A machine produces B unit goods in A hours...

Question

A machine produces $b$ unit goods in $a$ hours. How many hours does this machine need to produce $b\cdot c$ unit goods?

Answer 1

It takes $a\cdot c$ hours.

The rate of goods are being produced is $\frac{b}{a}$. We need to find the amount of time that has to pass for which we produce at this rate in order to achieve $b\cdot c$ units produced.

That is, you want to solve this for $t$:

$$b\cdot c = \frac{b}{a} t$$

We get $t= a\cdot c$.

Answer 2

$\because$ Machine produces $b$ unit goods in $a$ hours

$\therefore$ Machine would produce $1$ unit goods in $\frac{a}{b}$ hours

$\therefore$ Machine would produce $b\cdot c$ unit goods in $\frac{a}{{b}}\times b\cdot c$

Machine would need $a\cdot c$ hours.

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Alternate Method:

We observe that the number of goods produced $(N)$ is directly proportional to time $(t)$ i.e. in more time more number of goods will be produced.

$N \propto t$

$N = k\cdot t$ ($k$ is variation constant). (eq (1))

$\therefore k = \frac{N}{t} = \frac{b}{a}$

We have to find $t$, so $t= \frac{N}{k}$ (from eq(1)).

Now plugging the value of $N \& k$, we get $t= \frac{N}{k}=\frac{b\cdot c}{\frac{b}{a}} = \boxed{ac}$.

You should definitely check this method of variation.