Equivalent fractions are fractions that have different numerators and denominators but result in the same value after simplification. For example, 1/2 and 2/4 have different numerators and denominators but 2/4 after simplification gives 1/2 and hence the same value.
We know that we cannot compare apples to mangoes. We can compare apple to apple or mangoes to mangoes. Similarly, we cannot add or subtract fractions if they have different denominators. To compare them or add and subtract them we need to make their base the same.
To help better understand the topic say you have American and Indian coins. To figure out the total amount you need to convert Indian Rupees to American Dollars or American Dollar to Indian Rupees and then you can count them. If you want to purchase anything in these two countries you need to first convert them into their currencies and then only you will be able to spend it.
Lets add 1/2 and 2/3
Step 1: Base are 2 and 3. To make them equivalent we need to find common nearest multiple between them i.e. L.C.M also called as least common multiple.
Writing multiples of 2 and 3:
for 2 -> 2,4,6,8,10,12…..
for 3 -> 3,6,9,12……
Here 6 and 12 both are common and if we continue to write multiples of 2 and 3 we will find more multiples. Currently we need only the lowest common multiple between them which is 6.
step 2: Make denominator equal
2 x 3 and 3 x 2 gives 6
Important note: Whatever operation you perform on numerator, same has to be performed on denominator.
so 1/2 + 2/3 = (1×3)/(2×3) + (2 x2)/(3×2)
= 3/6 + 4/6
step 3: Add the numerator and keep denominator common
= (3 + 4) /6
Simplest Form with Variables
To put fractions that contain variables in simplest form, simply factor the numerator (using prime factors of the coefficient and factors of the variables) and similarly factor the denominator. Lets reduce
Prime factors of 12 = 2 x 2 x 3
Prime factors of 15 are 3 x 5
Then divide out common factors
12/15 = (2 x 2 x 3) / (3 x 5)
= (2 x 2)/5
= 4/ 5
Now find factors of variables and reduce it.
xyz3/ y2 = x.y.z.z.z/ y.y
thus the fraction is reduced to 4xz3/5y
Adding Fractions with common denominators:
Suppose we have fractions that we want to add and they all have a common denominator. Then we simply add the numerators and place the result over the common denominator. As an example, consider the following:
2/a + 1/a + 5/a + 11/a = 17/a
Subtraction works much the same way. Remember, in regular math, when you subtract fractions with common denominators, you just subtract the numerators and leave the common denominator as is (unless you’re simplifying). Consider the following subtraction problem
19/a – 2/a – 5/a – 11/a = 1/a
Different Numeric Denominators:
What if the denominators are different? You simply find a common denominator, then make equivalent fractions that all have the same denominator as you learned. Here is an example, let’s add these fractions that have different denominators.
3/24 + 5/12 + 1/6
we find L.C.M of 24,12,6 which comes out to be 24
5/12 and 1/ 6 as equivalent fractions with a denominator of 24:
5/12 • 2/2 = 10/24 and 1/6 • 4/4 = 4/24
Now we have all the terms with the same common denominator (or least common denominator), so we simply sum all the numerators:
3/24 + 10/24 + 4/24 = 17/24
Different Algebraic Denominators:
Now let’s add fractions that have different variables in the denominator, such as
11/𝑎 + 7/𝑏
First we need to find a common denominator. One way you can do so is by simply multiplying the two denominators together (in general, this will not always produce the least common multiple—but that is okay; it will always provide a common denominator). The least common
denominator of a and b in this case also turns out to be the product of the two denominators, or ab.
Next, looking at the 1st fraction, 11/𝑎, we must ask ourselves, by what factor must we multiply the denominator (a) to obtain ab? Well, this is not a difficult question, since given the denominator a, we must simply multiply it by the factor b to yield the result ab. Thus, we have determined
that we must multiply the 1st fraction by 𝑏 to yield an equivalent fraction that has ab as the denominator:
11/𝑎 • 𝑏/𝑏 = 11𝑏/ab
Similarly, for the 2nd fraction, 7/𝑏, in our addition problem above, we must ask ourselves, by what factor must we multiply the denominator (b) to obtain ab? Again, the answer is obvious; we must multiply by 𝑎/𝑎 to yield an equivalent fraction that has ab as the denominator:
7/b • 𝑎/𝑎 =7𝑎/𝑎b
So, now that we have created equivalent factors for all terms, we can rewrite the original
11𝑏/𝑎b + 7𝑎/𝑎b = (7𝑎+11𝑏)/ab