Content

• Steps
• Advice
• Warning

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

Steps

Method 1 of 5: Create Equivalent Fractions

1. 

   Multiply the numerator and denominator by the same number. By definition, two different but equivalent fractions have the numerator and the 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 on the new fractions will be different, they will have the same values.
   • For example, if we take the fraction 4/8 and multiply both the numerator and the 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 we multiply two fractions, we multiply horizontally, i.e. the numerator by the numerator and the denominator by the denominator.
   • Note that 2/2 equals 1 when you do the division. Hence, it's easy to see why 4/8 and 8/16 are equal because 4/8 × (2/2) is 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, big or small, to yield an equivalent fraction.

2. 

   
   
   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 the denominator of a fraction by the same number to get an equivalent fraction. However, the fraction obtained must have both the numerator and the sample be integers.
   • For example, look back at the fraction 4/8. Instead of multiplying, we divide both the numerator and the denominator by 2, we have (4 ÷ 2) / (8 ÷ 2) = 2/4. 2 and 4 are both integers, so this equivalent fraction is valid.
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Method 2 of 5: Using Basic Multiplication to Determine Equivalence

1. 

   
   
   Find the number in which the larger denominator is multiplied by the smaller denominator. Many fraction problems involve determining whether two fractions are equal or not. By calculating this number, you can return the fractions to the same term to determine equivalence.
   • For example, retrieve the fractions 4/8 and 8/16. The smaller denominator is 8, and we will have to multiply that number by 2 to get the larger denominator of 16. So, the number to look for in this case is 2.
   • For more complex numbers, you just need to 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 denominators are 2 and 7, then 7 divided by 2 equals 3.5.

2. 

   
   
   The numerator and denominator of the fraction are expressed in the lower term with the number identified in the above step. By definition, two different but equivalent fractions exist The 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 this new fraction will be different, their values ​​are the same.
   • For example, if we take the fraction 4/8 from step one and multiply both the numerator and the sample by the number 2 specified earlier, we have (4 × 2) / (8 × 2) = 8/16. That proves that these two fractions are equivalent.
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Method 3 of 5: Using Basic Division to Determine Equivalence

1. 

   Divide each fraction into a decimal. For simple fractions with no variables, you only need to represent each fraction as a decimal to determine equivalence. Since each 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 equal to 4 divided by 8, 4/8 = 0.5. You can divide that fraction like that, 8/16 = 0.5. Regardless of the format of the fractions, they are equivalent if the two numbers are equal when expressed in decimal.
   • Remember that the decimal representation can produce 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 find that these two fractions are not equivalent.

2. 

   Divide the numerator and denominator of a fraction by the same number to get an equivalent fraction. For more complex fractions, this division method requires additional steps. Like multiplying, you can divide the numerator and the denominator of a fraction by the same number to get an equivalent fraction. However, the fraction obtained must have both the numerator and the sample be integers.
   • Fraction example 4/8. Instead of multiplying, we are share Both the numerator and the denominator give 2, we get (4 ÷ 2) / (8 ÷ 2) = 2/4. 2 and 4 are both whole numbers, so this equivalent fraction is valid.

3. 

   
   
   Reduce the fraction to its minimal form. Most fractions are usually expressed in a minimal form, and you can return them to their minimal form by dividing by the largest common factor of the numerator and the sample. This step works in the same logic of representing equivalent fractions by converting them to the same denominator, but this method requires reducing each fraction to its minimal form.
   • When a fraction is in its minimal form, the numerator and its denominator are as small as possible. You cannot divide them by any integer to get a smaller number. To convert a fraction to its minimal form, we divide the numerator and denominator by greatest common factor.
   • The largest common factor of the numerator and the denominator is the maximum number they are divisible by. So, in the example 4/8, because 4 is the largest number that both 4 and 8 are divisible by, we will divide the numerator and the denominator of this fraction by 4 to get the simplified form. (4 ÷ 4) / (8 ÷ 4) = 1/2. In another example 8/16, GCF is 8, the result is also 1/2.
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Method 4 of 5: Using Cross Multiplication to Solve Variables Problem

1. 

   
   
   Put two fractions equal. We use cross multiplication for problems where we know fractions are equivalent, but one of the numbers has been replaced by the variable (usually x) that we have to solve the problem to find. In cases like these, cross-multiplication is a quick method.

2. 

   
   
   Take two equivalent fractions and cross them using an "X". In other words, you multiply the numerator of one fraction by the denominator of the other and vice versa, and then put these two results equal and solve the problem.
   • Take two examples, 4/8 and 8/16. These two fractions contain no variables, but we can prove they are equivalent. By cross multiplying, we get 4 x 16 = 8 x 8, or 64 = 64, which is obviously correct. If the two numbers are not the same, the fractions are not equivalent.

3. 

   Put in the variables. Since cross-multiplication is the easiest way to determine equivalent fractions when you have to solve the 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 put these two results equal:
     • 2 × 13 = 26
     • 10 × x = 10x
     • 10x = 26. By simple algebraic methods we can find variable x = 26/10 = 2.6, then the first two equivalent fractions are 2 / 2.6 = 10/13.

4. 

   Use cross multiplication for equations with multiple variables or variable expressions. One of the coolest things about cross-multiplication is that whether you have two simple fractions (like above) or more complex fractions, the solution is exactly the same. For example, if both fractions contain variables, simply remove them at the last step of the problem solving process. Likewise, if the numerators and denominators of fractions contain variable expressions (such as x + 1), simply cross-multiply and solve as you normally would.
   • 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 the sides for 2x
     • 2 = 2x + 12, to separate the variable we subtract the sides to 12
     • -10 = 2x, and divide the sides by 2 to find x
     • -5 = x
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Method 5 of 5: Using Quadratic Solution to Solve Variable Equations

1. 

   Cross multiply two fractions. For equivalence problems that require the use of quadratic solutions, we still start by using cross multiplication. However, any cross-multiplication involves multiplying the term containing a variable by the term containing another variable has the potential to yield an expression that cannot be easily solved by the algebraic method. In cases like these, you will need to use techniques like factorization and / or quadratic formulas.
   • For example, consider the following equation ((x +1) / 3) = (4 / (2x - 2)). Step 1, we cross multiply:
     • (x + 1) × (2x - 2) = 2x + 2x -2x - 2 = 2x - 2
     • 4 × 3 = 12
     • 2x - 2 = 12.

2. 

   Represent the equation as a quadratic equation. We must now represent the equation in quadratic form (ax + bx + c = 0), where we set the equation to zero. In this case, we subtract both sides by 12 to get 2x. - 14 = 0.
   • Some values ​​may be zero. Although 2x - 14 = 0 is the simplest form of equation, its quadratic is actually 2x + 0x + (-14) = 0. It helps to reflect Correct the form of a quadratic equation even if some values ​​are 0.

3. 

   Solve an equation by plugging in the known coefficients into the solution formula. The quadratic formula (x = (-b +/- √ (b - 4ac)) / 2a) will help us solve the problem of finding x at this point. Don't be afraid because the formula seems long. Simply take the values ​​from the quadratic equation in step two and replace them in their respective positions before solving.
   • x = (-b +/- √ (b - 4ac)) / 2a. In the equation, 2x - 14 = 0, a = 2, b = 0, and c = -14.
   • x = (-0 +/- √ (0 - 4 (2) (- 14))) / 2 (2)
   • x = (+/- √ (0 - -112)) / 2 (2)
   • x = (+/- √ (112)) / 2 (2)
   • x = (+/- 10.58 / 4)
   • x = +/- 2.64

4. 

   Check your answers by plugging the x back into your quadratic equation. By replacing the found x back into your quadratic equation from step two, you can easily determine if your answer is true or false. In this example, you would replace both 2.64 and -2.64 in the original quadratic equation. advertisement

Advice

• Converting fractions to fractions of equal value is actually the form of multiplying them by 1. When converting 1/2 to 2/4, we actually multiply the numerator and denominator by 2 or multiply. 1/2 with 2/2, which 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 (for example 1 3/4, 2 5/8, 5 2/3, etc.) can make the transition 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 number and consider the mixed number the answer.
  • To convert an irregular fraction, multiply the integer part of the mixed number by the denominator of the fraction and then add it to the numerator. For example, 1 2/3 = ((1 × 3) + 2) / 3 = 5/3. Then, if desired, you can convert to equivalent fractions as needed. For example, 5/3 × 2/2 = 10/6, which is still equal to 1 2/3.
  • However, we do not need to convert to the irregular fraction as above. Ignore the integer part, convert only the fraction part, then add the whole number part back to the converted fraction part. For example, for 3 4/16, we will only look at 4/16. 4/16 & divide; 4/4 = 1/4. Adding the integer part back, we have the new mixed number 3 1/4.

Warning

• Multiplication and division are used to create 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 fractional values. original. Addition and subtraction don't do that.
• Although you multiply the denominator and the denominator 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 instead I plus for 4/4, the answer will be completely different. 4/8 + 4/4 = 4/8 + 8/8 = 12/8 = 1 1/2 good 3/2, no answer is equal to 4/8.