Questions tagged [fractions]
Questions on fractions, i.e. expressions (not values) of the form $\frac ab$, including arithmetic with fractions. Not to be confused with the tag (rational-numbers): fractions denote rational numbers, but the same rational number may be written in different ways as a fraction.
3,049 questions
-1 votes
1 answer
63 views
I'm struggling with a logic problem and need some help understanding my mistake
The Problem A duck has two legs. When a duck folds one leg, only one leg is visible. When a duck is sitting, neither of its legs is visible. When Roman went to the lake, there were 33 ducks. He ...
1 vote
1 answer
171 views
Is there such thing as iterated division? [closed]
I'd an idea of it many years ago and it keeps coming back to me. I searched up iterated divisions and synonyms of it and never got anything. If it truly is something that was discussed before, how ...
- 19
2 votes
2 answers
114 views
Simplest Fraction in Interval
Consider an interval (open or closed) centered around a rational number $\frac ab$ with width $\frac1b$, where $a$ and $b$ integers (and $b$ a positive one) named $I$ ($I=\left[\frac{2a-1}{2b}, \frac{...
- 41
2 votes
2 answers
215 views
Is there an algorithm for finding the smallest value using a given set of fractions?
Imagine a game where you move around a circle. The object of the game is to end up as close as possible to where you started. You can move as many time as you want, but you do have to make at least ...
- 21
3 votes
0 answers
80 views
The multiplication of fractions of non-commutative ring is well-defined
In Lam's Lectures on modules and rings, page 302 (proof of Theorem 10.6), he defines the fraction of a ring $R$ with a multiplicative set $S$ which satisfies the following conditions: Right Ore ...
- 398
-1 votes
1 answer
141 views
Prove that $\frac{1}{1001}+\frac{1}{1002}+\frac{1}{1003}+\cdots+\frac{1}{3001}>1$ [duplicate]
Prove that $$\frac{1}{1001}+\frac{1}{1002}+\frac{1}{1003}+\cdots+\frac{1}{3001}>1$$ I tried in this way: Since there are total $2001$ fractions, I divided them into three groups each having $667$ ...
- 309
2 votes
5 answers
407 views
Concrete Definition for a Ratio
I had a question about ratios. I couldn't seem to find a rigorous definition on them. I understand the concept of fractions being numbers of the form a/b where a is divided by b (or a multiplied by 1/...
- 29
5 votes
2 answers
350 views
Forming a fraction equivalent to $1/3$, using each digit from $1$ to $9$ exactly once
This is from a mock test paper for a math contest intended for Year 7 students. Suppose that you have the digits $1$, $2$, $3$, $4$, $\ldots$ , $9$. Each digit must be used exactly once to create a ...
- 615
0 votes
4 answers
165 views
Finding the inverse of $g(x)=\frac{-8x+6}{-13x+11}$. How do I know which term I'm supposed to move?
Given the function $g(x)=\frac{-8x+6}{-13x+11}$, determine the inverse function $g^{-1}(x)$ in simplified form. I got down to $-13xy+11x=-8y+6$ and chose to move the -8y to the left instead of moving ...
- 137
5 votes
4 answers
276 views
Limit of $ e^{x(\ln2 - x)}$
Recently I came across the limit $$ \lim_{x\to\infty}\frac{2^x}{e^{x^2}} $$ I have organized it to be $$ \lim_{x\to\infty}\frac{e^{x\ln2}}{e^{x^2}} $$ Now, I want to say that by comparing the ...
- 143
3 votes
1 answer
166 views
Why do we draw one fewer line than the number of bars in area or bar models?
I’m learning multiplication and fractions using visual methods like bar models or area models. Something confuses me: If I want to divide a rectangle into 4 equal vertical bars, I draw 3 vertical ...
- 47
11 votes
3 answers
378 views
A dragon drinks $\frac{1}{N_1}$ of a lake and then $\frac{1}{N_2}$ of the remaining and so on. What fraction/percent remains after $N_n$ times?
The original question was A dragon drunk $\frac15$ of the water in a lake. After a break, he drank $\frac14$ of the remaining water. after another break he drank $\frac13$ of the remaining water. ...
- 615
-1 votes
2 answers
138 views
Why can't I multiply by the LCD here and cancel like-terms?
$$\frac{x+4}{x^2+12x+20}+\frac{x+1}{x^2+8x-20}$$ Factored: $$\frac{x+4}{(x+2)(x+10)}+\frac{x+1}{(x-2)(x+10)}$$ When I put this problem into Mathway, it says to multiply the first fraction by $(x-2)$ ...
- 137
2 votes
0 answers
113 views
Geometric interpretations of Farey addition and the role of $|ad-bc|=1$?
In Two Geometric Pictures of Farey Addition by Richard Evan Schwartz (here), the Farey sum $$\frac{a}{b} \oplus \frac{c}{d} = \frac{a + c}{b + d} $$ is interpreted in two elegant ways: It is defined ...
- 3,613
0 votes
1 answer
43 views
Minimizing or maximizing fractions of permutations
I have a sum of fractions of the form $$\sum_{i=1}^N \frac{A^i}{B^i}$$ where the $A^i$'s and $B^i$'s are some positive real numbers. I want to find two (possibly different) permutations $j$ and $k$ so ...
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