Approximating the product of two real numbers

Question

Let $a,b$ two positive real numbers. For example I want to calculate approximate value of $45.11\times 67.89$ only to 2 decimal places. Note that any calculators or other such devices aren't allowed. Also suppose I want to calculate $\frac{11789}{234558}$ only approximately to 2 decimal places then how to do it? One might ask why do I want to randomly calculate multiplication and division. Actually that isn't the case. There's an exam which I will be taking which has one of the sections as $\text{Data Interpretation}$. In this section there are problems related to the annual turnover ,profits of a company and many such things . Say that its given in the form of a pie chart the quarterly sales of a company for a certain year and we are asked to find the percentage increase /decrease in two consecutive quarters thus the question to quickly estimate the approximate value or if not that atleast giving a small range within which the value may fall. Any quick methods /suggestions will be appreciated! here I have provided one of the sample questions . What I am asking is a generalized method like using percentage(mostly for division).

Answer 1

It's $$(45+0.11)(68-0.11)=3060+23\cdot0.11-0.0121=$$ $$=3062.53-0.0121=3062.5179\approx3062.52$$

Answer 2

Have you asked the right question?

I think it unlikely that either of the calculations you want to do quickly to two decimal place accuracy is likely to come up on exam where data is

given in the form of a pie chart the quarterly sales of a company

and you can't use a calculator.

I think you might get more help here if you posted some sample questions along with how you tried to answer them and asked for better suggestions.

Answer 3

$$ \begin{align} 45.11\times 67.89 &= (45 + 11\cdot 10^{-2})(67 + 89\cdot 10^{-2}) \\&= 45\cdot 67 + (45\cdot 89 + 11\cdot 67)\cdot 10^{-2}+ (11 \cdot 89)\cdot 10^{-4} \\&\approx 45\cdot 67 + (45\cdot 89 + 11\cdot 67)\cdot 10^{-2} \\&= 3062.42 \end{align} $$ The exact value is $3062.5179$.

A better approximation is $$ \begin{align} 45.11\times 67.89 &= (45 + 11\cdot 10^{-2})(67 + 89\cdot 10^{-2}) \\&= 45\cdot 67 + (45\cdot 89 + 11\cdot 67)\cdot 10^{-2}+ (11 \cdot 89)\cdot 10^{-4} \\&\approx 45\cdot 67 + (45\cdot 89 + 11\cdot 67)\cdot 10^{-2}+ (10 \cdot 90)\cdot 10^{-4} \\&= 3062.51 \end{align} $$

Answer 4

$$\dfrac{11789}{234558} \approx \dfrac{12000}{235000}$$

In fact:

$$\left| \dfrac{11789}{234558} - \dfrac{12000}{235000}\right| < 0.001$$

It should be easy to round before you divide and still get very close.

Answer 5

For the sample question, first you use the “arithmetic sequence” information to fill in the missing monthly sales data for December $2016.$ (You have an arithmetic sequence with a known number of terms, you have the sum of the sequence, and you have the value of the first term, which is enough to determine every term of the sequence. The second quarter monthly sales can be found the same way although they do not seem to matter for the given questions.)

Use subtraction to find the sales for December $2017.$

You will then find that the December sales increased from $140$ to $180.$ The percentage increase is therefore $$\frac{40}{140}\times100\%.$$

Now realize that since this exam is multiple-choice, you don’t have to compute the percentage to two places past the decimal point; you only need to work it out precisely enough to eliminate three of the four choices. Since $.30\times 140=42,$ you can instantly eliminate two options and should quickly eliminate a third.

For the second question you need to fill in the missing quarterly data. Then determine which of the four ratios is largest. No percentages need be calculated, though rough estimates might help.