How to solve a fraction with a numerator in exponential form and a denominator in numerical form without a calculator?

Question

The question:

"Imagine unwinding (straightening out) all of the DNA from a single typical cell and laying it "end-to-end"; then the sum total length will be approximately $2$ meters. Assume the human body has $10^{14}$ cells containing DNA. How many times would the sum total length of DNA in your body wrap around the equator of the earth."

The Earth's equator is $40,075$ km

Now I got this question right by dividing the assumed total length of DNA by the distance of the equator:

$$\frac{10^{14} \cdot 2 \ m}{40,075,000 \ m} = 4,990,642$$

The answer key says the answer to the question is "about $5 * 10^6$ times around the equator". But my question is, can I solve this question with an equation that converts the distance of the equator to exponential form to arrive at the same formatted answer as the answer key? Is there a mnemonic that makes it simple to do in your head? For example, if I used the equation:

$$\frac{10^{14} \cdot 2}{10^7 \cdot 4}$$

Then solved that equation to this:

$$\frac{10^7 \cdot 2}{4}$$

From here is it possible to get $$10^6 \cdot 5$$ (the answer) without using a calculator?

Answer 1

Yes, it is possible. For your simpler example, $\frac{2 \cdot 10^7}{4}$, rewrite $10^7 $ as $10^1 \cdot 10^6 = 10 \cdot 10^6$. Then you have $\frac{20 \cdot 10^6}{4} = 5 \cdot 10^6$.

Now back to the original question: $$\frac{2 \cdot 10^{14}}{40,075,000}$$

First, convert the denominator to standard form (scientific notation), which is $4.0075 \cdot 10^7$. Then rewrite the numerator as $20 \cdot 10^{13}$ using the same process as before.

Then you have: $$\frac{20 \cdot 10^{13}}{4.0075 \cdot 10^7}$$

where you can now estimate the denominator as $4 \cdot 10^7$ since you will not lose any precision, except if you are using more than $3$ sig figs. Then use the laws of indices to calculate this expression (which one is it)?

Answer 2

You have to recognize that $10=2 \cdot 5$, so $\frac{10 \cdot 2}4=5$. You can borrow a $10$ from the $10^7$ by subtracting $1$ from the exponent.

Mental arithmetic, like so many skills, rewards practice. Depending on the calculations you want to do, it also rewards having facts memorized so they are easy. Do you see $1001$ and immediately think $7 \cdot 11 \cdot 13?$ Or $1000(1+0.1\%)?$ For calculations like this, approximations are acceptable. I answered an earlier question here with the types of things I have at my fingertips.

Answer 3

I think you are talking about scientific notation.

$40,075 $kilometers is $40,075,000$ meters. And $40,075,000=4.0075\times 10^7$.

And the dna being $2$ meters times $10^{14}$ is $2\times 10^4$.

So you want to solve $\frac {2\times 10^{14}}{4.0075 \times 10^7}=$

$\frac {2}{4.0075}\times 10^{14-7}\approx$

$\frac 12 \times 10^{7}=$

$0.5 \times 10^{7}=$

$0.5 \times 10 \times 10^6$

$5 \times 10^{6}$.

Why did I round $\frac 2{4.0075}$ to $\frac 12$?

The degree of accuracy of "about" $2$ meters, makes the accuracy of the equater of the earth far more accurate then is necessary. The molecule is not $2$ meters long it is about $2$ meters long and the degree of error will be greater than $75$ kilometers.

Note: this is not being lazy or inaccurate. It would actually be wrong and inaccurate to include the $75$ kilometers.

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Anyhoo.... scientific notation is all about these type of multiplication and division problem of numbers of extreme scale.

Any number can be written, within some degree of accuracy, as as a single ones value decimal times $10$ to some power. So to multiply or divide you deal with just the ones value decimals and add or subrtact the tens power.

Example if hair grows at $5$ inches a year, how fast does it grow in miles per hour.

Well there are $12$ inches in a foot and $3$ feet in a yard and $1760$ yards in a mile so there are $12\times 3\times 1760$ inches in a mile.

$12=1.2\times 10^1$ and $3 = 3.0\times 10^0$ and $1760=1.8\times 10^3$ (that's as accurate as we need) so there are $1.2\times 3\times 1.8\times 10^{1+0+3}=6.48\times 10^4\approx 6.5 10^4$ inches in a mile.

And there are $24$ hours in a day, and $364.5$ days in a year so $2.4\times 10^1 \times 3.645\times 10^2 \approx 2.5\times 3.5 \times 10^3=8.75 \times 10^4$ hours in a year.

so $5 \frac {inches}{year} = 5\frac {\frac 1{6.5 10^4} miles}{8.75\times 10^4 hours}=$

$\frac {5}{6.5\times 8.75 \times 10^{4+4}}=$

$\frac {5}{56.875 \times 10^8}\approx$

$\frac {5}{57\times 10^8}\approx \frac 1{11}\times 10^{-8}\approx$

$0.90909090.... \times 10^{-8}\approx$

$0.91 \times 10^{-8}= 9.1\times 10^{-9}$.

So hair grows approximately $9$ billionths of a mile per hour.