# Confusion in fraction notation

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

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. $1,2,\dfrac{3}{5},\dfrac{8}{5}$

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 $\frac{xy}{z}$.

• 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? Dec 28 '16 at 12:01
• @Bakuriu: Never seen it? Haven't you heard of this Fellini film? In any case, it's not an Americanism $-$ I am British. Dec 28 '16 at 14:26
• @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 $x\times\frac{y}{z}$. Dec 28 '16 at 16:37
• I can confirm that in elementary schools in Croatia (metric units) $3\frac 14 = 3+\frac 14$. Dec 28 '16 at 16:39
• Also in Poland (metric) - I'm quite sure $3\frac{1}{4} = 3 + \frac{1}{4}$ while $x\frac{y}{z} = x \times \frac{y}{z}$. Dec 28 '16 at 18:50

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.

• 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. Dec 28 '16 at 1:27
• @JorgeFernándezHidalgo I would agree with you, except that mixed fractions occur often enough in real life that students must be prepared to encounter them. Dec 28 '16 at 2:18

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 $3+\frac14$.

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 $$a\frac{c}{d}:=a\cdot\frac{c}{d}$$

• But how does it depend on context? That's the whole point of the question, isn't it? Dec 28 '16 at 12:02

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. Dec 28 '16 at 20:23

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

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)

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.