22

$$a_n = n\dfrac{n^2 + 5}{4}$$

In the above fraction series, for $n=3$ I think the answer should be $26/4$, while the answer in the answer book is $21/2$ (or $42/4$). I think the difference stems from how we treat the first $n$. In my understanding, the first number is a complete part and should be added to fraction, while the book treats it as part of fraction itself, thus multiplying it with $n^2+5$. So, I just want to understand which convention is correct.

This is from problem 6 in exercise 9.1 on page 180 of the book Sequences and Series.

Here is the answer sheet from the book (answer 6, 3rd element):

  1. $3,8,15,24,35$
  2. $\dfrac{1}{2},\dfrac{2}{3},\dfrac{3}{4},\dfrac{4}{5},\dfrac{5}{6}$
  3. $2, 4, 8, 16 \text{ and } 32$
  4. $-\dfrac{1}{6},\dfrac{1}{6},\dfrac{1}{2},\dfrac{5}{6},\dfrac{7}{6}$
  5. $25,-125,625,-3125,15625$
  6. $\dfrac{3}{2},\dfrac{9}{2},\dfrac{21}{2},21,\dfrac{75}{2}$
  7. $65, 93$
  8. $\dfrac{49}{128}$
  9. $729$
  10. $\dfrac{360}{23}$
  11. $3, 11, 35, 107, 323$; $3+11+35+107+323+...$
  12. $-1,\dfrac{-1}{2},\dfrac{-1}{6},\dfrac{-1}{24},\dfrac{-1}{120}$; $-1+(\dfrac{-1}{2})+(\dfrac{-1}{6})+(\dfrac{-1}{24})+(d\frac{-1}{120})+...$
  13. $2, 2, 1, 0, -1$; $2+2+1+0+(-1)+...$
  14. $1,2,\dfrac{3}{5},\dfrac{8}{5}$
Deepak Suwalka
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skjoshi
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7 Answers7

34

in elementary school math the fraction $x\frac{y}{z}$ usually means $x+\frac{y}{z}$ and is called a mixed fraction.

However these are almost never used after junior high.

Most of the time when you see $x\frac{y}{z}$ the two terms should be multiplied, so it is equal to $\frac{xy}{z}$.

Asinomás
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  • Or just $\frac{x.z + y}{z}$ – Fawad Dec 28 '16 at 01:35
  • @Ramanujan I do not understand. Do you mean $\frac{x(z+y)}{z} = x\frac{z+y}{z}$? – Jeppe Stig Nielsen Dec 28 '16 at 08:15
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    @JeppeStigNielsen I think Ramanujan just meant that $x + \frac{y}{z} = \frac{xz+y}{z}$. – Robin Saunders Dec 28 '16 at 09:45
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    This answer misses the point, which is this: the fraction e.g. $3\frac14$ does mean $3+\frac14$. Unambiguously. But the expression $x \frac{y}{z}$ means $x \times \frac{y}{z}$. Also unambiguously. How to explain this discrepancy? – TonyK Dec 28 '16 at 12:01
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    @TonyK I've never seen the $3\frac{1}{4}$ notation before. I believe this is some US-centric thing... To me $3\frac{1}{4} = 3 \cdot \frac{1}{4} = \frac{3\cdot 1}{4} =\frac{3}{4}$. – Bakuriu Dec 28 '16 at 14:09
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    @Bakuriu: Never seen it? Haven't you heard of this Fellini film? In any case, it's not an Americanism $-$ I am British. – TonyK Dec 28 '16 at 14:26
  • @TonyK I think it's a mostly an imperial vs metric thing. With imperial units I've seen a lot more usage of mixed fractions. However doing this with metric would undermine why the metric system is useful, so decimal notation is used pretty excursively instead. It's part of the culture. Personally when I hear "three and a half" I instantly picture 3.5 not $3\frac{1}{2}$. I have never really used that mixed fraction notation in my life, at least since early primary school. It is all just decimal and maybe improper fractions. – Christer Dec 28 '16 at 15:55
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    @Bakuriu It's the same in the Netherlands; in elementary school and high school the notation $3\frac14$ means $3+\frac14$, but $x\frac{y}{z}$ means $x\times\frac{y}{z}$. – Servaes Dec 28 '16 at 16:37
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    I can confirm that in elementary schools in Croatia (metric units) $3\frac 14 = 3+\frac 14$. – Ennar Dec 28 '16 at 16:39
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    Also in Poland (metric) - I'm quite sure $3\frac{1}{4} = 3 + \frac{1}{4}$ while $x\frac{y}{z} = x \times \frac{y}{z}$. – Maja Piechotka Dec 28 '16 at 18:50
  • I don't get why people don't use mixed fraction in high school, It is easier to add in mixed fraction. just try $\displaystyle {1125\over 8}\left(140{5 \over 8 }\right)$ and $\displaystyle {51683\over 8}\left(6460{3 \over 8 }\right)$. –  Dec 28 '16 at 18:58
  • "are almost never used after junior high" - huh, I use tools, and I'm more likely to encounter $3\frac12$ than $\frac72$ in measurements. I haven't seen a carpenter who prefers improper fractions to mixed numbers myself, but maybe you know a few. – J. M. ain't a mathematician Dec 28 '16 at 19:17
  • I recognize $3\tfrac14$ to mean $\3+\tfrac14$ from elementary school in Israel as well. But I don't recall seeing it in any mathematical text beyond that. If I did see $3\tfrac14$, I would probably assume it means $\tfrac34$. In a non-mathematical context, mixed fractions appear occasionally. – Meni Rosenfeld Dec 28 '16 at 23:52
  • @Bakuriu Definitely not just the US. See, for example, the running times on page 3 of this official London Underground timetable http://content.tfl.gov.uk/wtt-41-bakerloo-11-dec-2016.pdf – Au101 Dec 29 '16 at 00:19
  • @Bakuriu I have posted this question from an Indian textbook and the confusion stemmed due to the thing we studied in Elementry days. – skjoshi Dec 29 '16 at 06:13
  • Same in Spain. Could be, in two or three centuries from now, someone decides to remove the useless concept of mixed fractions from primary school books. – pasaba por aqui Dec 29 '16 at 13:52
28

I don't think I've ever seen $x \frac{y}{z}$ used to mean $x + \frac{y}{z}$ except when $x$, $y$ and $z$ are literal integers (e.g. $2 \frac{3}{4}$). That's not to say it never happens, but it would be terribly confusing.

Robert Israel
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    in my opinion the whole miced fraction notation should be avoided, the condfusion could be avoided by placing a plus sign. Besides, kids learn so little math that it makes no sense to replace some cool stuff with teaching them stupid notation. – Asinomás Dec 28 '16 at 01:27
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    @JorgeFernándezHidalgo I would agree with you, except that mixed fractions occur often enough in real life that students must be prepared to encounter them. – Robert Israel Dec 28 '16 at 02:18
17

Here is my attempt at a helpful rule:

The expression $x\frac{y}{z}$ always means $x\times\frac{y}{z}$ except when $x,y,$ and $z$ are all integers written in decimal notation; then it means $x+\frac{y}{z}$.

So $n\frac{n^2+5}{4}$ means $n\times\frac{n^2+5}{4}$, but $3\frac14$ means $3+\frac14$.

TonyK
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5

It just depends on context.

In some rare cases $$ a\frac{c}{d}:=a+\frac{c}{d} $$ which is the interpretation in your answer, but mostly $$ a\frac{c}{d}:=a\cdot\frac{c}{d} $$

5

I don't think there can ever be a mixed fraction of the form $n\frac{n^2+5}{4}$ if $n$ $\in$ $\mathbb{N}$. Please note that if it were a mixed fraction then $n^2+5$ would denote the remainder while $4$ is the divisor and this would never be possible as for $n$ $\in$ $\mathbb{N}$, $n^2+5 \gt 4$ always.

Hence, this expression would definitely denote $n\times$$\frac{n^2+5}{4}$.

  • This may be true, but you're assuming that mixed fractions are normalized somehow, but it's not that uncommon to see something like $1\frac35+2\frac45 = 3\frac75 = 4\frac25$. Anyway, having to perform any computation / estimation before you can even start reading a trivial formula is too impractical. In a perfect world, one meaning should die. – maaartinus Dec 28 '16 at 20:23
0

$3 \times \frac{3^2+5}{4} = 3 \times \frac{9+5}{4} = 3\times \frac{14}{4} = 3\times \frac{7}{2} = \frac{21}{2}$

Normally (even though in calculator this is often not true) the convention is that a number on the side of a fraction is multiplying that fraction. There should be a "times" either a cross or a dot, however often you can omit it as a shortcut (in veritas, it is almost always omitted apart from very specific cases when someone wants to emphasise the steps as in my answer above)

Euler_Salter
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-1

Here's the Cliff Notes version. Actually cut and pasted from Cliff Notes: "•Two variables (letters) next to each other: ab means a times b"

This is always the case, even if the variable is identified as an integer. If a=3 and b=4 then ab=12, NOT 34. To understand why 3 1/4 is not 3 x 1/4 it is actually simple - it follows a different rule of notation. when a number is in front of a fraction they are considered parts of a total number that consists of a whole and a fractional part. One final example of notation. Intuitively we know that 34 is thirty-four. it's not 7, ie 3+4, and it's not 12, ie 3x4. The notation is our common base 10 system: so 34 is just an abbreviation for (3x10) + (4x1). Since we do it so often we don't think about it, it's intuitive and we don't look for exceptions. That's how you should treat ab: it's a times b always.

infraprods
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