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Mathematics LibreTexts

5.4: Add and Subtract Rational Expressions

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A fraction is a proportion. The fraction communicates the number of parts out of the whole.

abThe part is the numerator.The whole is the denominator.

For example, the egg carton pictured below contains 10 eggs. Part of the eggs are brown (7 eggs) while the rest are white (3 eggs). A fraction ab quickly communicates the proportion of eggs that are brown or are white. Proportions can also be given as a decimal or percentage.

clipboard_e3c5408e1d01cd0704f558548ac275e13.png

Proportion of brown eggs: 710=7÷10=0.7=70%

Proportion of white eggs: 310=3÷10=0.3=30%

The proportion of brown and white eggs describes all of the eggs. Intuitively, we know this proportion is 100%. The math should match our intuition, so let’s investigate.

70%+30%=100%710+310=1010

Our investigation leads us to deduce that we add the numerators, but not the denominators.

To Add or Subtract Fractions (Denominator is the Same)
  1. Add or subtract the numerators.
  2. Keep the same denominator.
  3. Reduce if necessary.

ac+bc=a+bc

What works for real numbers also works for algebraic quantities. Algebra follows the same rules, which is an excellent anchor from which to perform your own algebraic investigations.

Example 5.4.1

Add 2xx+3+xx+3

Solution

The denominators match. Add the two numerators, but keep the denominator.

2xx+3+xx+3=2x+xx+3=3xx+3

Since the result cannot be further simplified, the answer is 3xx+3

Example 5.4.2

Subtract y2+2y+8y22y15y2+2y22y15

Solution

The denominators are the same. Subtract the numerators. Remember that each numerator is a quantity.

y2+2y+8y22y15y2+2y22y15=y2+2y+8(y2+2)y22y15Numerator: subtract the quantity (y2+2).=y2+2y+8y22y22y15Each term of the quantity is subtracted.=2y+6y22y15The subtraction is complete.=2(y+3)(y5)(y+3)Factor numerator and denominator. Cancel.=2y5The answer is most simplified.

Addition and Subtraction with Different Denominators

In order to add rational numbers or rational expressions, we need to find a common denominator. Create common denominators using equivalent fractions. That is, adbd=adbd. Then use the process for adding fractions with common denominators.

Example 5.4.3

Add 775a2b+13180ab

Solution

To find the least common denominator, we play a game of “who has more?” for each denominator. Break down denominators into prime factors. Imagine that each denominator is a set of cards, and each card is a factor of the denominator. Because we have two denominators, we have two players who will play the game, “who has more?”. Whichever player has the most of a factor “wins,” and they place those cards into the winning pile. Ready to play?

Player #1: 75a2 clipboard_e82900e65b6134c70d503b2fed4be7096.png

Player #2: 180ab clipboard_ec69093ffe99e21d390dcc4c691e250bf.png

Who has more? Player #1 Player #2 Winning Pile
2's   22
3's   33
5's   55
a's   aa
b's TIE TIE b

The Winning Pile =223252a2b

The Least Common Denominator =900a2b

Use the least common denominator to create a common denominator. Determine what factors are missing in each denominator. Multiply the missing factors to the denominator and also to the numerator. Then add.

775a2b+13180ab=71275a2b12+135a180ab5a=84900a2b+65a900a2b=84+65a900a2b.

The answer is: 84+65a900a2b

Example 5.4.4

Add 2y+1(4y+1)2(y+6)+5(4y+1)(y+6)

Solution

In the game of “who has more?” the quantities (4y+1) and (y+6) act as single numbers.

Player #1: (4y+1)2(y+6) clipboard_ee200f708cfb0a13a36f0a1987aee86af.png

Player #2: (4y+1)(y+6) clipboard_e96fc9cb8c18dcb8c3ca506179e29e468.png

Winning Pile = (4y+1)2(y+6) = Least Common Denominator

2y+1(4y+1)2(y+6)+5(4y+1)(y+6)=2y+1(4y+1)2(y+6)+5(4y+1)(4y+1)(y+6)(4y+1)=2y+1+(20y+5)(4y+1)2(y+6)Simplify the numerator, but leave the denominator in factored form.=22y+6(4y+1)2(y+6)The numerator is simplified. This is the answer!

Tip: The numerator can be factored using GCF =2: 22y+6=2(11y+3). However, this factorization does not lead to a cancellation. Therefore, the stated numerator is preferred.

Example 5.4.5

Subtract u2+2uu2+3u102u3u6

Solution

To compare the factors of each denominator, we need to factor.

u2+2uu2+3u102u3u6=u2+2u(u+5)(u2)2u3(u2)The denominators are factored.

Play the game of “who has more?” to find the Least Common Denominator. Quantities act as single numbers.

Player #1: clipboard_e5b29d8142e8dc9709ca5f8f83c112282.png

Player #2: clipboard_e9db21ccf70da655263da0e7093153b56.png

Winning Pile =3(u+5)(u2)= Least Common Denominator

u2+2u(u+5)(u2)2u3(u2)=3(u2+2u)3(u+5)(u2)2u(u+5)3(u2)(u+5)Determine the missing factors & multiply.=3u2+6u3(u+5)(u2)2u2+10u3(u+5)(u2)Simplify numerators.=3u2+6u(2u2+10u)3(u+5)(u2)Numerator: subtract the quantity.=3u2+6u2u210u3(u+5)(u2)Subtract each term of the quantity.=u24u3(u+5)(u2)The numerator is simplified. This is the answer!

Tip: The numerator can be factored using GCF =u: u24u=u(u4). However, this factorization does not lead to a cancellation. Therefore, the stated numerator is preferred.

Example 5.4.6

Subtract 3c49c21c27c

Solution

To compare the factors of each denominator, we need to factor.

3c(7c)(7+c)1c(c7)The denominators are factored.

Play the game of “who has more?” to find the Least Common Denominator. Quantities act as single numbers.

Player #1: clipboard_e536c760e466e1d3499db5678220afecb.png

Player #2: clipboard_e078fec96a07e13d83128820663fd76ea.png

Note: 7c and c7 are opposites. Since they are opposites, one quantity can be made to “look like” the other.

7c=(c7)

Next, bounce the negative to the numerator. Voilà! This is how we make a Least Common Denominator.

Winning Pile =c(c+7)(c7)= Least Common Denominator

3c(7c)(7+c)1c(c7)=3c(c7)(7+c)1c(c7)Replace 7c with (c7).=3c(c7)(7+c)1c(c7)Since ab=ab, put the negative on top.=3ccc(c7)(c+7)1(c+7)c(c7)(c+7)Construct the LCD. Note: 7+c=c+7.=3c2(c+7)c(c7)(c+7)Numerator: subtract the quantity.=3c2c7c(c7)(c+7)The numerator has been simplified.=3c2+c+7c(c7)(c+7)Factor out 1.

Either of the last two lines could be your final answer. Which do you prefer? It’s your choice!

Note

After simplifying the numerator, pause and ask yourself, “what more can I do to simplify?” If the numerator is factorable, check to see if the factors are the same as those in the denominator. If so, cancel the common factors. Keep the denominator factored to remind yourself (and your reader) what factors do not cancel.

Try It! (Exercises)

For #1-20, add or subtract the rational expressions as indicated.

  1. 5x3y2+x3y2
  2. 14a5b4a5b
  3. 4r9qp+13q2p
  4. 1n6n+2n+13n
  5. n+212m323m3n
  6. t+6t+5+t+4t+5
  7. 6c+1(5c+2)3c1(5c+2)3
  8. 2x3+310
  9. 4y+2+23y+6
  10. 3x+33x3
  11. h2(h+1)2+23h+3
  12. 5y13y21
  13. 2a25a+6+2a3
  14. 4d35d12d26d+9
  15. 5v25vv5v25
  16. 1nn24n2n22n8
  17. 3+xx+2
  18. 3p1p2+5p+6+2p3p2+3p+2
  19. 3q+3+2qq2+8q+15
  20. w+24w+162w2+4w

For #21-30, opposite quantities occur. Add or subtract the rational expressions as indicated.

  1. 38+58
  2. y2y5+4y+55y
  3. 6w2+w+32w
  4. 2a75a8+6+10a85a
  5. u222u2u3+u232u
  6. b23b2b2025b
  7. t+1t13t211t2
  8. 263x+xx24
  9. 5r12r2r2r+1+34r21
  10. 2yy291+43y

This page titled 5.4: Add and Subtract Rational Expressions is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Jennifer Freidenreich.

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