UPSKILL MATH PLUS

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Let us consider a group of social workers who planned to plant $$100$$ trees by digging $$100$$ pits at a certain location. The following will be true if one person can dig one pit:

If $$20$$ person dig for one hour, they will take $\frac{100}{20}$ $$=$$ 5 hours to complete.

If $$10$$ person dig for one hour, they will take $$=$$ $\frac{100}{10}$ $$=$$ 10 hours to complete.

If $$5$$ person dig for one hour, they will take $$=$$ $\frac{100}{5}$ $$=$$ 20 hours to complete.
Now, in this condition, are the number of person, and the number of hours to complete the work is in direct proportion?
If your answer is NO, that is correct. Because when the number of workers is increasing, the hours are decreasing accordingly at the same rate as the number of employees grow.

These quantities are now said to be in inverse proportion.
Let's use the notation $$X$$ for the number of social workers and $$Y$$ for the number of hours. Now observe the following table.

 Number of Social Workers $$X$$ $$20$$ $$10$$ $$5$$ Number of hours $$Y$$ 5 10 20

We can observe from the table that when the values of $$X$$ decrease, the corresponding values of $$Y$$ increase in such way that the ratio of $\frac{X}{Y}$ in each case has the same value which is a constant (say $$k$$).

Derivation:
Consider each of the value of $$X$$ and the corresponding value of $$Y$$. Their products are all equal say $\mathit{XY}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}100\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}k$ ($$k$$ is a constant), and it can be expressed as $\mathit{XY}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}k$ ($$k$$ is a constant).

If ${X}_{1}{X}_{2}$ are the values of $$X$$ is corresponding to the values of ${Y}_{1}{Y}_{2}$ of $$Y$$, respectively.

Therefore, ${X}_{1}{Y}_{1}={X}_{2}{Y}_{2}=k$ constant.

That is $\frac{{X}_{1}}{{X}_{2}}=\frac{{Y}_{2}}{{Y}_{1}}$.

As a result, $$X$$ and $$Y$$ are inversely proportional.
From the above table, we should take ${X}_{1}$ and ${X}_{2}$ from the values of $$X$$. Similarly, take ${Y}_{1}$ and ${Y}_{2}$ from the values of $$Y$$.

 Number of Social Workers  $$X$$ ${X}_{1}$ ${X}_{2}$ ${X}_{3}$ ${X}_{4}$ Number of hours $$Y$$ ${Y}_{1}$ ${Y}_{2}$ ${Y}_{3}$ ${Y}_{4}$

From the above table, we can learn that we need at least $$3$$ variables to determine the other value.
Do you know how to determine the values of ${Y}_{2}$, ${Y}_{3}$ and${Y}_{4}$?

Step 1:

Let's consider that ${X}_{1}$ and ${Y}_{1}$ are in $$1$$ series, ${X}_{2}$ and ${X}_{3}$ are in $$2$$ series and so on.

If  ${X}_{1}$, ${X}_{2}$ and ${Y}_{1}$ values are provided using these values, we can determine ${Y}_{2}$.

Step 2:

Similarly to find ${Y}_{3}$ value first, you have to make sure that you determine the values of  ${X}_{2}$${X}_{3}$ and ${Y}_{2}$.

If you are unsure, you must use data from earlier in the series to determine the unknown value.

After that using the $$3$$ variables ${X}_{2}$, ${X}_{3}$ and ${Y}_{2}$, we can find out the value of  ${Y}_{3}$.

Step 3:

Now we know the value of ${Y}_{3}$ then using ${X}_{3}$${X}_{4}$ and ${Y}_{3}$ values we can calculate the value of ${Y}_{4}$.