9. Get it on Tape (Diagrams)

Tape Diagrams are yet another method for graphically or visually solving ratio-type word problems. It is slightly different than Double Lines or Equivalence Tables.

Tape-type word problems typically include a ratio of two things (two of These to three of Those) plus a bonus of one other number which is typically the total of one of the items.  For example:  The ratio of Apples to Oranges is 4:3.  We have 24 Apples….how many Oranges do we have?  See?  What did I tell you?  This problem gave us the ratio of the two items plus a bonus number of the total of one of the items.

When you set up a Tape Diagram for this type of problem, you simply show the ratio in the form of a bunch of boxes (it’s supposed to resemble some sort of  piece of tape).  

The drawing at right shows two Tape Diagrams, one for a ratio of 3:2 and the other a ratio of 4:3.   

To diagram the 4:3 ratio of apples-to-oranges problem above, we would draw four boxes in a row to resemble the apples, and then three boxes in a row for the oranges. 

Important Safety Tip:  Always make the boxes the same size, and always line them up on the left.

Next, label the rows of boxes.  We can also add the bonus number we know, which from the problem is 24 apples.  Show that number across the top of the Apples. 

The problem is asking for the total number of Oranges…which is the bottom number.

Now here’s the tricky part:  We want to place a number in all of the boxes, and that “magic" number will be the same number in all boxes.  There is only one number which fits (that’s why it is a magic number).  To figure out that one magic number, we look at the row of boxes for the item which we know the total number - in this case it is the total of 24 apples.  Since we have four boxes, and the number is always the same number in all boxes, we divide (in our head) the total of 24 by the number of boxes in that row (4).  

Another way to think of this is:  you have a total of 24 apples in a pile.  You need to put an equal number of apples in each of the four boxes.  You start by putting  one apple in each box and keep repeating until all of the apples are gone.  You would end up with 6 apples in each box.  Hence,  24 divided by four is six.  So the magic number is 6.

Now here’s the fun part: Since the magic number is 6, put a "6" in each of the Oranges boxes.  There are 3 boxes of 6. 

So, again, we do in our head “3 times 6 equals 18” , and write 18 in the bottom total for Oranges.

Another important safety tip:  These types of Tape Diagram problems always lend themselves to using numbers which are easily divisible by other numbers.  We usually do not see a ratio of 4 1/2 to 3 (where it’s tough to draw four and half boxes on top), or a ratio of 4:3 with a total of 23 (where 4 doesn’t divide into 23 very easily).  

Now try a few problems (answers to follow in another follow-on post):

Tape Q1:  We’re making pizza!  The ratio of pepperoni to sausage is 3 to 1.  We used up 24 slices of pepperoni.  How many pieces of sausage did we use?

Tape Q2:   It’s Pizza Day at the cafeteria (can’t get enough of that Pizza).  There are two cafeteria lines - Sixth graders in one line and Fifth graders in the other.  There are two Sixth graders for every three Fifth graders.  You count 40 Sixth graders in your line.  How any Fifth graders are in line?

Tape Q3:  You have major math homework tonight - 30 math problems…YIKES!.  You are also starving for those homemade Chocolate Chip cookies.  Mom says you can have two cookies for every 6 math problems you work out correctly.  You finish all of the math homework correct.  How many cookies do you get?

Good Luck!