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I have these question, a is always c+d and b is e+f ?.

Thanks.

valfar
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    No. Fractions $\frac{a}{b}$ and $\frac{x}{y}$ can be equal even if $a\ne x$ and $b\ne y$. For example $\frac{1}{2}=\frac{2}{4}$. – anon Jul 21 '16 at 00:15
  • For real numbers, $a,b,x,y$ with $b\neq 0$ and $y\neq 0$, way say $\frac{a}{b}=\frac{x}{y}$ iff $ay=bx$. – JMoravitz Jul 21 '16 at 00:27
  • $\frac ab =\frac{c+d}{e+f}\iff a=m(c+d)\text{ and }b=m(e+f)$ for all $m$ – Piquito Jul 21 '16 at 01:08

2 Answers2

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NO.

Here is an counterexample.

$$\frac{1}{2} = \frac{1+2}{3+3}$$

Obviously $1 \not = 1+2 = 3$, $2 \not = 3+3 =6$

cokecokecoke
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There are plenty of easy counterexamples, of course. $\frac{1}{1} = \frac{2 + 3}{2 + 3}$, but $2 + 3 \neq 1$. But what you suggest is true if both fractions are in reduced form - that is, if $a$ and $b$ are relatively prime and $c + d$ and $e + f$ are relatively prime.

Reese Johnston
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    One also needs to account for signs, e.g. by assuming the denominators have the same sign, else e.g. $,-1/2 = 1/(-2)\ \ $ – Bill Dubuque Jul 21 '16 at 02:07