# Math Help - Applied math problem: library collection development

1. ## Applied math problem: library collection development

First of all, this is my first post here. I struggled with forum selection. If there is a more appropriate forum, I hope mods will assist me in moving the thread.

I am using "relative use" to determine a "demand" variable to assist me in allocating my library materials budget between various component collections.

u = number of times books are picked up or "used" for each component collection
v = number of volumes, or books, in a component collection
U = total number of uses for all component collections
V = total number of volumes in all component collections
r = relative use, or (u/v)/(U/V)
d = demand, or (u*r) for each component collection.

The problem is that it is not behaving how I would expect: the sum total of the demands of all component collections does not equal the demand for the entire collection as a whole. Where is my math going wrong, if this expectation is correct?

Here are my numbers (space delimited, please tell me if there is a better way to display this). I've been using a spreadsheet for this.

Collection u v r d
Fiction 15716 8818 1.13 17765
LP Fiction 948 725 0.83 786
Mystery 6150 3822 1.02 6276
LP Mystery 220 219 0.64 140
SciFi 1639 1704 0.61 1000
SS 358 403 0.56 202
Western 458 475 0.61 280

Total Demand: 25,489
Sum of component Demands: 26,449
Shouldn't these be the same?

2. Hello, istril!

I am using "relative use" to determine a "demand" variable to assist me
in allocating my library materials budget between various component collections.

$\,u$ = number of times books are picked up or "used" for each component collection
$\,v$ = number of volumes, or books, in a component collection
$\,U$ = total number of uses for all component collections
$\,V$ = total number of volumes in all component collections
$\,r$ = relative use, or $\frac{u}{v}$
$\,d$ = demand, or $u\cdot r$

The problem is that it is not behaving how I would expect:
the sum total of the demands of all component collections does not equal
the demand for the entire collection as a whole.
Where is my math going wrong, if this expectation is correct?

Here are my numbers.

. . $\begin{array}{|c||c|c|c|c|}
\text{Collection} & u & v & r & d \\ \hline \hline
\text{Fiction} & 15716 & 8818 & 1.13 & 17765 \\
\text{LP Fiction} & 948 & 725 & 0.83 & 786 \\
\text{Mystery} & 6150 & 3822 & 1.02 & 6276 \\
\text{LP Mystery} & 220 & 219 & 0.64 & 140 \\
\text{SciFi} & 1639 & 1704 & 0.61 & 1000 \\
\text{SS} & 358 & 403 & 0.56 & 202 \\
\text{Western} & 458 & 475 & 0.61 & 280 \\ \hline \end{array}$

Where did the $\,r$- and $\,d$-values come from?
. . They are all wrong!

You said: . $d \,=\,u\cdot r$

Then in the first row (Fiction), we should have:

. . $d \:=\:u\cdot r \:=\:15,\!716 \times 1.13 \:\approx\:17,\!759$

Even if the $\,r$-values are correct, the $d\,$-values are off.
. . But all the $\,r$-values are wrong!

You said: / $r \,=\,\dfrac{u}{v}$

In the first row (Fiction), we should have:

. . $r \:=\:\dfrac{u}{v} \:=\:\dfrac{15,\!716}{8,\!818} \:\approx\:1.78$

3. Hello Soroban, thank you for replying.

Allow me to clarify; I am using a spreadsheet for this, and only the first two decimal places are shown. r=1.13 should actually be r=1.13037281097007. This should account for the discrepancy you saw in d.

Secondly, r does not equal u/v; it equals (u/v)/(U/V), or, (for the first row

r = (15716/8818)/(25489/16166) = 1.13 approx

Also, which tags are you using to format the numbers so nicely? Is there a help file for this?

EDIT: Allow me to further clarify, the descriptions of the variables U and V I gave above are correct, but their values are as follows:

U=25489
V=16166