Two fractions are said to be “equivalent” when they have the same value. Knowing how to convert a fraction into another equivalent fraction is a mathematical skill that is useful both in college and university. The object of this article is to show you how to obtain equivalent fractions both with simple tools (like multiplication and division) and with more complex techniques.
Method 1 of 5: Obtain equivalent fractions
Step 1. Multiply the numerator and denominator by the same number
Two equivalent but different fractions have this in common that their numerators are multiple between them, as well as their denominators. In other words, if you multiply the numerator and denominator of a fraction by the same value, you get a fraction equivalent to the first. In the end, these two fractions are not alike, but their ratios are identical.
- As an example, take the fraction 4/8. Multiply its numerator and its denominator by 2. We obtain: (4 x 2) / (8 x 2) = 8/16. The two fractions, 4/8 and 8/16, are equivalent.
- (4 x 2) / (8 x 2) can be written as: 4/8 x 2/2. We remind you that the product of two fractions is done by multiplying the numerators between them and by doing the same with the denominators, the fraction mark remains unchanged.
- You know that the fraction 2/2 is equal to 1. It is therefore easier to understand why 4/8 and 8/16 are equivalent. We did: 4/8 × (2/2) = 8/16. If we rewrite this equality, we have: 4/8 × 1 = 4/8. Conclusion: 4/8 = 8/16.
- Any fraction has countless equivalent fractions. Each time you multiply the numerator and denominator of a fraction by the same integer, you get a new fraction equivalent to the starting fraction. This is to say if there are many!
Step 2. Divide the numerator and denominator by the same number
Instead of multiplying, we can also divide a fraction by the same value to obtain an equivalent fraction. It is enough to divide the numerator and the denominator of a fraction by the same whole, you obtain a new fraction equivalent to the starting fraction. In the case of division, there is one precaution to be taken - the divisions of the numerator and denominator must be integers, otherwise no equivalent fraction!
Let's take the fraction 4/8 again. If, instead of multiplying, you divided the numerator and denominator by 2, you would get: (4 ÷ 2) / (8 ÷ 2) = 2/4. 2 and 4 being integers, this last fraction is equivalent to 4/8
Method 2 of 5: Using simple multiplication to determine equivalence
Step 1. Find the factor (= number) by which to multiply the smallest denominator to find the largest
In many exercises, we are led to ask ourselves whether or not two fractions are equivalent. By finding this number that allows you to go from the small to the large denominator, you are on the way to reducing the two fractions to the same denominator and being able to easily compare them.
- Let's go back to fractions 4/8 and 8/16. The most denominator is 8 and if we multiply it by 2, we get the second denominator, 16. This famous number we were talking about is here:
- If the denominators are larger numbers, just divide the larger by the smaller. Here, 16 divided by 8, gives
2nd step.: it's the same answer!
- This number is not necessarily an integer. Thus, with 2 and 7 as denominators, we obtain a ratio of 1 to 3, 5 (7 = 2 x 3, 5).
Step 2. Multiply the numerator and denominator of the fraction reduced to its simplest expression by the number found previously
Two different but equivalent fractions have, by definition, numerators and multiple denominators of each other. In other words, the fact of multiplying the numerator and the denominator of a fraction by the same number gives an equivalent fraction. Just because the elements of equivalent fractions are different doesn't mean they don't have the same value.
- So our fraction from step 1, 4/8, can be transformed into an equivalent fraction by multiplying the two terms of the fraction by the value 2 found previously. This gives: (4 × 2) / (8 × 2) = 8/16. The two fractions are equivalent.
Method 3 of 5: Using simple division to determine equivalence
Step 1. Express each fraction as a decimal
If you are dealing with two fractions without unknowns, it is possible to express them in decimal form in order to see if they are equivalent. Since fractions are nothing more than divisions, it is possible to do these operations to simply establish whether or not two fractions are equivalent.
- Let's go back to our fraction 4/8. This fraction can be considered as the simple division of 4 by 8, i.e. 0.5). More generally, if two different fractions have the same decimal value after division, then they are equivalent.
- When you express fractions in decimal form, you must divide to the end, that is to say with all the decimals. Thus, 1/3 = 0, 333…, while 3/10 = 0, 3. Admittedly, the first decimal is the same, but not the following ones: the two fractions are therefore not equivalent!
Step 2. Divide the numerator and denominator of a fraction by the same number and you will get an equivalent fraction
This method applies to all fractions, although for more complex fractions some additional operations are required. We have seen that we can multiply by the same number, know that we can also divide a fraction by the same number in order to obtain an equivalent fraction. The only precaution to respect: the fraction obtained must have in numerator and denominator only integers.
- Let's take our fraction 4/8 again. If, instead of multiplying, we divided the numerator and the denominator by 2, we would obtain: (4 ÷ 2) / (8 ÷ 2) = 2/4. 2 and 4 being integers, 2/4 is an equivalent fraction of 4/8.
Step 3. Reduce each fraction to its simplest expression
Most fractions can be reduced to their simplest expression and for that, we must divide them by what is called the "Greatest Common Divisor" or GCD. This method is in the same spirit as the reduction to the same denominator, except that this reduction will be done with the lowest common denominator.
- In a fraction reduced to its simplest expression, the numerator and denominator can no longer be reduced simultaneously or if you prefer, then there is no integer capable of dividing the two components of the fraction at the same time. To transform any fraction into an equivalent, but irreducible fraction, divide the numerator and denominator by their "Greatest Common Divisor".
- By “Greatest Common Divisor” (GCD) of the numerator and the denominator, we mean “the largest integer which simultaneously divides these two integers”. Thus, for the fraction 4/8,
Step 4. is the largest integer which simultaneously divides 4 and 8: it is the GCD. If we divide each element of the fraction 4/8 by 4, we obtain a fraction reduced to its simplest expression (we also say "irreducible"). We therefore have: (4 ÷ 4) / (8 ÷ 4) = 1/2. For the fraction 8/16, the GCD is 8, which amounts to saying that (8/16) ÷ 8 = 1/2, irreducible fraction.
Method 4 of 5: Using the cross product to find an unknown
Step 1. Make your two fractions even
We are going to use the cross product in problems of fractions which we know in advance that they are equivalent, but which contain an unknown ("x"). The objective is then to find the value of "x". We know from the start that the fractions are equivalent since they have an “=” sign between them. On the other hand, solving this equation, because it is, is a little difficult at first glance, but it is not. Indeed, we have at our disposal a very practical procedure for this kind of equation: the cross product!
Step 2. The cross product is to multiply the elements of the fractions by a cross, an "X" if you prefer
Concretely, we multiply the numerator of one of the fractions with the denominator of the other and vice versa. We put these two products on an equal footing and we solve.
Let's go back to our two previous examples: 4/8 and 8/16. Of course, we have no unknowns, but we can still use this method, now that we know that these two fractions are equivalent. The cross product gives: 4 x 16 = 8 x 8, or 64 = 64. CQFD! If with other fractions, this equality had not been verified, then the fractions would not have been equivalent
Step 3. Enter an unknown
Since the cross product makes it possible to quickly verify that two fractions are equivalent, you can complicate the problem by introducing an unknown factor.
- As an example, consider the following equation: 2 / x = 10/13. With the cross product, you need to multiply 2 by 13 and 10 by x, then make these two quantities equal:
- 2 × 13 = 26
- 10 × x = 10x
- 10x = 26. Finding "x" is therefore child's play. x = 10/26 = 2, 6, which gives us the following equivalence: 2/2, 6 = 10/13.
Step 4. Use the cross product to solve equations containing several times the same unknown or different unknowns
The great advantage of the cross product is that it works just as well with simple fractions (as we have seen!) As with much more complicated fractions. Thus, for two fractions containing an unknown, the objective will consist, during the calculations, to group the unknown. In the same way, if you have, in numerator or in denominator, expressions in parentheses containing an unknown (of the type: x + 1), it is necessary to resort to the distributivity to develop, then to classically solve your equation.
- Take, for example, the following equation: ((x + 3) / 2) = ((x + 1) / 4). To solve it, we therefore use the cross product:
- (x + 3) × 4 = 4x + 12
- (x + 1) × 2 = 2x + 2
- 2x + 2 = 4x + 12, we can simplify by removing 2x on each side
- 2 = 2x + 12, then isolate "x" by removing 12 on each side
- -10 = 2x, finally you just have to divide by 2 each side to have "x"
- - 5 = x or x = - 5
Method 5 of 5: Using the discriminant to find an unknown
Step 1. Make the cross product of the two fractions
With the problems which require to use the resolution via the discriminant, it is necessary first and always to begin by making the product in cross. Most often, when we do the cross product of somewhat complex fractions containing the unknown several times, we obtain a somewhat complicated equation, often of the quadratic, which cannot be solved as easily as the previous case. It is then necessary to resort to more elaborate methods, such as factorization and / or the discriminant method.
- Consider, for example, the following equation: ((x +1) / 3) = (4 / (2x - 2)). We start with the cross product:
- (x + 1) × (2x - 2) = 2x2 + 2x -2x - 2 = 2x2 - 2
- 4 × 3 = 12
- 2x2 - 2 = 12.
Step 2. Arrange the equation as a quadratic equation
Indeed, at this stage, we must rewrite the equation in the classical form: ax2 + bx + c = 0. It suffices that the second member of the equation is equal to 0. Here, you will subtract 12 from the two members of the equation: 2x2 - 2 (-12) = 12 (-12), or 2x2 - 14 = 0.
- Certain coefficients are equal to 0. Certainly, 2x2 - 14 = 0 is an equation in its simplest form, but we could just as easily write it like this: 2x2 + 0x + (-14) = 0. At the beginning, you can write the equation in all its extension, with 0 as a coefficient; later, you won't need it anymore.
Step 3. Find the solutions (or roots) of the quadratic equation
Starting from the formula for the roots, let x = (-b +/- √ (b2 - 4ac)) / 2a), just make the digital application with your values. Don't be afraid of the length of this formula! Locate the values "a, b, c" of your equation to solve and place them, without making a mistake, in the formula. Example:
- x = (-b +/- √ (b2 - 4ac)) / 2a. Our equation is: 2x2 - 14 = 0, so 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 by doing the numeric application by taking this value of "x" and putting it back into the starting equation
If in this equation you give "x the value (s) found and the equation holds, then your calculation is correct." In our example, we have two solutions: x = 2, 64 and x = - 2, 64. We replace “x” and we see that the equation is indeed verified.
- On closer inspection, converting to equivalent fractions is nothing more than multiplying by 1. To transform 1/2 into 2/4, multiply the numerator and denominator by 2, which is the same as multiplying 1 / 2 by 2/2, or by 1.
- If you want, in order to simplify the calculations, you can first transform your mixed numbers into improper fractions. Not all the fractions that we can give you will always be as easy to convert as the fraction, 4/8, studied previously. Thus, some mixed numbers (1 3/4, 2 5/8, 5 2/3, etc.) are more difficult to convert to equivalent fractions. To obtain an equivalent fraction from a mixed number, there are two possibilities: that is you turn the mixed number into an improper fraction, then you find the equivalent fraction, as shown, that is you keep the form of the mixed number and you give another mixed number as an answer.
- To convert a mixed number into an improper fraction, multiply the whole part of the number by the denominator of the fractional part, then add the numerator, all reduced to the denominator. So, with 1 2/3, we do: 1 2/3 = ((1 × 3) + 2) / 3 = 5/3. Then you can do calculations to find equivalent fractions. For example, you can do: 5/3 × 2/2 = 10/6, which fraction is perfectly equivalent to 1 2/3.
- In some cases, it is not necessary to convert the mixed number to an improper fraction. At this point, we ignore the whole part and only transform the fractional part. Take the example of 3 4/16, we only take into account 4/16 that we divide by 4/4: 4/16 ÷ 4/4 = 1/4. We keep the whole part and we put the new reduced fraction: we then obtain a new mixed number, 3 1/4.
- Obtaining equivalent fractions can only be achieved by multiplying or dividing the starting fractions. We always multiply or divide these by 1 (or rather by a fractional form of 1, like 2/2, 3/3, etc.) Never, we cannot go through addition or subtraction!
- In a product of fractions, we multiply the numerators between them, then the denominators in the same way and we have the result. In the case of an addition or a subtraction of fractions, it is out of the question to do the same: it is first necessary to reduce the fractions to the same denominator
- We saw above that: 4/8 ÷ 4/4 = 1/2. If, instead of dividing, we had added 4/4, we would have obtained a radically different answer, i.e. 4/8 + 4/4 = 4/8 + 8/8 = 12/8 = 1 1/2 or 3/2. Neither of these values is equivalent to 4/8!