Solve the Algebraic Fraction: (12a+3b)/(a·b) Simplification

Fraction Simplification with Variable Cancellation

(12a+3b)/(ab)=? (12a+3b)/(a\cdot b)=\text{?}

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Step-by-step video solution

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00:00 Solve
00:03 Write division as a fraction
00:09 Break down the fraction into two fractions
00:17 Reduce what's possible
00:31 And this is the solution to the question

Step-by-step written solution

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1

Understand the problem

(12a+3b)/(ab)=? (12a+3b)/(a\cdot b)=\text{?}

2

Step-by-step solution

Let's first rewrite the exercise as a fraction:

12a+3ba×b= \frac{12a+3b}{a\times b}=

Since the numerator of the fraction contains an addition operation and the denominator of the fraction contains a multiplication operation, we'll break down the exercise into an addition of fractions:

12aa×b+3ba×b= \frac{12a}{a\times b}+\frac{3b}{a\times b}=

Finally, we can cancel out the a a s in the numerator and denominator of the first fraction and the b b s in the numerator and denominator of the second fraction, giving us:

12b+3a \frac{12}{b}+\frac{3}{a}

3

Final Answer

12b+3a \frac{12}{b}+\frac{3}{a}

Key Points to Remember

Essential concepts to master this topic
  • Rule: Split fractions with addition in numerator into separate fractions
  • Technique: Cancel matching variables: 12aa×b=12b \frac{12a}{a \times b} = \frac{12}{b}
  • Check: Verify by expanding back: 12b+3a=12a+3bab \frac{12}{b} + \frac{3}{a} = \frac{12a + 3b}{ab}

Common Mistakes

Avoid these frequent errors
  • Canceling variables incorrectly from addition terms
    Don't cancel the 'a' from 12a+3b to get 12+3b/b = wrong answer! You can't cancel variables across addition operations in the numerator. Always split the fraction first, then cancel variables within each separate fraction.

Practice Quiz

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\( 70:(14\times5)= \)

FAQ

Everything you need to know about this question

Why can't I just cancel the 'a' from both 12a and the denominator directly?

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You can only cancel variables that are factors of the entire numerator or denominator. Since we have 12a + 3b (addition), the 'a' is not a factor of the whole numerator, so it can't be canceled directly.

How do I know when to split a fraction?

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Split fractions when the numerator has addition or subtraction and you want to simplify. Each term in the numerator becomes its own fraction with the same denominator.

What if both terms had the same variable to cancel?

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Great! If both terms could cancel the same variable, you'd factor it out first. For example: 12a+9aa×b=21aab=21b \frac{12a + 9a}{a \times b} = \frac{21a}{ab} = \frac{21}{b}

Can I simplify this fraction any further?

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No, 12b+3a \frac{12}{b} + \frac{3}{a} is fully simplified! The terms have different denominators and no common factors to cancel.

What's the difference between this and regular fraction addition?

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This is actually fraction subtraction in reverse! We started with one fraction and split it into two simpler fractions. It's the opposite of finding a common denominator.

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