### Theory:

General rule: Rounding each factor of product/quotient of numbers to its greatest place, then multiply/divide the rounded off factors.
Example:
1. Estimate: $$387 × 42$$

Let us round of $$387$$ and $$42$$ separately and then multiply to get the answer.

Consider $$387$$,

It is a three digit-number. We can round off this to a maximum of the nearest $$100's$$.

Let us round off $$387$$ to $$100's$$.

Let us follow the steps to find the estimated number.

 Step Applying it $$387$$ to hundreds 1 Find the digits to the hundreds place $$3$$87 2 Determine the digit to its right. $$3$$87 3 If this digit is $$5$$ or greater add $$1$$ to it.  If it is lesser, then leave it as it is. $$3$$87($$8>5$$). Add $$1$$ to it. 4 Make the digits to the right of hundreds place $$5$$ to zeros. 400

Thus, estimating the number $$387$$ to the nearest hundreds is $$400$$.

Consider $$42$$,

It is a two-digit number. We can round off this to a maximum of the nearest $$10's$$.

Let us round off $$42$$ to $$10's$$.

Let us follow the steps to find the estimated number.

 Step Applying it $$42$$ to tens 1 Find the digits to tens place $$4$$2 2 Determine the digit to its right. $$4$$2 3 If this digit is $$5$$ or greater add $$1$$ to it.  If it is lesser, then leave it as it is. $$4$$2(2<5). Leave it as it is. 4 Make the digits to the right of tens place $$5$$ to zeros. 40

Thus, estimating the number $$42$$ to the nearest tens is $$40$$.

Now multiply the estimated numbers.

That is $$400 × 40 = 16000$$.

Therefore, the estimated product is $$16000$$.

The actual product is $$387 × 42 = 1596$$. Thus, the estimated product is roughly nearer to the actual product.

Therefore, it is a meaningful estimation.

2. Find the estimated value of $5160÷392$.

Let us round of $$5160$$ and $$392$$ separately and then divide to get the answer.

Consider $$5160$$,

It is a four-digit number. We can round off this to a maximum of the nearest $$1000's$$.

Now try to round off evenly by $$100's$$.

Let us round off $$5160$$ to $$100's$$.

Let us follow the steps to find the estimated number.

 Step Applying it $$5160$$ to hundreds 1 Find the digits to the hundreds place 5$$1$$60 2 Determine the digit to its right. 5$$1$$60 3 If this digit is $$5$$ or greater add $$1$$ to it.  If it is lesser, then leave it as it is. 5$$1$$60(6>5). Add $$1$$ to it. 4 Make the digits to the right of hundreds place $$5$$ to zeros. 5200

Thus, estimating the number $$5160$$ to the nearest hundreds is $$5200$$.

Let us follow the steps to find the estimated number.

 Step Applying it $$392$$ to hundreds 1 Find the digits to the hundreds place $$3$$92 2 Determine the digit to its right. $$3$$92 3 If this digit is $$5$$ or greater add $$1$$ to it.  If it is lesser, then leave it as it is. $$3$$92(9>5). Add $$1$$ to it. 4 Make the digits to the right of hundreds place $$5$$ to zeros. 400

Thus, estimating the number $$392$$ to the nearest hundreds is $$400$$.

Now divide $$5200$$ by $$400$$.

Estimated quotient.

$\begin{array}{l}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{1.176em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}13\\ 400\overline{|5200}\\ \phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{1.323em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\underset{¯}{5200}\\ \phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{1.029em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.735em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\underset{¯}{\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}0\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}}\phantom{\rule{0.147em}{0ex}}\end{array}$

Actual quotient

$\begin{array}{l}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.882em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}13\\ 392\overline{|5160}\\ \phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{1.029em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\underset{¯}{5096}\\ \phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{1.176em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\underset{¯}{\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}64\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}}\end{array}$

Thus, the estimated product is roughly nearer to the actual product.

Therefore, it is a meaningful estimation.