How to Find Equivalent Fractions

Two fractions are said to be equivalent if they have the same value. Knowing how to convert a fraction to its equivalent forms is a necessary math skill for everything from basic algebra to advanced math. This article will cover several ways to calculate equivalent fractions from basic multiplication and division to more complex methods for solving equations with equivalent fractions.

Table of Contents

Creating Equivalent Fractions

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Multiply the numerator and denominator by the same number. By definition, two different but equivalent fractions whose numerator and denominator are multiples of each other. In other words, multiplying the numerator and denominator of a fraction by the same number yields an equivalent fraction. Although the numbers in the new fraction will be 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 exactly the same as 4/8×2/2. Remember that when multiplying two fractions, we are multiplying horizontally, i.e. numerator by numerator and denominator by denominator.
Note that 2/2 equals 1 when you do the division. Therefore, it is easy to see why 4/8 and 8/16 are equal because 4/8 × (2/2) still = 4/8. Likewise 4/8 = 8/16.
Any fraction has an infinite number of equivalent fractions. You can multiply the numerator and denominator by any integer, large or small, to produce an equivalent fraction.

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Divide the numerator and denominator by the same number. Like multiplication, division is also used to find a new fraction that is equivalent to the original fraction. Simply divide the numerator and denominator of a fraction by the same number to get an equivalent fraction. However, the resulting fraction must have both numerator and denominator being integers.

For example, look back at the fraction 4/8. Instead of multiplying, we divide both numerator and denominator by 2, we get (4 ÷ 2)/(8 ÷ 2) = 2/4. 2 and 4 are both integers, so this equivalent fraction is valid.

Using Basic Multiplication to Determine Equivalence

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Find the number that, when multiplied by the smaller denominator, gives the larger denominator. Many fraction problems involve determining if two fractions are equivalent. By calculating this number, you can reduce the fractions to the same terms to determine equivalence.

For example, get back the fractions 4/8 and 8/16. The smaller denominator is 8, and we will have to multiply that by 2 to get the larger denominator 16. So, the number to look for in this case is 2.
For more complex numbers, you can simply divide the large denominator by the small denominator. In the above example 16 divided by 8, the result is 2.
This number is not always an integer. For example, if the two denominators are 2 and 7, then 7 divided by 2 equals 3.5.

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Multiply the numerator and denominator of the fraction represented in the lower term by the number determined in the previous step. By definition, two different but equivalent fractions whose numerator and denominator are multiples of each other . In other words, multiplying the numerator and denominator of a fraction by the same number yields the equivalent fraction. Although the numbers in this new fraction will be different, their values are the same. ^{[1] X Research Source}

For example, if we take the fraction 4/8 from step one and multiply both the numerator and the denominator by the 2 determined earlier, we have (4×2)/(8×2) = 8/16 . That proves that these two fractions are equivalent.

Using Basic Division to Determine Equivalence

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Divide each fraction into decimal form. For simple fractions with no variables, you can simply represent each fraction as a decimal to determine equivalence. Since every fraction is essentially a division, this is the simplest way to determine equivalence.

For example, take the fraction 4/8 above. The fraction 4/8 is equivalent to 4 divided by 8, 4/8 = 0.5. You can divide the other fraction like so, 8/16 = 0.5. Regardless of the form of the fraction, they are equivalent if the two numbers are equal when expressed as a decimal.
Remember that representing it as a decimal can yield many digits before concluding that they are not equivalent. A basic example is 1/3 = 0.333… while 3/10 = 0.3. Just more than one digit, we see that these two fractions are not equivalent.

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Divide the numerator and denominator of a fraction by the same number to get the equivalent fraction. For more complex fractions, this method of division requires additional steps. Like multiplication, you can divide the numerator and denominator of a fraction by the same number to get an equivalent fraction. However, the resulting fraction must have both numerator and denominator being integers.

For example the fraction 4/8. Instead of multiplying, we divide both numerator and denominator by 2, we get (4 ÷ 2)/(8 ÷ 2) = 2/4 . 2 and 4 are both integers so this equivalent fraction is valid.

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Reduce fractions to their simplest form. Most fractions are usually expressed in the simplest form, and you can reduce them to the simplest form by dividing by the greatest common factor of the numerator and denominator. This step works by the same logic of representing equivalent fractions by converting them to the same denominator, but this method requires reducing each fraction to its simplest form.

When a fraction is in its simplest form, its numerator and denominator are both as small as possible. You cannot divide them by any integer to get a smaller number. To convert a fraction to its simplest form, we divide the numerator and denominator by the greatest common factor .
The greatest common factor of the numerator and denominator is the largest number that they are divisible by. So, in the 4/8 example, since 4 is the largest number that both 4 and 8 are divisible by, we’ll divide the numerator and denominator of this fraction by 4 to get the simplest form. (4 ÷ 4)/(8 ÷ 4) = 1/2 . In another example 8/16, GCF is 8, the result is also 1/2.

Using Cross Multiplication to Solve the Variable Finding Problem

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Set two equal fractions. We use cross multiplication for problems where we know the fractions are equivalent, but one of the numbers has been replaced by the variable (usually x) that we have to solve to find. In cases like these, cross-multiplication is a quick method. ^{[2] X Research Source}

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Take two equivalent fractions and cross multiply them in the shape of an “X”. In other words, you multiply the numerator of one fraction by the denominator of the other and vice versa, then set the two results equal and solve the problem. ^{[3] X Research Sources}

Take two examples, 4/8 and 8/16. These two fractions contain no variables, but we can show that they are equivalent. By cross multiplying, we get 4 x 16 = 8 x 8, or 64 = 64, which is obviously true. If these two numbers are not the same, then the fractions are not equivalent.

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Enter the variable. Since cross multiplication is the easiest way to determine equivalent fractions when you have to solve a problem of finding variables, add variables.

For example, consider the following equation 2/x = 10/13. To cross multiply, we multiply 2 by 13 and 10 by x, then set these two results equal:
- 2 × 13 = 26
- 10 × x = 10x
- 10x = 26. By simple algebraic method we can find the variable x = 26/10 = 2.6 , then the two initial equivalent fractions are 2/2.6 = 10/13.

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Use cross multiplication for multivariable equations or variable expressions. One of the coolest things about cross multiplication is that whether you have two simple fractions (as above) or more complex fractions, the solution is the same. For example, if both fractions contain variables, you can simply remove these variables at the last step in the problem-solving process. Likewise, if the numerators and denominators of the fraction contain variable expressions (such as x + 1), you can simply cross multiply and solve as usual. ^{[4] X Research Sources}

For example, consider the following equation ((x + 3)/2) = ((x + 1)/4). As above, we solve by cross multiplying two fractions:
- (x + 3) × 4 = 4x + 12
- (x + 1) × 2 = 2x + 2
- 2x + 2 = 4x + 12, subtract both sides for 2x
- 2 = 2x + 12, to separate the variable we subtract both sides by 12
- -10 = 2x, and divide both sides by 2 to find x
- -5 = x

Using Quadratic Root Formula to Solve Variable Finding Equations

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Cross multiply two fractions. For equivalence problems that require the use of the quadratic root formula, we still start by using cross multiplication. However, any cross multiplication that involves multiplying the term containing the variable by the term containing another variable has the potential to yield an expression that cannot be easily solved algebraically. In cases like these, you will need to use techniques such as factoring and/or quadratic formula. ^{[5] X Research Sources}

For example, consider the following equation ((x +1)/3) = (4/(2x – 2)). Step 1, we cross multiply:
- (x + 1) × (2x – 2) = 2x ² + 2x -2x – 2 = 2x ² – 2
- 4 × 3 = 12
- 2x ² – 2 = 12.

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Express the equation as a quadratic equation. Now we have to represent the equation in quadratic form (ax ² + bx + c = 0), where we set the equation to 0. In this case, we subtract both sides by 12 to get 2x ² – 14 = 0.

Some values can be zero. Although 2x ² – 14 = 0 is the simplest form of equation, its correct quadratic equation is 2x ² + 0x + (-14) = 0. It will help. reflects the correct form of a quadratic equation even if some values are 0.

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Solve the equation by substituting the known coefficients into the solution formula. The quadratic root formula (x = (-b +/- √(b ² – 4ac))/2a) will help us solve the problem of finding x at this point. ^{[6] X Research Source} Don’t be afraid because the recipe seems long. You simply take the values from the quadratic equation in step two and put them in their respective places before solving.

x = (-b +/- √(b ² – 4ac))/2a. In the equation, 2x ² – 14 = 0, a = 2, b = 0, and c = -14.
x = (-0 +/- √(0 ² – 4(2)(-14)))/2(2)
x = (+/- √( 0 – -112))/2(2)
x = (+/- (112))/2(2)
x = (+/- 10.58/4)
x = +/- 2.64

Advice

Converting fractions to equivalent fractions is actually the form of multiplying them by 1. When converting 1/2 to 2/4, we are actually multiplying the numerator and denominator by 2, or the multiplication itself. 1/2 with 2/2, exactly equals 1.
If desired, convert the mixed number to an irregular fraction to make the conversion easier. Obviously, not every fraction you come across is as easy to convert as our 4/8 example above. For example, mixed numbers (eg 1 3/4, 2 5/8, 5 2/3, etc.) can make the conversion a bit more complicated. If you need to convert a mixed number to an equivalent fraction, you can do it in two ways: convert the mixed number to an irregular fraction, then convert as usual, or keep the mixed form. and consider mixed numbers as the answer.
- To convert an irregular fraction, multiply the integer part of the mixed number by the denominator of the fractional part then add it to the numerator. For example, 1 2/3 = ((1 × 3) + 2)/3 = 5/3. Then, if you want, you can convert to equivalent fractions as needed. For example, 5/3 × 2/2 = 10/6 , which is still equivalent to 1 2/3.
- However, we do not need to convert to the non-canonical fraction form as above. Skip the integer part, convert only the fraction part, then add the integer part back to the converted fraction part. For example, for 3 4/16, we will only look at 4/16. 4/16 ÷ 4/4 = 1/4. Adding the integer part back, we have a new mixed number of 3 1/4 .

Warning

Multiplication and division are used to produce equivalent fractions because multiplying and dividing by the fractional form of the number 1 (2/2, 3/3, etc.) by definition has no effect on the fractional value. initial. Adding and subtracting doesn’t do the same.
Although you multiply the numerator and denominator together when multiplying fractions, you cannot add or subtract the denominator when adding or subtracting fractions.
- As the example above, we see that 4/8 ÷ 4/4 = 1/2. If we added 4/4 instead, the answer would be completely different. 4/8 + 4/4 = 4/8 + 8/8 = 12/8 = 1 1/2 or 3/2 , no answer equals 4/8.

Steps