This e-mail answers a series of questions that can be asked now that we have completed our survey of citrus fruits.
In my e-mail of 25 Jan 2015, I commented on the small size of the Murcott tangerines by saying that they are "more trouble than they're worth." In addition, in the e-mail on mandarins of 7 Mar 2015, I wondered whether this fruit is "too small to fool with." Exactly what is meant by these quoted statements?
The essence is that the effort that one expends in peeling tiny fruits is not rewarded with sufficient fruit to repay that effort. One way to look at this is that, for these roughly spherical fruits, the area of peel to remove is proportional to the square of the diameter of the fruit, while the volume of the fruit is proportional to the cube. This means that as the fruit gets smaller, you obtain less fruit per unit of peel removed. This in turn means that if the fruit is small enough, the ratio of peel to fruit is so high that it is not worthwhile to spend the time to peel the fruit.
Let's now look into this quantitatively.
Question 1: For each citrus fruit, for each minute spent peeling a fruit, on average how many ounces of fruit are made available for eating?
Answering this question will throw light on assertions such as whether an inordinate amount of time is spent peeling a Murcott or a mandarin. To answer this question, between Valentines Day and San Jacinto Day of this year, I accumulated data on each of the 147 citrus fruits that I ate. (Texas won its independence from Mexico at the Battle of San Jacinto, 21 Apr 1836.) Calculations made with this data are reported in Table 1, which contains the following columns of information about each type of citrus fruit that I ate during the study period.
- (1): The PLU code for the fruit. If one wants to know more about the fruit that is reported on, one can look up a PLU code at the PLU site or go the the relevant Fruit Explorer e-mail shown in Table 2.
- (2): The sample size. This is the number of fruits of this type that I ate during the study period and for which I gathered data.
- (3): Peeling method. I used two different methods, depending on the fruit.
- Quarter: I cut off the ends of the fruit, made four equally spaced cuts through the peel from the top of the fruit to the bottom, and peeled off each section of peel. I used this method whenever the peel could not be easily removed with just my thumbs..
- Spiral: I used my thumbnail to break the peel at the top and worked my thumb around the peel while working down in a spiral pattern until I had removed the entire peel. This left me with a peel star. In some cases, I used this method because it was faster, e.g., the gold nugget. In other cases, I was forced to use this method since the peel clung so tightly to the fruit.that I could not use the quartering method, e.g., mandarin and temple orange.
- (4) The average weight of the unpeeled fruit. I measured this on my kitchen scale to the nearest twentieth of an ounce.
- (5): The average weight of the edible fruit. I measured this on my kitchen scale to the nearest twentieth of an ounce. The edible fruit was obtained by discarding the peel as well as any inedible portion of the fruit, which for the most part was the central column (14 Mar 2015) and the secondary fruit of a navel orange (27 Mar 2015); see picture below. Note that only edible fruit is considered; this this is the payload that one eats.
- (6): The average number of seconds it took me to peel the fruit, as measured by my wristwatch. I peeled the fruit at my normal speed and did not go into my ginzu knife routine to speed up the peeling.
- (7): The average weight of the edible fruit that was obtained with one minute of peeling. This is obtained by dividing column (5) by column (6) and dividing by 60. The data in this column will be referred to as peeling efficiency.
Note that the first three columns are general information, the next three are raw data that I collected separately for each fruit, and the last, which is calculated from the raw data, is the answer to Question 1.
Table 1: Data on Peeling Citrus Fruits, 14 Feb - 21 Apr 2015
Fruit
|
PLU #
(1)
|
Sample Size
(2)
|
Peeling Method
(3)
|
Average Unpeeled Weight Per Fruit (oz)
(4)
|
Average Edible Citrus Per Fruit
(oz)
(5)
|
Average Peeling Time Per Fruit
(sec)
(6)
|
Average Edible Citrus Produced by One Minute of Peeling (oz/min)
(7)
|
Mandarin
|
?
|
44
|
Spiral
|
1.9
|
1.5
|
34
|
2.7
|
Large Navel Orange
|
4012
|
31
|
Quarter
|
11.2
|
7.5
|
44
|
10.4
|
Cara cara Navel Orange
|
3110
|
41
|
Quarter
|
12.3
|
8.4
|
45
|
11.1
|
Small Navel Orange
|
4013
|
18
|
Quarter
|
6.0
|
4.1
|
46
|
5.3
|
Jaffa Navel Orange
|
4390
|
3
|
Quarter
|
4.9
|
3.4
|
56
|
3.6
|
Gold Nugget
|
3386, 4055
|
4
|
Spiral
|
7.4
|
5.3
|
26
|
12.2
|
Childhood Tangerine
|
?
|
2
|
Spiral
|
4.4
|
3.4
|
71
|
2.9
|
Minneola tangelo
|
4383
|
1
|
Quarter
|
7.7
|
5.4
|
47
|
6.8
|
Tangelo
|
4456
|
1
|
Spiral
|
7.0
|
5.8
|
65
|
5.4
|
Temple orange
|
4387
|
1
|
Spiral
|
5.8
|
4.8
|
70
|
4.1
|
Pixie Tangerine
|
?
|
1
|
Spiral
|
4.4
|
3.3
|
21
|
9.3
|
The reader is invited to inspect the raw data, which is shown in the attached spreadsheet; this spreadsheet contains the data that I collected for each of the 147 test fruits as well as the calculations of the averages and other figures not used in this e-mail. (For the analysis in this e-mail I have at every step used the simplest assumption. There is more in this data than is reported here, and the interested reader is invited to re-analyze the data.) This spreadsheet also explains how I dealt with the complications that always arise when working with real data.
Compare the weight of the unpeeled fruit in column (4) to the amount of edible fruit produced by a minute of peeling in column (7). As expected, the tiny mandarin is by far the most inefficient fruit since each minute of peeling produces only a pittance of 2.7 ounces of edible fruit. As expected, the large navel orange and the cara cara are highly efficient in that a minute of peeling produces 10.4 and 11.1 ounces, respectively, of edible fruit. The surprise is that the most efficient fruit is the medium-sized gold nugget, which produces 12.2 ounces of edible fruit per minute of peeling. The explanation is that the peel of the gold nugget is very loose, and one can easily rip it off with one's bare hands. If you are hungry and really want to tear into a citrus fruit, the gold nugget cannot be beat.
It is obvious, however, that peeling efficiency is not all that matters. The cost of the citrus fruit also matters. (By "cost" I mean the cost of purchasing the fruit.)
Question 2: For each citrus fruit, what is the average cost of an ounce of edible citrus?
This question is answered by Table 2, which contains the following columns of information about each type of citrus fruit that I ate during the study period. .
- (1) The date of the Fruit Explorer e-mail that covers each fruit.
- (2): The average amount of edible citrus that one gets from one fruit; this is taken from column (5) of Table 1.
- (3): The average cost per fruit. These costs are taken from my e-mails given in column (1), except for the large navel and cara cara, for which I used the typical prices found in stores.
- (4): The average cost of an ounce of edible citrus. This is calculated by dividing column (3) by column (2).
The last column of Table 2 answers Question 2.
Table 2: Average Cost Per Ounce of Edible Citrus Fruit
Date of E-mail Covering this Fruit
(1)
|
Average Edible Citrus Per Fruit (oz)
(2)
|
Average Cost per Fruit ($)
(3)
|
Average Cost per Ounce of Edible Citrus ($/oz)
(4)
| |
Mandarin
|
7 Mar 2015
|
1.54
|
0.16
|
0.10
|
Large Navel Orange
|
None
|
7.52
|
1.00
|
0.13
|
Cara cara Navel Orange
|
8 Feb 2015 (but not used)
|
8.36
|
1.25
|
0.15
|
Small Navel Orange
|
4 Apr 2015
|
4.06
|
0.36
|
0.09
|
Jaffa Navel Orange
|
4 Apr 2015
|
3.37
|
0.71
|
0.21
|
Gold Nugget
|
4 Apr 2015
|
5.25
|
1.15
|
0.22
|
Childhood Tangerine
|
4 Apr 2015
|
3.35
|
0.46
|
0.14
|
Minneola tangelo
|
28 Feb 2015
|
5.35
|
0.81
|
0.15
|
Tangelo
|
4 Apr 2015
|
5.80
|
0.99
|
0.17
|
Temple orange
|
4 Apr 2015
|
4.75
|
0.32
|
0.07
|
Pixie Tangerine
|
28 Feb 2015
|
3.25
|
0.97
|
0.30
|
If we set aside the temple orange, which I reported as having an unappealing taste, then the small navel orange at 9 cents per ounce and the mandarin at 10 cents per ounce are the cheapest fruits. My workhorse fruits, the large navel and the cara cara, come in at 13 and 15 cents per ounce, respectively. Sad to say, the gold nugget, which column (7) of Table 1 showed to be the most efficient orange to peel, is the second most expensive orange at 22 cents per ounce. It should be stressed that this cost data is only illustrative since the prices of fruits vary over the season. Moreover, even at a given time, prices vary significantly between stores. (Quality also differs between stores, but discussing quality is beyond the scope of this e-mail.) Finally, prices vary over time within the same store since sales on citrus fruits are common. For example, this table is based on prices of $1.99/pound for gold nuggets and $1.00 each for large navel oranges; at Market Basket on 23 Apr 2015, however, I got gold nuggets for $1.49/pound and large navel oranges for 69 cents each. In short, opportunistic shopping can improve upon these figures.
The purchase price of the fruit, however, is not the only cost. If your time has value, then there is an implied cost due to the value of the time that you spend peeling the fruit. Then the total cost is defined by:
Total cost = cost of the fruit + value of the time spent peeling the fruit.
Question 3: For each citrus fruit, on average what is the total cost of an ounce of that fruit?
We will answer this question in two steps. The basic data needed to answer this question is shown in Table 3, which has the following columns.
- (1): The average cost of an ounce of edible fruit. This is from column (4) of Table 2.
- (2): The average number of minutes needed to peel an ounce of edible fruit. This column in obtained by taking the inverse of column (7) of Table 1. (That is, divide one by an entry in column (7) of Table 1 to get the corresponding entry in column (2) of Table 3.
- (3): The average number of hours that it takes to peel one ounce of edible fruit. This is obtained by dividing column (2) by 60. Note: All calculations reported in this e-mail are carried out to eight significant digits and then rounded.
Table 3: Basic Data Needed to Calculate Average Total Cost
Average Cost per Ounce of Edible Citrus Fruit ($/oz)
(1)
|
|Average Number of Minutes Needed to Peel One Ounce of Edible Fruit (min)
(1)
|
Average Number of Hours Needed to Peel One ounce of Edible Fruit (hr)
(3)
| |
Mandarin
|
0.10
|
0.37
|
0.0062
|
Large Navel Orange
|
0.13
|
0.10
|
0.0016
|
Cara cara Navel Orange
|
0.15
|
0.09
|
0.0015
|
Small Navel Orange
|
0.09
|
0.19
|
0.0031
|
Jaffa Navel Orange
|
0.21
|
0.28
|
0.0046
|
Gold Nugget
|
0.22
|
0.08
|
0.0014
|
Childhood Tangerine
|
0.14
|
0.34
|
0.0057
|
Minneola tangelo
|
0.15
|
0.15
|
0.0025
|
Tangelo
|
0.17
|
0.19
|
0.0031
|
Temple orange
|
0.07
|
0.24
|
0.0041
|
Pixie Tangerine
|
0.30
|
0.11
|
0.0018
|
Now we are in a position to answer the question on the total cost of an ounce of fruit. A complication is that the total cost depends on the value that is placed on one's time. Table 4 shows the total cost of a ounce of fruit for various values of time ranging from $0 per hour, which means that you put no value on your time, to $50 per hour. To get an entry in Table 4, multiply the corresponding entry in column (3) of Table 3 by the value of time and add to column (1) of Table 3. Table 4 answers Question 3.
Table 4: Average Total Cost of One Ounce of Citrus Fruit as a Function of the Value of Time
Fruit
|
Average Total Cost of One Ounce of Citrus Fruit as a Function of the Value of Time ($)
| |||||
$0/hr
|
$10/hr
|
$20/hr
|
$30/hr
|
$40/hr
|
$50/hr
| |
Mandarin
|
0.10
|
0.16
|
0.23
|
0.29
|
0.35
|
0.41
|
Large Navel Orange
|
0.13
|
0.15
|
0.17
|
0.18
|
0.20
|
0.21
|
Cara cara Navel Orange
|
0.15
|
0.16
|
0.18
|
0.19
|
0.21
|
0.22
|
Small Navel Orange
|
0.09
|
0.12
|
0.15
|
0.18
|
0.21
|
0.25
|
Jaffa Navel Orange
|
0.21
|
0.26
|
0.30
|
0.35
|
0.40
|
0.44
|
Gold Nugget
|
0.22
|
0.23
|
0.25
|
0.26
|
0.27
|
0.29
|
Childhood Tangerine
|
0.14
|
0.19
|
0.25
|
0.31
|
0.37
|
0.42
|
Minneola tangelo
|
0.15
|
0.18
|
0.20
|
0.23
|
0.25
|
0.27
|
Tangelo
|
0.17
|
0.20
|
0.23
|
0.26
|
0.29
|
0.33
|
Temple orange
|
0.07
|
0.11
|
0.15
|
0.19
|
0.23
|
0.27
|
Pixie Tangerine
|
0.30
|
0.32
|
0.33
|
0.35
|
0.37
|
0.39
|
To see how to use this table, compare the mandarin and the cara cara. If you put no value on your time, then column (1) applies, and the mandarin's total cost of 10 cents per ounce is much less than the cost of the cara cara of 15 cents per ounce. However, as column (7) of Table 1 shows, the cara cara can be peeled much more efficiently than the mandarin, so as the value that you put on your time increases, the total cost of the mandarin increases much more rapidly than that of the cara cara. This means that if you place a value of $50 per hour on your time, then the mandarin with a total cost of 41 cents per ounce is much more expensive that the cara cara at 22 cents per ounce. As the value of time goes from $0 to $50 per hour, the mandarin goes from being the second cheapest ot the second most expensive. This shows the importance of the value of time.
For another example, consider the large and small navel oranges. Table 4 shows that if the value of time is zero, then the small navel has a total cost advantage per ounce over the large navel of 4 cents, i.e., 13 cents minus 9 cents. As the value of time rises, this cost advantage narrows, until the total cost of these two oranges is the same at a value of time of $30/hr. As the value of time continues to increase, the total cost advantage shifts to the large navel orange; for example, if the value of time is $50/hr, then the large navel has an advantage in total cost over the small navel of 4 cents per ounce, i.e., 25 - 21 cents per ounce. Again, this shows the importance of peeling efficiency. The lower peeling efficiency of the small fruit means that its cost rises more rapidly as the value of time increases.
Also notice that as the value of time increases, in the limit all that matters in minimizing total cost is peeling efficiency. For example, one can calculate that at any value of time above $700 per hour, the gold nugget, which has the highest peeling efficiency, is the fruit with the lowest total cost even though it is the second most expensive in terms of cost per edible ounce. Yet again, this shows the importance of the value of time. This result, however, is probably of limited applicability in the real world since a person who valued his time at more than $700 per hour probably would not peel his own fruit.
Another consideration is that the value that you place on your time might not be a constant but might be a variable that depends on the situation. For example, if you peel oranges while watching TV, it might be that you assign a value of zero to your time since the peeling does not take you away from any activity.
Of course, total cost per ounce isn't everything. Even if you place a lofty value on your time, perhaps you only want a small taste of citrus, and a mandarin can provide it. Perhaps you especially like the taste of a mandarin. Perhaps, after peeling a mandarin, you like to study the peel star and play with it; I myself sometimes hurl one around like a frisbee. Therefore, the results in Table 4 are to be used as a factor in your decision making but not as an ironclad command.
I did not eat a pomelo during the study period, but my e-mail of 19 Oct 2015 implies that the cost of an edible ounce of peeled fruit was 15 cents for the one that I bought. This is a competitive cost, as one can see from column (4) of Table 2. If, however, one puts much value on one's time, then the total cost of the pomelo soars since, with its three-peel construction, it takes a maddeningly long time to peel. The pomelo is an example of a large fruit with a very low peeling efficiency; it is anomalous since its structure makes it unusually hard to peel.
A further aspect that perhaps should have been taken into account is the eating efficiency. Once a fruit is prepared and ready to eat, how much of that fruit can you eat per minute? The main thing I have in mind is that for fruits that are seed-laden, eating them takes a long time since you have to deal with the seeds, and, at least for me, this detracts significantly from the enjoyment of the fruit. Seeds, however, are not the only impediment. Recall that the fruit of the uniq (27 Jan 2015) is so fibrous that it takes a long time to chew a segment, even after correcting for size. I did not follow up on this idea of eating efficiency since almost all the fruits eaten during my study period were seedless with approximately the same consistency.
I have not been able to find anything in the literature to which I can compare my results.
Here's your party tip. Provide your guests with fruits and pads of paper so that as they consume the fruit they can compile their own raw data on unpeeled weight, peeled weight, and peeling time. They can then calculate peeling efficiency. Tell them the price of the fruit so that they can calculate the cost of an ounce of edible fruit. If they assume a value for their time, they can calculate the total cost of an ounce of fruit. Your guests will be delighted to see that all it takes is addition, subtraction, multiplication, and division to give them a deep quantitative understanding of their eating habits. Challenge your guests to apply this new knowledge to their everyday lives.
This is the last planned e-mail on citrus. It is always possible that I will stumble upon some as-yet untasted citrus fruit, but it looks like our long day's journey into citrus has come to an end. Other types of fruit, however, beckon.
Rick
P.S. On 24 April 2015, when I entered the shower stall at the gym after working out, I found that a previous user had left a squeeze bottle of Family Dollar Invigorating Apricot Scrub. The front of the bottle stated, "Exfoliates to reveal smoother and healthier skin." The bottle also stated, "100% Guaranteed." I carefully inspected the bottle but could not discover what was guaranteed. The bottle said that this product should be compared to St. Ives Apricot Scrub, which is pictured in the apricot e-mail of 1 Feb 2015. One is left to wonder what the result of such a comparison would be. I thought that my readers might like a report on what this product was like, but I resisted the temptation to use it.
