Content

• To step
• Method 1 of 2: Create equivalent fractions
• Method 2 of 2: Solving equivalent fractions with variables
• Tips
• Warnings

Two fractions are "equivalent" if they have the same value. For example, the fractions 1/2 and 2/4 are equivalent because 1 divided by 2 has the same value as 2 divided by 4 (0.5 in decimal form). Knowing how to convert a fraction to another, but equivalent fraction, is an essential mathematical dignity you need, from basic algebra to rocket science. See Step 1 to get started!

To step

Method 1 of 2: Create equivalent fractions

1. Multiply the numerator and denominator of a fraction by the same number to get an equivalent fraction. Two fractions that are different, but have equivalent 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. Even though the numbers in this new fraction are different, it still has 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 the same as 4/8 × 2/2. Remember, multiplying two fractions is like this - numerator times numerator and denominator times denominator. Note that 2/2 equals 1. So it's easy to see why 4/8 equals 8/16 - the second fraction is the first fraction multiplied by 2!
2. Divide the numerator and denominator or a fraction by the same number to get an equivalent fraction. Like multiplication, division can also be used to find a new fraction that is equivalent to the given fraction. Simply divide the numerator and denominator of a fraction by the same number to get an equivalent fraction. There is a catch here - the resulting fraction must consist of integers in both the numerator and denominator to be valid.
   • For example, let's take 4/8 again. If, instead of a multiplication, we divide both the numerator and denominator by 2, we get (4 ÷ 2) / (8 ÷ 2) = 2/4. 2 and 4 are both whole numbers, so this equivalent fraction is valid.
3. Simplify your fraction using the greatest common divisor (GCD). Any given fraction has an infinite number of equivalent fractions - you can multiply numerator and denominator by any integer, large or small to get an equivalent fraction. But the simplest form of a given fraction is usually the one with the smallest terms. In that case, the numerator and denominator are both as small as possible - they can no longer be divided by any integer to make the term even smaller. To simplify a fraction, we divide both the numerator and the denominator by the greatest common denominator.
   • The greatest common divisor (GCD) of the numerator and denominator is the greatest integer, so that both numerator and denominator are divisible. So in our 4/8 example, because 4 is the largest divisor of both 4 and 8, we divide the numerator and denominator of our fraction by 4 to get the simplest terms. (4 ÷ 4) / (8 ÷ 4) = 1/2.
4. If desired, convert mixed numbers to improper fractions to make conversion easier. Of course, not every fraction you come across will make sense as easily as 4/8. For example, mixed numbers (eg 1 3/4, 2 5/8, 5 2/3, etc.) can make this conversion a bit more difficult.If you want to make a fraction of a mixed number, you can do this in two ways: make the mixed number an improper fraction, and then continue, or keep the mixed number and give a mixed number as answer.
   • To convert an improper fraction, multiply the integer of the mixed number by the denominator of the fraction and then add the product to the numerator. For example, 1 2/3 = ((1 × 3) + 2) / 3 = 5/3. Then you can convert this again if necessary. For example, 5/3 × 2/2 = 10/6, still the same as 1 2/3.
   • However, converting an improper fraction is not necessary. We can ignore the whole number and just convert the fraction and then add the whole number to it. For example, at 3 4/16, we only look at 4/16. 4/16 ÷ 4/4 = 1/4. So now we add the whole number again and get a new mixed number, 3 1/4.
5. Never add or subtract to get equivalent fractions. When converting fractions to their equivalent form, it is important to remember that the only operations you are applying are multiplication and division. Never use addition or subtraction. Multiplication and division work for getting equivalent fractions because these operations are actually forms of the number 1 (2/2, 3/3, etc.) and give answers equal to the fraction you started with. Addition and subtraction do not have this option.
   • For example, above we found that 4/8 ÷ 4/4 = 1/2. If we added 4/4 to this instead, we would have gotten a completely different answer. 4/8 + 4/4 = 4/8 + 8/8 = 12/8 = 1 1/2 or 3/2, and none of these are equal to 4/8.

Method 2 of 2: Solving equivalent fractions with variables

1. Use cross multiplication to solve equivalence problems with fractions. A tricky type of algebra problem dealing with equivalent fractions involves equations with two fractions, where one or both contain a variable. In cases like this, we know that these fractions are equivalent because they are the only terms on either side of the equation sign of an equation, but it's not always obvious how to solve for the variable. Fortunately, with cross multiplication, we can solve this type of problem without any problems.
   • Cross multiplication is just what it sounds like - you are multiplying crosswise over the equal sign. In other words, you multiply the numerator of one fraction by the denominator of the other fraction and vice versa. Then you solve the equation further.
   • For example, we have the equation 2 / x = 10/13. Now cross multiply: multiply 2 by 13 and 10 by x, and work out the equation further:
     • 2 × 13 = 26
     • 10 × x = 10x
     • 10x = 26. Now we work out the equation further. x = 26/10 = 2.6
2. Use cross multiplication in the same way as multi-variable comparisons or variable expressions. One of the best features of cross multiplication is that it works much the same whether you're dealing with two simple or complex fractions. For example, if both fractions contain variables, nothing changes - you just have to cancel these variables. Likewise, if the numerators or denominators of your fractions contain variable expressions, just "continue multiplying" using the distributive property and solving as you usually do.
   • For example, suppose we have the equation ((x + 3) / 2) = ((x + 1) / 4). In this case, we solve it with cross multiplication:
     • (x + 3) × 4 = 4x + 12
     • (x + 1) × 2 = 2x + 2
     • 2x + 2 = 4x + 12
     • 2 = 2x + 12
     • -10 = 2x
     • -5 = x
3. Make use of polynomial solving techniques. Cross multiplication does not matter always a result that you can solve with simple algebra. If you are dealing with variable terms, you will quickly get a second-degree equation or other polynomial as a result. In such cases, you use, for example, squaring and / or the squared formula.
   • For example, we take the equation ((x +1) / 3) = (4 / (2x - 2)). Cross multiply first:
     • (x + 1) × (2x - 2) = 2x + 2x -2x - 2 = 2x - 2
     • 4 × 3 = 12
     • 2x - 2 = 12. At this point, we want to convert this to a second-order equation (ax + bx + c = 0) by subtracting 12 from both sides, giving us 2x - 14 = 0. Now we use the formula (x = (-b +/- √ (b - 4ac)) / 2a) to find the value of x:
       • x = (-b +/- √ (b - 4ac)) / 2a. In our 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 At this point, we check our answer by substituting 2.64 and -2.64 in the original second-degree equation.

Tips

• Converting fractions to an equivalent form is basically the same as multiplying by a fraction such as 2/2 or 5/5. Since this ultimately equals 1, the value of the fraction remains the same.

Warnings

• Addition and subtraction of fractions is different from multiplication and division of fractions.