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If $3$ workers assemble $5$ computers parts in $3$ hours. How many computer parts it will take $9$ workers to assemble in $5$ hours.

I always get stuck with three term proportionality problems. What is the general formula to solve this?

The answer is:

$$\frac{9\cdot5\cdot5}{3\cdot3} = 25 $$

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  • $\begingroup$ The proportion must be $\dfrac 5 {3 \cdot 3 } = \dfrac x {9 \cdot 5}$ from which ... $\endgroup$
    – Mauro ALLEGRANZA
    Dec 4, 2016 at 16:10
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    $\begingroup$ If the $A$ computers are produced with total work $B$ (the product of the workers for the time they are working) then $x$ computers will be produced by total work $D$, i.e. : $\dfrac A B = \dfrac x D$ and then solve for $x$. $\endgroup$
    – Mauro ALLEGRANZA
    Dec 4, 2016 at 16:18

3 Answers 3

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You can do this in two steps. In each step, change just one of the variables to the desired final value.

If $3$ workers assemble $5$ computers parts in $3$ hours. How many computer parts it will take $9$ workers to assemble in $5$ hours.

We need to increase the number of workers from $3$ to $9$ and the number of hours from $3$ to $5.$ Choose one thing to do first.

Suppose you choose to increase the number of workers first from $3$ to $9.$ Then $9$ workers can assemble $\frac{9}{3}\times5=15$ parts in $3$ hours.

Now increase the number of hours. We know $9$ workers can assemble $15$ parts in $3$ hours, so in $5$ hours, the same $9$ workers can assemble $\frac{5}{3}\times15=25$ parts.

To do the whole thing in one equation, just apply the second ratio without first simplifying the first multiplication. So instead of $\frac{9}{3}\times5=15$ and then $\frac{5}{3}\times15=25,$ you have $$ \frac{5}{3}\times\frac{9}{3}\times5= \frac{5\times9\times5}{3\times3}=25. $$

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You need to figure out how many computer parts one person can do in an hour. Since it takes 3 worker to build 5 in 3 hours, 3 workers can build $\frac{5}{3}$ of a computer part in an hour so one worker can build $\frac{5}{9}$ Since you have 9 workers and 5 hours you multiply that by 9 and 5

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There are 3 quantities. They are interdependent. They are related by proportionality and inverse proportionality. One has to express the wanted quantity by the other quantities.

The requested quantity is the number of computer parts. The number of computer parts (p) per number of workers (w) and number of hours (h) is: $\frac{p}{w\cdot h}=\frac{5}{3\cdot 3}=\frac{5}{9}$. Solving for the wanted $p$ yields $p=\frac{5}{9}\cdot w\cdot h$.

The result for your example is: $p=\frac{5}{9}\cdot 9\cdot 5=25$.

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