Pikudei- Mixed Measures

"All the gold that was used for the work- for all the labor of the Sanctuary- the offered-up gold was twenty-nine kikar and seven hundred thirty shekel, in the shekel of the Sanctuary. The silver of the accountings of the assembly, a hundred kikar, one thousand seven hundred seventy-five shekel, in the shekel of the Sanctuary." ~Shemot 38;24-25

"The offered-up copper was seventy kikar and two thousand four hundred shekel." ~Shemot 38;29

Rashi on 38;24 explains that a kikar of the Sanctuary is three thousand shekalim, and, therefore, the individual shekalim that are listed are mentioned because together they don't add up to one kikar.

What is a "mixed measure"?:
Understanding how to measure with standard measurements is a basic skill that students begin practicing with from an early age. How many feet long is the room? How many cups of water are in the pitcher? But, what happens when you don't have enough to complete an exact measurement with the measure that you're using? What if the room is between 10 and 11 feet long? What if the pitcher has between 5 and 6 cups of water? Sometimes we use fractions to indicate these middle measures. However, we also have other, smaller measurements that can be used to measure these partial portions. We have a standard equivalency that 12 inches are equivalent to 1 foot, for example. So, if my room is between 10 and 11 feet long, I can measure the "extra" length using inches. Maybe my room is 10 feet and 3 inches. This type of measurement is known as a mixed measure. To simplify a measurement out of a mixed measure, we could convert the mixed measure into the larger measurement using a mixed number (a whole number with a fraction) or decimal, or we could convert the mixed measure into the smaller measurement. With our room example, 10 ft 3 in would be 10 1/4 ft or 10.25 ft when measured only in feet, or 123 inches when measured only in inches. For younger students, the calculation for converting the mixed measure into the smaller unit (from feet and inches to just inches) is an easier one and also helps to give them a better sense of the full meaning of the number, where mixed measures and mixed numbers are often harder for them to visualize and fully understand.

Let's calculate:
So how do we convert to the smaller measure unit? Let's use our 10 ft 3 in room example. We know that 1 foot is made up of 12 inches. So, we multiply 10 x 12 (10 feet, with 12 inches in each) = 120 inches. Then, we add the extra 3 inches that were left over, so 120 + 3 = 123 inches.

What does this have to do with the parsha?:
In this week's parsha, Rashi helps us understand that the weight measurements that we're given for the gold, silver, and copper that were donated to the mishkan are mixed measures using kikar and shekalim. Rashi explains that 1 kikar = 3,000 shekalim.

The amounts that we are given in the parsha are:
Gold: 29 kikar and 730 shekel
Silver: 100 kikar and 1,775 shekel
Copper: 70 kikar and 2,400 shekel

How can we convert these to all shekalim measures?
We know that 1 kikar has 3,000 shekalim, so:
Gold: 29 x 3,000 (29 kikar, with 3,000 shekalim in each) = 87,000 + 730 shekel = 87,730 shekel
Silver: 100 x 3,000 (100 kikar, with 3,000 shekalim in each) = 300,000 + 1,775 shekel = 301,775 shekel
Copper: 70 x 3,000 (70 kikar, with 3,000 shekalim in each) = 210,000 + 2,400 shekel = 212,400 shekel

So, our final calculations after converting from mixed measures to just shekalim are:
Gold: 87,730 shekel
Silver: 301,775 shekel
Copper: 212,400 shekel

A further thought:
The silver calculation is particularly meaningful, since we are told "a beka for every head, a half-shekel in the shekel of the Sanctuary for everyone who passed the counters, from twenty years of age and up, for the six hundred three thousand, five hundred fifty." (28;26) So, our silver calculation should be accurate for 603,550 men 20 years of age and older having each given a 1/2 shekel. Does this work? 
301,775 shekel of silver x 2 (two 1/2-shekel pieces for each shekel) = 603,550 1/2-shekel pieces.
So, it seems that there isn't even estimation here, but rather the calculations work exactly!