Verify the Fraction Equation: Is (a-b)/(c-b) = a/c True?

Q: Why can't I just cancel out the b terms from both fractions?

You can only cancel terms that are multiplied (like factors), not terms that are added or subtracted. Since b is subtracted in both numerator and denominator, it cannot be canceled.

Q: How do I know if two fractions with variables are equal?

Use cross-multiplication! If $ \frac{A}{B} = \frac{C}{D} $, then $ A \cdot D = B \cdot C $. If this equation is true for all values, the fractions are equal.

Q: What happens when I cross-multiply this specific problem?

Cross-multiplying gives: $ (a-b) \cdot c = a \cdot (c-b) $Left side: $ ac - bc $Right side: $ ac - ab $Since $ -bc \neq -ab $, the fractions are not equal!

Q: Could there be special values where this equation is true?

Yes! For specific values like when a = c or b = 0, the equation might work. But since we're asking if it's always true for any values of a, b, and c, the answer is false.

Q: Is there a quicker way to see this is false?

Try simple numbers! Let a=5, b=2, c=3:$ \frac{5-2}{3-2} = \frac{3}{1} = 3 $$ \frac{5}{3} ≈ 1.67 $Since 3 ≠ 1.67, the equation is false!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine if the equation is correct

00:05 Comparing the expressions as they are, we can see they are not equal

00:10 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Indicate whether the following expression is true or false:

$\frac{a-b}{c-b}=\frac{a}{c}$

2

Step-by-step solution

Let's begin by examining the problem:

$\frac{a-b}{c-b}\stackrel{?}{= }\frac{a}{c}$

Note that the expression on the left side cannot be simplified, despite the fact that both in the numerator and denominator there is the term $b$ . This is due to the fact that it is missing from the second term (both in numerator and denominator) and does not multiply it. Therefore the simplification - which is in fact - applying the division operation which is the inverse operation of multiplication, is not possible, and therefore the current form of the expression on the left side:

$\frac{a-b}{c-b}$

is its final and most simplified form,

The term on the right side is:

$\frac{a}{c}$

Therefore the expressions on both sides of the (assumed) equality are not equal, meaning:

$\frac{a-b}{c-b}\stackrel{!}{\neq }\frac{a}{c}$

(In other words, there is no identical equality- that is true for all possible values of the parameters $a,b,c$ )

Therefore, the correct answer is answer B.

3

Final Answer

Not true

Key Points to Remember

Essential concepts to master this topic

Rule: Fractions are equal only if cross-multiplication gives same result
Technique: Cross-multiply: $(a-b) \cdot c = a \cdot (c-b)$
Check: Expand both sides and verify if $ac - bc = ac - ab$ ✓

Common Mistakes

Avoid these frequent errors

Incorrectly canceling the b terms
Don't cancel b from (a-b) and (c-b) thinking they simplify = wrong equality! The b terms are being subtracted, not multiplied, so they cannot be canceled. Always cross-multiply to check fraction equality properly.

Practice Quiz

Test your knowledge with interactive questions

Determine if the simplification shown below is correct:

$\frac{7}{7\cdot8}=8$

FAQ

Everything you need to know about this question

Why can't I just cancel out the b terms from both fractions?

+

You can only cancel terms that are multiplied (like factors), not terms that are added or subtracted. Since b is subtracted in both numerator and denominator, it cannot be canceled.

How do I know if two fractions with variables are equal?

+

Use cross-multiplication! If $\frac{A}{B} = \frac{C}{D}$ , then $A \cdot D = B \cdot C$ . If this equation is true for all values, the fractions are equal.

What happens when I cross-multiply this specific problem?

+

Cross-multiplying gives: $(a-b) \cdot c = a \cdot (c-b)$
Left side: $ac - bc$
Right side: $ac - ab$
Since $-bc \neq -ab$ , the fractions are not equal!

Could there be special values where this equation is true?

+

Yes! For specific values like when a = c or b = 0, the equation might work. But since we're asking if it's always true for any values of a, b, and c, the answer is false.

Is there a quicker way to see this is false?

+

Try simple numbers! Let a=5, b=2, c=3:
$\frac{5-2}{3-2} = \frac{3}{1} = 3$
$\frac{5}{3} ≈ 1.67$
Since 3 ≠ 1.67, the equation is false!

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Verify the Fraction Equation: Is (a-b)/(c-b) = a/c True?

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