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# Canola oil and olive oil mixture problem

While shopping for groceries, you notice a new cooking oil product consisting of a blend of canola oil and olive oil. The label does not say in what proportions the oils are mixed, but the Nutrition Facts panel says that a 14 gram serving has 1 gram of saturated fat, 9 grams of monounsaturated fat, and 4 grams of polyunsaturated fat. What percentage of olive oil does the blend contain?

Hint: Use the fatty acid composition table to solve the problem.

To calculate the proportion of the constituents of the blend, we need to find out the fatty acid profiles of canola oil and olive oil, and then determine what combination would yield the fatty acid profile of the mixture. We start by looking at the fatty acid composition table:

Percent by weight of total fatty acids.
 Oil or Fat Unsat./Sat.ratio Saturated Monounsaturated Polyunsaturated Capric Acid C10:0 Lauric Acid C12:0 Myristic Acid C14:0 Palmitic Acid C16:0 Stearic Acid C18:0 Oleic Acid C18:1 Linoleic Acid (ω6) C18:2 Alpha Linolenic Acid (ω3) C18:3 Canola Oil 15.7 - - - 4 2 62 22 10 Olive Oil 4.6 - - - 13 3 71 10 1

We calculate the percentage of saturated (%sat), monounsaturated (%mono), and polyunsaturated (%poly) fat for the mixture, and tabulate the saturation profiles for the three oils.

Calculation of the fatty acid percentages for the oil mixture:

%sat = 100×1/14 = 7%
%mono = 100×9/14 = 64%
%poly = 100×4/14 = 29%
Saturation profiles for Canola oil, Olive oil, and the unknown mixture
Oil  %sat   %mono   %poly
Canola 6 62 32
Olive 16 71 11
Mixture 7 64 29

We notice that the percentages do not add to 100% for olive oil due to rounding and other constituents not listed. This just means that our answer will not be as accurate as it could be with better analytical data.

If X represents the fraction of olive oil in the mixture, (1-X) is the fraction of canola oil. The proportion of the components can be calculated for the saturated, monounsaturated, and polyunsaturated components separately to arrive at three different answers.

Calculation of the proportion of olive oil and canola oil required to produce the specific percentage of fatty acids in the unknown mixture. The coefficients correspond to the percentages in the columns.

Calculation based on saturated fat:
6(1-X) + 16X = 7
6 - 6X + 16X = 7
10X = 1
X = 1/10 = 0.10

Calculation based on monosaturated fat:
62(1-X) + 71X = 64
62 - 62X + 71X = 64
9X = 2
X = 2/9 = 0.22

Calculation based on polyunsaturated fat:
32(1-X) + 11X = 29
32 - 32X + 11X = 29
-21X = -3
X = -3/-21 = 0.14

The three different calculations indicate that the fraction of olive oil may range from 0.10 to 0.22. An average of the three values yields:

(0.10 + 0.22 + 0.14)/3 = 0.15

A fractional value of 0.15 corresponds to 15 percent of olive oil. This is as good an estimate as we can make with the data that we have.