# Solving $\frac{x}{5}+\frac{x}{10}+\frac{x}{4}+75=x$

I had this equation in my mathbook (Pitkä SIGMA 1: Funktiot ja yhtälöt)

$\frac{x}{5}+\frac{x}{10}+\frac{x}{4}+75=x$

I first start with transforming 75 to a fraction

$\frac{x}{5}+\frac{x}{10}+\frac{x}{4}+\frac{750}{10}=x$

Then we expand these to get same denominators

$\frac{^{4)}x}{5}+\frac{^{2)}x}{10}+\frac{^{5)}x}{4}+\frac{^{2)}750}{10}=x$ so I have $\frac{4x}{20}+\frac{2x}{20}+\frac{5x}{20}+\frac{1500}{20}=x$

Now I multiply the $x$ located right with 20 to make all denominators to disappear, so we have

$4x+2x+5x+1500=20x$.

I move the $20x$ to the left and 1500 to the right. As I do this, positive goes to negative and negative goes to positive etc...

$4x+2x+5x-20x=-1500.$

Little more magic and we have

$-9x=-1500.$ I will now divide $-1500$ with $-9$, so we have $166,666$, which is $\frac{1500}{9}$

We cancel that and the anwser is drum roll $\frac{500}{3}$

If I look the original question (which I'm not intended to translate), the answer is at least strange. There cannot be 166.6 fishes in the aquarium tank. Well anyhow, is it right to multiply the $x$ with 20 and is the answer correct?

• That equation indeed leeds to $x=\frac{500}{3}$, see e.g. WolframAlpha. So your answer is correct, if $x$ is supposed to represent number of fish in an aquarium tank, then the question might indeed be strange. – Eric S. Jan 13 '16 at 13:33
• @EricS. Or, of course, the equation is not set up properly. – 5xum Jan 13 '16 at 13:33
• Would be good to have the original question. – Integral Jan 13 '16 at 13:34
• Or, there is a decapitated fish in the fish bowl. He starts with I had this equation in my mathbook, so I work under that assumption. Even so, doesn't the question might indeed be strange account for the equation is not set up properly? – Eric S. Jan 13 '16 at 13:35
• For a resolution of the mystery, see my comment and edit to the answer below. – Did Jan 13 '16 at 17:21

## 1 Answer

It is correct that if $\frac x5 + \frac x{10} + \frac x4 + 75=x$, then $x=\frac{500}{3}$. The math of solving the problem is all correct.

However, there still may be a mistake either in the original question or in your transformation of the question into the equation...

Edit: From the comments below, it appears that the text of the exercise is actually the following.

There is a large aquarium tank at the lobby. From all the fishes, one fifth are basses, and 30% are perches. The rest 75 of the fishes are tetras. How many fishes are in the tank?

Calling $x$ the total number of fishes in the tank, this translates into $x/5$ basses, $3x/10$ perches and $75$ tetras, hence $x=(x/5)+(3x/10)+75$, which leads to $x=150$ (an integer).

• No, the question is right and this is not the first time there is errors or silly mistakes in my mathbook. Still though this makes people to doubt theis anwsers. EDIT: Sorry, my transformation is right. The equation is written "AS IS" in the book. – Kasperi Koski Jan 13 '16 at 13:30
• @KasperiKoski Well, I cannot confirm that your equation is correct because you did not provide the question. As I said, I can guarantee that your solution is correct and that, given the equation, $x$ must be equal to $\frac{500}{3}$. – 5xum Jan 13 '16 at 13:31
• The question was the equation $\frac x5 + \frac x{10} + \frac x4 + 75=x$. It was a "connect these equations and questions". There was this equation and I connected it to: There is a large aquarium tank at the lobby. From all the fishes, one fifth are basses, and 30% are perches. The rest 75 of the fishes are tetras. How many fishes are in the tank? It was THE only option! Thank you guys! So no doubts for my math haha ;) – Kasperi Koski Jan 13 '16 at 13:39
• @ArchisWelankar First of all, that is no reason to downvote. Second of all, comments are intended to be comments. Even if the answer is short, it should be posted as an answer, where it can get accepted. Answers are answers, and comments are comments. And answers, no matter the length, belong in the "answer" section, and comments belong in the comment section. – 5xum Jan 13 '16 at 13:39
• @KasperiKoski $30\%$ means $\frac{3}{10} x$, so you probably connected your question to the incorrect equation... – 5xum Jan 13 '16 at 13:40