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bekas [8.4K]
3 years ago
15

Write .112112 as a fraction

Mathematics
1 answer:
tresset_1 [31]3 years ago
8 0

Answer:

112/999

Explanation:

Here's how to convert 0.112112 to a fraction...

The decimal part of your number seems to have the digits 112 repeating in it.

Your original number to convert is 0.112112. Let's slide the decimal point in this number to the right 3 place(s) (the same number of digits in the number 112).

If we do this, we'll get a 112.112000 (slide the decimal in the 0.112112 right 3 places, you'll get 112.112000).

So what? Well now, we have two numbers with the same repeating decimal parts, 112.112000 and 0.112112.

Now let's just work a little algebra into all of this. Let's call your original number x. And in this case, x=0.112112. The number with the decimal point slid over can be called 1000x, because 1000x=112.112000

What if we subtracted these two equations (that is, subtract the items on the left of the equal sign

from the stuff on the right of the equal sign)?

1000x = 112.112

- x = 0.112112

---------------

999x = 112.

Now here's the important result of doing all of this: Notice how all of the repeating decimal parts have subtracted away to zero! We are left with a nice, simple 112 on the right side of the equal sign.

Now, solving 999x=112 for x by dividing both sides of it by 999, we'll get that x=112/999. And this is your answer.

How is this your answer? Well remember that above, x was originally set equal to 0.112112 via x=0.112112, and now we have that x is also equal to 112/999, so that means 0.112112=112/999..and there's 0.112112 written as a fraction!

So your final answer is: 0.112112 can be written as the fraction 112/999

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hichkok12 [17]

Answer:

0.007 = 0.7% probability that the proportion of Rolls Royce owners in a sample of 595 Americans would differ from the population proportion by more than 3%

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean \mu and standard deviation \sigma, the z-score of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean \mu and standard deviation \sigma, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean \mu and standard deviation s = \frac{\sigma}{\sqrt{n}}.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean \mu = p and standard deviation s = \sqrt{\frac{p(1-p)}{n}}

A statistician calculates that 8% of Americans own a Rolls Royce.

This means that p = 0.08

Sample of 595:

This means that n = 595

Mean and standard deviation:

\mu = p = 0.08

s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.08*0.92}{595}} = 0.0111

What is the probability that the proportion of Rolls Royce owners in a sample of 595 Americans would differ from the population proportion by more than 3%?

Proportion above 8% + 3% = 11% or below 8% - 3% = 5%. Since the normal distribution is symmetric, these probabilities are equal, and so we find one of them and multiply by 2.

Probability the proportion is less than 5%:

P-value of Z when X = 0.05. So

Z = \frac{X - \mu}{\sigma}

By the Central Limit Theorem

Z = \frac{X - \mu}{s}

Z = \frac{0.05 - 0.08}{0.0111}

Z = -2.7

Z = -2.7 has a p-value of 0.0035

2*0.0035 = 0.0070

0.007 = 0.7% probability that the proportion of Rolls Royce owners in a sample of 595 Americans would differ from the population proportion by more than 3%

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