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# Mixing Concrete

Alignments to Content Standards: 6.RP.A.3

A mixture of concrete is made up of sand and cement in a ratio of 5 : 3. How many cubic feet of each are needed to make 160 cubic feet of concrete mix?

## IM Commentary

In order to solve this problem, students must assume that if you mix a cubic foot of sand with a cubic foot of cement, you will have 2 cubic feet of mix. In reality, the volume of the mixture may actually be less than that as cement particles settle into the spaces between the grains of sand. It is important for students to understand that they must explicitly make this assumption, and that for some contexts this is a reasonable assumption (e.g. mixing water with juice concentrate) and others it is completely inappropriate (e.g. mixing water and salt).

## Solutions

Solution: Ratio table

Constructing a ratio table that shows the amount of sand, cement, and concrete mix (assuming the volumes add):

 Sand 5 10 20 30 40 50 60 70 80 90 100 Cement 3 6 12 18 24 30 36 42 48 54 60 Concrete mix 8 16 32 48 64 80 96 112 128 144 160

We can see in the last column that one needs 100 cubic feet of sand and 60 cubic feet of cement to make 160 cubic feet of concrete mixture.

Solution: Using the scale factor

We know that to make
$k \times 8$ cubic feet of concrete mix, we need
$k \times 5$ cubic feet of sand and
$k \times 3$ cubic feet of cement.

We need 160 cubic feet of concrete mix and
20 x 8 = 160, so we need to use
20 x 5 = 100 cubic feet of sand and
20 x 3 = 60 cubic feet of cement.

In other words, 100 ft3 of sand and 60 ft3 of cement will make 160 ft3 of concrete mix.

#### Nancy says:

over 3 years

Could someone please explain the A.3? I understand the 6.RP. but am trying to interpret the A.3 suffix. Is there a key for this? Thank you in advance for your reply.

#### Cam says:

over 3 years

Hi msnanmac,

It's just the (increasingly universal) shorthand for the standard. As you know, 6.RP puts us in the 6th grade "Ratio and Proportional Relationships" area, then the "A" puts us in the first cluster (there's only one cluster under that particular domain), and the 3 specifies the standard ("Use ratio and rate reasoning...."). Some times there will even be one more part to the designation, a lower-case letter to designate a part of a standard, e.g., 6.RP.A.3b for part b of the standard just cited.

Hope that helps!