(8÷2) = 2/4. 2 and 4 are both integers, so this equivalent fraction is valid.
Step 3. Reduce fractions to their minimum terms
Most fractions should normally be expressed in their minimum terms, and it will be possible to convert them to these minimum terms by dividing them by their greatest common factor (MFC). This step operates using the same logic in expressing equivalent fractions by converting them to have the same denominator, but this method seeks to reduce each fraction to its minimum expressible terms.
- When a fraction is in its simplest terms, its numerator and denominator are both as small as they can be, and neither can they be divided by any whole number to get a smaller number. To convert a fraction that is not in its simplest terms to one that is, we divide the numerator and denominator by their greatest common factor.
- The greatest common factor (MFC) of the numerator and denominator equals the greatest number that divides them both to get an integer result. Thus, in our 4/8 copy, since
Step 4. is the largest number that divides both 4 and 8, we'll divide the numerator and denominator of our fraction by 4 to get its simplest terms: (4÷4)/(8÷4) = 1/2. In the other example, 8/16, the MFC is 8, whereby we also arrive at the result 1/2 as the simplest expression of the fraction.
Method 4 of 5: Using Cross Multiplication to Solve a Variable
Step 1. Match the two fractions
We use cross-multiplication in mathematical problems that we know are equivalent, but where one of the numbers in one of them has been replaced by a variable (usually x) that must be solved. In cases like this, we know that fractions are equivalent because they are the only terms on opposite sides of the equal sign, but this resolution is not always obvious. Fortunately, in cross-multiplication, solving these problems is easy.
Step 2. Take both equivalent fractions and multiply them crosswise, in an “X” shape
In other words, one must multiply the numerator of one fraction by the denominator of the other and vice versa, then finding these two answers equal to each other and solving the problem.
Let's take the two examples 4/8 and 8/16. They don't contain a variable, but it's possible to prove the concept since we already know they're equivalent. By cross-multiplication, we have that 4×16 = 9×9, or 64 = 64, which is indisputably true. If the two numbers are not identical, the fractions are not equivalent
Step 3. Input a variable
Since cross-multiplication is the easiest way to determine equivalent fractions when solving a variable, let's introduce an unknown.
- For example, consider the equation 2/x = 10/13. To cross-multiply, we'll multiply 2 by 13 and 10 by x, then setting the answers equal to each other:
- 2×13 = 26
- 10×x = 10x
- 10x = 26
- From here, getting an answer to our variable is a matter of simple algebra. X = 10/26 = 2, 6, defining the initial equivalent fractions as 2/2, 6 = 10/13.
Step 4. Use cross multiplication in equations with multiple variables or expressions with unknowns
One of the best things about cross multiplication is the fact that it works essentially the same whether you're dealing with two simple fractions (as above) or with more complex fractions. For example, if both fractions contain variables, they should only be eliminated at the end of the resolution process. Similarly, if the numerators or denominators of fractions contain expressions with variables (such as x+1), just “multiply” through the distributive property and solve them normally.
- For example, consider the equation [(x+3)/2] = [(x+1)/4)]. In this case, as before, we will solve it with cross multiplication:
- (x+3)×4 = 4x+12
- (x+1)×2 = 2x+2
- 2x+2 = 4x+12
We'll simplify the equation by subtracting 2x from both sides
- 2 = 2x+12
Here, we will isolate the variable by subtracting 12 from both sides
- -10 = 2x
We'll divide both numbers by 2 to unravel x
- -5 = x
Method 5 of 5: Using the Quadratic Formula to Solve Variables
Step 1. Multiply the two fractions crosswise
In equivalence problems that require the quadratic formula, we will still start with cross-multiplication. However, any multiplication that involves multiplying variable terms by other variable terms will likely result in an expression that will not easily be solved with pure algebra. In cases like this, you may need to use techniques such as factorization and quadratic formulas.
- For example, let's look at the equation [(x+1)/3] = [4/(2x-2)]. Initially, we will perform cross-multiplication:
- (x+1)×(2x-2) = 2x2+2x-2x-2 = 2x2-2
- 4×3 = 12
- 2x2-2 = 12
Step 2. Express the equation as a quadratic equation
At this point, we want to express this equation in quadratic form (ax2+bx+c = 0), which can be done by setting it to zero. In this case, we'll subtract 12 from both sides to get 2x2-14 = 0.
- Some values can equal 0. Although 2x2-14 = 0 is the simplest form for the equation, the true quadratic equation is represented by 2x2+0x+(-14) = 0. It helps to look at the quadratic form of an equation even when some of its values are equal to 0.
Step 3. Solve it by entering the numbers of your equation into the quadratic formula
The quadratic formula x = [-b±√(b2-4ac)]/2a will help us figure out the x value. Don't be intimidated by the size of the formula. You are simply taking the values of the quadratic equation from step two and entering them at the appropriate points before solving it.
- [x = (-b±√(b)2-4ac)]/2a
- In our equation, 2x2-14=0, a = 2, b = 0 and c = -14.
- x = [-0±√(02-4(2)(-14))]/2(2)
- x = [±√(0-(-112))]/2(2)
- x = [± 112]/2(2)
- x = ±10, 58/4
- x = ±2, 64
Step 4. Check the answer by entering the x value back into the quadratic equation
By entering the calculated value into the quadratic equation from step two, you can easily determine if you've arrived at the correct answer. In this example, you will place both 2, 64 and -2, 64 into the quadratic equation.
- Converting fractions to equivalent form is a way of multiplying them by 1. When converting 1/2 to 2/4, multiplying the numerator and denominator by 2 is the same as multiplying 1/2 by 2/2, resulting in 1.
- If you prefer, convert mixed numbers into inappropriate fractions to facilitate the conversion. Obviously, not all fractions will be as simple to convert as the 4/8 example above. For example, mixed numbers (such as 1 3/4, 2 5/8, 5 2/3, etc.) can make the conversion process a little more complicated. If you need to convert a mixed number to an equivalent fraction, you can do it in two ways: transforming the mixed number into an improper fraction and converting it normally or keeping the mixed number and getting a mixed number in response.
- To convert it to an improper fraction, multiply the integer component by the denominator of the fractional component, adding it to the numerator. For example, 1 2/3 = [(1×3)+2]/3 = 5/3. Then, if you prefer, you can freely convert it. For example, 5/x×2/2 = 10/6, which is equivalent to 1 2/3.
- However, it is not necessary to convert it to an improper fraction as described above. If we don't, we'll ignore the integer component, convert the isolated fractional component, and then add the unchanged integer component. For example, in the case of 3 4/16, we will only look at 4/16. 4/16÷4/4 = 1/4. So when we add the integer component, we have a new mixed number, or 3 1/4.
- Multiplication and division work by getting equivalent fractions because multiplying and dividing by fractional forms of the number 1 (2/2, 3/3, etc.) result, by definition, in answers equivalent to the initial fraction. Addition and subtraction do not allow for this possibility.
- Although you multiply numerators and denominators together when multiplying fractions, you cannot add or subtract denominators when adding or subtracting fractions.
- For example, above, we found that 4/8÷4/4 = 1/2. If we add 4/4 instead, we get a completely different answer: 4/8+4/4 = 4/8+8/8 = 12/8 = 1 1/2 or 3/2, neither of which is equal to 4/8.