Video transcript
- What I want to do in this video is really make sure that we feel comfortable manipulating algebraic expressions that involve fractions. So we'll start with some fairly straightforward ones. So let's say that I had, let's say I had A over B plus C over D, and if I actually wanted to add these things, so it is just one fraction, how would I do that? Well, what we could do is, we could find a common denominator. Well, over here, we don't know what B is, we don't know what D is, but we know a common denominator is just going to be B times D. That is going to be a common multiple of B and D. So we could rewrite this as two fractions, with the common denominator BD, so, plus, BD, actually, let me color code it a little bit. So A over B is going to be the same thing as what over BD? Well, to get BD, I multiplied the denominator by D, so let me multiply the numerator by D as well, then I haven't changed the value of the fraction, I'm just multiplying by D over D. So this is going to be A times D over B times D. Notice I could divide the numerator and the denominator by D, and I'm going to get back to A over B. And then we can look at the second fraction, C over D, to go from D to BD, we multiplied by B and so, if I multiply the denominator by B, if I don't want to change the value of the fraction, I have to multiply the numerator by B as well. So let's multiply the numerator by B as well, and it's going to be BC, BC. BC over BD. This is C over D. So what I have here in magenta, this fraction is equivalent to this fraction. I just multiplied it by D over D, which we can assume is one, if we assume that D is not equal to zero, and then, if we just multiply C over D times one, which is the same thing as B over B, if we assume that B is not equal to zero, then this fraction and this fraction are equivalent. Now, why did I go through all of this trouble? Well, now, I have a common denominator, so I can add these two fractions. So what's this going to be? Well, common denominator is BD, so let me just, so the common denominator is BD, and I could just add the numerators, just like you would've done if these were numbers, if this wasn't an algebraic expression. So this is going to be, this is going to be AD plus BC, all of that over BD.
Numbers represented in the form of m/n are called Fractions. Here, ‘m’ or the upper part of the fraction is the numerator, and ‘n’ or the lower part of the fraction is called the denominator. Fractions with numerator lesser than Denominator, are Proper Fraction. Fractions with numerator greater than Denominator, fall in the category of Improper Fraction. Improper fractions are often denoted by Mixed Fractions, where there is a whole part and fractional part.
Rational Numbers
Fractions in the form of m/n where n!=0 fall in the category of Rational Numbers. So, any fractions 4/5, 2/4, 1/8 fall in the category of rational numbers. Rational numbers can be positive or negative. The only pre-requisite for any fractional number to be called rational number is that the denominator of the fraction must not be zero.
For same Denominators, Addition and Subtraction of Rational Expressions is relatively easier as the mathematical operation are performed straight on the numerators. When denominators are different or the rational numbers have different variables then simply addition of terms doesn’t work. For different denominators, to perform addition and subtraction we need to find LCM of the given terms first and then perform Mathematical operations. For different variables, we can’t perform addition and subtraction on them, only like variable can combine to perform Mathematical operations.
Add and Subtract Rational Expressions with Unlike Denominators and Variables
Below steps must be followed when we add or subtract rational expressions containing variables with different denominators:
Step 1: Since the denominators are different, mathematical operations such as addition/ subtraction can’t be performed directly. For this, we need to first find the LCM of different fractional terms and equalise the denominators.
Step 2: Calculations can be performed easily on the numerator as the denominators are equal. Perform addition and subtraction operations on the numerator part of rational numbers as desired.
Step 3: Simplify the result and reduce the expression to the lowest possible form.
Sample Problems
Question 1: Add 11b/6 and 19b/6.
Solution:
Since the denominator is same, we will directly add the numerators.
= 11b/6 + 19b/6
= 30b/6
= 5b
Question 2: Subtract 11b/6 from 19b/6.
Solution:
Since the denominator is same, we will directly subtract the numerators.
= 19b/6 – 11b/6
= 8b/6
= 4b/3
Question 3: Add 10s/4 and 10s/3.
Solution:
Since, the denominator is not the same, we will take the LCM of denominators.
= 10s/4 + 10s/3
The LCM of 4 and 3 is 12.
So, (10s × 3)/(4 × 3) + (10s × 4)/(3 × 4)
= 30s/12 + 40s/12
= 70s/12
Question 4: Subtract 10s/4 from 10s/3
Solution:
Since, the denominator is not the same, we will take the LCM of denominators.
= 10s/3 – 10s/4
The LCM of 4 and 3 is 12.
So, (10s × 4)/(3 × 4) – (10s × 3)/(4 × 3)
= 40s/12 – 30s/12
= 10s/12
= 5s/6
Question 5: Add 3z/4 + 10y/3 + 4z/3.
Solution:
Since, the denominator is not the same, take the LCM of denominators.
= 3z/4 +10y/3 +4z/3
The LCM of 4 and 3 is 12.
So, (3z × 3)/(4 × 3) + (10y × 4)/(3 × 4) + (4z × 4)/(3 × 4)
= 9z/12 + 40y/12 + 16z/12
Combine terms with Like Variables i.e add terms with Like Variables
= 25z/12 + 40y/12
Question 6: Subtract 7z/4 from 10z/3.
Solution:
Since the denominator is not the same, take the LCM of denominators.
= 10z/3 – 7z/4
The LCM of 4 and 3 is 12.
So, (10z × 4)/(3 × 4) – (7z × 3)/(4 × 3)
= 40z/12 – 21z/12
= 19z/12
Question 7: Subtract 10i/4 from 10i/3
Solution:
Since, the denominator is not the same, take the LCM of denominators.
= 10i/3 – 10i/4
The LCM of 4 and 3 is 12.
So, (10i × 4)/(3 × 4) – (10i × 3)/(4 × 3)
= 40i/12 – 30i/12
= 10i/12
= 5i/6