Using Pictures Tables and Venn Diagrams in Word Problems Study Guide
Find practice problems and solutions for these concepts at Using Pictures, Tables, and Venn Diagrams in Word Problems Practice Problems.
We can apply the eight-step process we developed in the last chapter to solve any word problem, but sometimes it is easier to draw a picture or create a table. Pictures and tables can help us see exactly what a question is asking us.
Drawing a Picture
Sketching a word problem can help us decide which operation to use and what numbers to plug into an equation.
A piano has 88 keys that are either black or white. If 36 keys are black, how many are white?
First, let's use the eight-step process:
- Read the entire word problem.
- Identify the question being asked.
- Underline the keywords.
- Cross out extra information and translate words into numbers.
We are given the total number of keys on a piano and the number of keys that are black.
We're looking for the number of keys that are white.
There are no keywords in this problem.
There is no extra information to cross out, and there are no words that need to be translated into numbers.
So far, the eight-step process hasn't moved us much closer to solving the problem. Let's try drawing a picture. It's fine if you're not an artist; the picture just needs to represent accurately the problem. We are told that the piano has 88 keys, so start by drawing a piano with 88 keys:
We know that all the keys are either black or white, and we know that 36 keys are black, so color 36 of the keys black. It doesn't matter if the picture looks exactly the way a real piano would look; we just need 36 of the 88 keys colored black:
Now we can see that the remaining keys on the right side of the piano are the white keys. By counting them, we can find that there are 52 white keys.
After coloring 36 of the keys black, you may have realized that this was a subtraction problem. The number of white keys is equal to the total number of keys minus the number of black keys: 88 – 36 = 52. The picture is meant to help you as far as you need help. Once you realize what must be done to solve a problem, you can leave your picture and find the answer.
A bookshelf has six shelves. If there are 16 books on each shelf, how many books are on the bookshelf?
The keyword each tells us that this is likely either a multiplication or division problem, but a picture may help us solve this problem faster than the eight-step process. Draw a bookshelf with six shelves and put 16 books on each shelf:
We can now see that there are many books on this shelf—many more than 16. We won't use division or subtraction; we need an operation that increases 16 by 6 times. Counting the books on the shelf would give us the total number of books, but the bookshelf itself represents 16 multiplied by 6: 16 × 6 = 96. There are 96 books on the shelf.
Building a Table
We don't always need to draw a picture. Often, a table can be just as helpful. In the bookshelf example, we drew six shelves and placed 16 books on each shelf. We could have drawn a table and placed the numbers of books in the table:
The table, just like the picture, helps us to see that we have six sets of 16. The best way to find the total number of books is to multiply 6 by 16: 6 × 16 = 96 books.
The type of table we build depends on the information in the word problem. Tables are especially good for organizing multiplication problems, as we just saw. But they are also good for solving other kinds of problems.
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