Answers to the Pyramid Fraction Exercise

 The problem is to build a Pyramid using small blocks of tan and white starting with a base of 9 x 9 blocks and building up from there.  There must be a ratio of 1 out of 5 blocks in the Pyramid being tan, and the remaining white.

The problem also states that the blocks will be manufactured using a batch method.  The "Mixed" batch will make 3 tan blocks and 1 white one.

The other batch will make all white blocks.

The question is then how many "Mixed" batches will be required to make enough Tan blocks?...and correspondingly, how many regular batches will be required to make up the remaining white blocks?

The first step is to calculate the total number of blocks required.  The problem almost gave us the answer:

Adding up each layer, the total is 140 blocks.

The second step is to calculate the total number of tan blocks required.  Since the ratio is 1 tan block for every 5 blocks, the total number of tan blocks required is 28.

Now the hard part...how many mixed batches are required?

The are 3 tan blocks per one mixed batch.  Divide this into the total number of tan blocks required to find the number of mixed batches. (we are trying to find the number of "groups of three" fit inside the number 28).

Since there are a little over 9 mixed batches required, we best make 10 batches...and eat the left over blocks...unless you painted them, then never mind.

Now to find the white ones:  We know there are 140 total blocks, and 28 of those are tan.  hence, the total number of white blocks are 

140 - 28 = 112 white blocks

We previously made 10 "mixed" batches which each had 1 white block.  No fraction division required here, we know we have 10 white ones from the mixed batches.

112 - 10 = 102 white blocks remaining

We repeat the division process with the regular batches using the ratio of 4 white blocks per batch:

We need 25 1/2 regular batches, so we might as well make 26 and find a use for the left overs.

Just a Fraction of a Fraction

Fractions can be tough, particularly when you are trying to add, subtract, multiply, or heaven forbid divide them!  …and speaking of divide…why in the world would you want to divide two different fractions?  We’ll look at a few examples below.

The key to remember with fraction division is that the fraction below the line (that is, the denominator) is inverted then multiplied against the other fraction which is above the line.  For example:

Fractions Q1:  

Dad is barbecuing ribs again, and he sends you to the store for his favorite BBQ sauce:  "Bubba’s Down Home That’s What I’m Talking About BBQ Sauce”.  He already pre-paid for the ribs, but you need to buy enough sauce for the quantity of ribs he ordered.  He says is uses 3/4 of a cup of BBQ sauce for every 1/2 pound of ribs.  When you get to the store, you find out he ordered 5 1/2 pounds of ribs.  

How many cups of sauce do you need to buy?  What is the final ratio of cups per pound?

A:  The question is for a quantity of sauce, so set up the fraction with the cups of sauce on top to get cups/pound:  

We now know there are 1 1/2 cup for every pound of ribs.  Now find how much sauce we need for 5 1/2 pounds of ribs:

We need 8 and 1/4 cups of sauce.

Fractions Q2:  For every 1/2 cup of "Bubba’s Down Home That’s What I’m Talking About BBQ Sauce” you buy, Dad also adds 1/4 tablespoon of chili powder.    He has about 4 table spoons of chili powder at home, and asks you to pick up the remaining amount at the store.  How many table spoons of chili powder do you need to buy?

A:  The question is asking for an amount of chili powder, so set up the fraction with Chili powder on top:

Total amount of chili powder required = 

16 1/2 cups of sauce x 1/2 tbls of chili / cup of sauce = 8 1/4 tbls of chile

Amount of chili powder to buy = 8 1/4 tbls (total required)  - 4 tbls (at home) 

= 4 1/4 tbls of chili powder to buy


Fractions Q3:  Pyramid Construction

The class is building a pyramid using small blocks (such as sugar cubes) to resemble the stones.  The Pyramid will start with a base of 9 x 9 blocks, then 8 x 8, 7 x 7, and so on.

Levels of the Pyramid:

The Pharaoh for which this Pyramid is being constructed was well respected, so we will honor him with a Pyramid having  some color

consisting of a few tan blocks mixed in with the white ones (the class can use natural or brown sugar cubes, or paint, etc. for the tan colored blocks).

The ratios are:

• 1 Tan block in every 5 blocks
• 4 white blocks in every 5 blocks

(in other words, out of every group of 5 blocks, 1 of those will be tan, and the remaining four will be white)

We're going to emulate the ancient method of manufacturing blocks (actually used to make old adobe missions) by using a batch method - 4 blocks per batch.  One batch of blocks (the "mixed" batch) will make 3 tan blocks and 1 white block .  The other batch will make all 4 white blocks.

Hence, there is a ratio of 3 Tan blocks per 1 Mixed Batch.

One half of the class will be responsible for the "mixed" batches.  The other half of the class will be responsible for the regular all-white batches.

How many "mixed" batches will be required to make enough tan blocks to meet the required quantity? 

How many regular batches of white blocks will be required to make the remaining amount of blocks for the Pyramid?

Note:  if you come up with an answer which is not an integer, but instead includes a fraction, then just round up to the next integer.