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3 years ago

7

List 0.5 1/5 0.35 12/25 4/5 from least to greatest

2 answers:

igomit [66]3 years ago

7 0

The answer is as follows:
1/5
0.35
12/25
0.5
4/5

Just for a bit of explanation, 1/5 is equal to 0.2, 12/25 is just barely under 0.5, 4/5 is equal to 0.8.
I hope this helps!

Anika [276]3 years ago

3 0

One way is to make them all decimals or all fractions:

.5 .2 .35 .48 .80 are the numbers above changed to decimals.
I will make them the same length (two digit past the decimal point) to compare
.50 .20 .35 .48 and .80

Now it is easy to put them in order

.2 .35 .48 .5 and .80

Let's put them into all fractions

5/10 1/5 35/100 12/25 and 4/5

Instead of finding a common denominator let's try this:
Think of them as test scores - the person who got 1 out 5 did the worst
Next came the person who got only 35 out of 100 - they only did better than the one above (1/5)
12/25 is almost 1/2 but a little below half the questions right
5/10 is exactly half the questions right (so it is bigger than 12/25 by a little).
4/5 is the best score (and the only one who passed).
So 1/5, 35/100, 12/25, .5 and 4/5 are in order least to greatest and match the answer above in bold.

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Answer:

$\begin{aligned}\int{\frac{1}{1 + e^{x}}\cdot dx}= x - \ln(1 + e^{x}) + C\end{aligned}$ .

Step-by-step explanation:

The first derivative of the denominator $1 + e^{x}$ is $e^{x}$ . Rewrite the fraction to obtain that expression on the numerator.

$\begin{aligned}\frac{1}{1 + e^{x}} &= \frac{1 + e^{x}}{1 + e^{x}} - \frac{e^{x}}{1+e^{x}}\\&=1-\frac{e^{x}}{1+e^{x}}\end{aligned}$ .

In other words,

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Therefore

$\begin{aligned}\int{\frac{1}{1 + e^{x}}\cdot dx} &= \int{dx} - \int{\frac{e^{x}}{1+e^{x}}\cdot dx}\\ & = x - \ln{(1 + e^{x})}+C\end{aligned}$ .

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