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Which of these two ways of simplifying $y=\dfrac{a\times\frac{b}{2}\times c}{d}$ is correct, and why?

1st way:

$$\begin{align} y &=\frac{a \times \frac{b}{2} \times c}{d} \\[4pt] \implies\quad y &=\frac{\frac{abc}{2}}{d} \tag1 \\[4pt] \implies\quad y &=\frac{abc}{2} \div d \tag2 \\[4pt] \implies\quad y &=\frac{abc}{2d} \tag3 \end{align}$$

2nd way:

$$\begin{align} y&=\frac{a \times \frac{b}{2} \times c}{d} \\[4pt] \implies\quad y&=\frac{\frac{abc}{2}}{d} \tag4 \\[4pt] \implies\quad y&=abc \div \frac{2}{d} \tag5 \\[4pt] \implies\quad y&=\frac{abcd}{2} \tag6 \end{align}$$

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    $\begingroup$ the first one is correct $\endgroup$ Commented May 16, 2021 at 1:23

3 Answers 3

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In the expression $y = \frac{a \times \frac{b}{2} \times c}{d}$, the top part (the numerator) is one big chunk, and the bottom part (the denominator) is $d$. Group the top: You multiply $a$, $\frac{b}{2}$, and $c$ together to get $\frac{abc}{2}$. The "Big" Division: Now you have $(\frac{abc}{2})$ divided by $d$.The Flip: When you divide a fraction by a whole number, that number joins the bottom. It’s like sharing half a pizza ($\frac{1}{2}$) with $d$ people; everyone gets an even smaller slice ($\frac{1}{2d}$). Math-speak: $\frac{abc}{2} \div d = \frac{abc}{2} \times \frac{1}{d} = \frac{abc}{2d}$.

The 2nd way starts off fine but trips at Step 5. It tries to turn $\frac{\frac{abc}{2}}{d}$ into $abc \div \frac{2}{d}$. It basically kicked the $abc$ out of the fraction and tried to flip the $2$ and the $d$ into their own separate fraction.In math, you can't just pick and choose which parts to "flip" like that. The $2$ belongs to the $abc$ (it's the denominator of the top chunk), while the $d$ is the "divisor" for the whole thing. You can't swap their roles!

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The correct method is the first way. It all comes down to order of operations. Recall that mathematical expressions in parentheses come before multiplication and division. So when we write $$\frac{\frac{abc}{2}}{d}$$ we're saying: $$(abc/2)/d$$ because the fraction bar between the $2$ and $d$ is largest.

This fractional expression is equivalent to $$\frac{\frac{abc}{2}}{\frac{d}{1}} = \frac{abc}{2}\cdot\frac{1}{d} = \frac{abc}{2d}$$

The second way is evaluating $$abc/(2/d),$$ which changes the location of the parentheses.

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For convenience, represent your fractions as division in parentheses:

$$ \frac{a\times \frac{b}{2}\times c}{d} = \left(a\times \left(b \div 2\right) \times c\right)\div d $$

Division has a rule $\boxed{A\times (B\div C) = (A\times B)\div C}$ (note that this works in both directions), therefore:

$$ \left(a\times \left(b \div 2\right) \times c\right)\div d = \left(a\times b \times \left(1 \div 2\right) \times c\right)\div d $$

Multiplication is commutative, therefore we can change the order of multiplication as we like:

$$ \left(a\times b \times \left(1 \div 2\right) \times c\right)\div d = \left(a\times b \times c \times \left(1 \div 2\right)\right)\div d $$

The rule boxed above gives us:

$$ \left(a\times b \times c \times \left(1 \div 2\right)\right)\div d = \left(1 \div 2\right)\times \left(a\times b \times c\right)\div d $$

This gives us the right answer:

$$ \frac{a\times \frac{b}{2}\times c}{d} = \frac{1}{2}\times \frac{a\times b\times c}{d} = \frac{a\times b\times c}{2\times d}. $$

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