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I'm a very simple man living his life. I don't know much math. This is a real world scenario of me trying to apply math and trying to find how much approximately I will be paid this month. I know some math terms and basic theorems but I don't know how to apply them and I can't do much more than basic math operations.

I'm working 30 days a month. In the first 9 days of the month I've been paid $1,313. What is the correct way to linearly extrapolate this quantity to approximate how much I will be paid in 30 days?

I get different results when I try different methods (that I thought of myself). Why do they all give different results? (I'm very confused and excited, I thought I was doing the same thing, yet results are different! Not that I will ever understand what I did but I'm still curious.)

Attempt 1

The first method is I notice that if today was the 30th day then I'd have been paid in full, so I'd have been paid 30/30 of my salary. But since in reality today is the 9th day, I've been paid only 9/30 of my salary. There is 21/30 remaining to be paid.

Therefore in 30 days I will be paid what I've already been paid, plus 21/30 more:

$$\$1,313 (already \space paid) + \frac{21}{30}\$1,313 = \$1,313 + $919 = \$2,232$$

Something doesn't seem right here. Using my math my salary for the reamaining 21 days will be smaller than for the first 9 days. My paycheck doesn't lie and I always get paid more than \$3,000 a month. I think I've made a mistake here but I will leave this result to indicate my thinking. I've no idea what my error is because following my thinking I can't see a mistake.

Attempt 2

For the second method my thinking goes like this. I work for 30 days a month. I've worked 9/30 so far. Similarly to the previous method, I've 21/30 to work. But somehow if I reverse this fraction to 30/21 I get some kind of a multiplier that I can apply to $1,313. This multiplier "feels like" portion of money I will still be paid, so I get:

$$\$1,313 (already \space paid) + \frac{30}{21}\$1,313 = \$1,313 + $1,875 = \$3,188$$

This method seems better as I will be paid over \$3,000 but I've no idea how it worked. Why did I reverse the 21/30 to 31/21 to get this result. I've no idea. But I believe this result more than the previous one because I always get paid more than \$3,000.

Attempt 3

I use algebra from high school to solve!

Let $x$ be total salary this month. Now $\frac{9}{30}x$ is what I've been paid so far:

$$\frac{9}{30}x = \$1,313$$

Then I can simply transport $\frac{9}{30}$ to the other side of equation and get:

$$x = \$1,313\frac{30}{9}$$

And arrive at:

$$x = \$4,376$$

Wow. Will I get paid \$4,376 this month? Can't be right, I've never been paid more than \$4,000 so far in my life. Something doesn't seem right again but what... I've no idea. High school algebra seems right but the paycheck never lies.

Attempt 4

I think this way: I've been paid \$1,313 for 9 days of work. There are 30 days of work this month. So I've worked 9/30 of all the time. $1-\frac{9}{30}$ is what I haven't worked. I introduce proportions:

$$\frac{9}{30} = \$1,313 (paid \space so \space far)$$

And

$$1 - \frac{9}{30} = (not \space yet \space paid)$$

Now I substitute the $\frac{9}{30}$ from first equation into second and I get:

$$1 - \$1,313 (paid \space so \space far) = (not \space yet \space paid)$$

Somehow I arrive at negative number $-\$1,312$. I just drop the minus sign as I don't know what it is, and it seems I will be paid $$\$1,312$$ for the remaining 21 days. My total salary using this method would be:

$$ \$1,313 + \$1,312 = \$2,625 $$

This is not correct as my paycheck doesn't lie. My paycheck is $3,000+. I've made a mistake somewhere, but where...? I've no idea. I followed all math steps but somehow the result ended up wrong.

Final thoughts

I'd appreciate input from someone who really gets mathematics to explain how to extrapolate 9 days of salary to 30 days. I pretty much give up at this point as it's just making my mind explode figuring out who the results are wrong and what is the meaning of 21/30 and why is 30/21 a better multiplier and why I get negative results. Take it easy on me. I'm a simple guy living paycheck to paycheck and math is not my strong side. Thanks!

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    $\begingroup$ $1313/9=\$ 145.89$ per day$\times 30$ days$=\$4376.67$ for $30$ days $\endgroup$ – Vikram Dec 15 '14 at 8:16
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$$\begin{align} \text{Salary for 1 day} &= \frac{1313}{9} \\ \text{Salary for 30 days} &= \frac{1313}{9}\times 30=4376.66 \end{align} $$ You could rather follow this logic : $$\text{9 days of work is to 1313 what 30 days of work is to x}$$ $$9:1313:\::30:x$$

So, according to me, your Attempt 1 is wrong. Your Attempts 2 and 4 are wrong too.

You can graph it to convince yourself :

enter image description here

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  • $\begingroup$ I like your method of finding how much I made in 1 day, and then multiplying by 30. I didn't even think of that. The graph is nice. $\endgroup$ – bodacydo Dec 15 '14 at 14:24

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