Questions on fractions, numbers of the form $p/q$ where $p$ and $q$ are integers, and $q$ is not zero.

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5
votes
7answers
1k views

Best way to exactly solve a linear system (300x300) with integer coefficients

I want to solve system of linear equations of size 300x300 (300 variables & 300 equations) exactly (not with floating point, aka dgesl, but with fractions in ...
6
votes
1answer
3k views

Ratios make me feel like an idiot - help me mix up some Coca-Cola

I may be over-complicating things, but something doesn't seem right (and I swear this isn't homework, I'm friggin 30 years old :P). I want to see what it will cost me to make a TWO liter of Coca-Cola ...
4
votes
3answers
505 views

Computing a rational number between two others, minimizing numerator and denominator

Given two positive rational number $\frac{a_1}{b_1}$ and $\frac{a_3}{b_3}$ (written in lowest terms) such that $$\frac{a_1}{b_1} < \frac{a_3}{b_3},$$ I want to find a rational number ...
6
votes
2answers
183 views

How to solve this rational equation for y

Greetings! On a test recently I ended up having to solve this for y: $$ x = \frac{2y}{y + 1} $$ But I kept getting stuck in cirlces... $$ \begin{aligned} x(y + 1 ) = 2y \\ xy + x = 2y \\ ...
16
votes
3answers
1k views

Why is the decimal representation of 1/7 “cyclical”?

1/7 = 0.(142857)... with the digits in the parentheses repeating. I understand that the reason it's a repeating fraction is because 7 and 10 are coprime. But this...cyclical nature is something ...