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## Brainteaser: Egg-cellent farming

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Difficulty: Fun

Farmer Wallace packages his eggs in cartons that hold a dozen eggs (there are 12 eggs in a dozen).

In March, he sells 5 standard orders of eggs to the local grocery store. Each standard order consists of 4 grosses of eggs. A gross is 144 eggs.

How many cartons of eggs did Farmer Wallace sell?

## Need a hint?

Try working backwards. How many cartons are in a gross? And then from there, how many cartons are in a standard order?

Farmer Wallace sold 240 cartons of eggs.

How many cartons does it take to make a gross? To find this, we need to divide 144 eggs by 12 eggs, which equals 12. It takes 12 cartons to make a gross.

Next, we can figure out how many cartons are in a standard order. A standard order is 4 times a gross. We know 1 gross is equal to 12 cartons, so we can multiply 4 times 12 to get 48 cartons in a standard order.

Finally, we can figure out how many cartons are in 5 standard orders by multiplying 5 by 48, which equals 240 cartons!

This sort of calculation can be tricky. How do you know when to divide or multiply? Here’s a trick using fractions.

To start, we can write our conversion factor as an equation: 1 gross = 12 dozen
Next, we will turn this into a fraction by putting one number on top of the other:

1 gross/12 dozen

And we know that the top and bottom of this fraction are the same, so the fraction equals 1.
We can also do it the other way, and we’ll still get 1:

12 dozen/1 gross = 1

Now we’re ready to convert! Write the number you have to convert and the new unit you want:

4 gross = ? dozen

Next, we can multiply by our number by one of our fractions. Because the fractions are equal to 1, this keeps the quantity the same while changing the units. Here’s the trick: put the units you want on top and the units you don’t want on the bottom:

4 gross * 12 dozen/1 gross

The unit “gross” cancels out, giving you the answer in dozens:

4 * 12 dozen ÷ 1 = 48 dozen

We can use this fraction trick to solve the whole problem working forwards from 5 standard orders:
5 standard orders * 4 gross/1 standard order * 144 eggs/1 gross * 1 dozen/12 eggs * 1 carton/1 dozen

Which simplifies to: 5 * 4 * 144 ÷ 12 * 1 carton = 240 cartons

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