# How to calculate (predict) how much I will be paid in 30 days if I know how much I was paid in the first 9?

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!

• $1313/9=\$ 145.89$per day$\times 30$days$=\$4376.67$ for $30$ days Dec 15, 2014 at 8:16

\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 : $$\text{9 days of work is to 1313 what 30 days of work is to x}$$ $$9:1313:\::30:x$$