Drawing Models: More Examples

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Here are more questions to show you how to apply the Drawing Models method. I hope that the explanations give you enough insight into the method for you to apply to other types of questions.

I bought 6 pens and some books. Each pen costs $3 and each book costs $7. How many books did I buy if I paid $74 in all?

This question can be solved in several ways. I will show you how to solve it using the drawing models method. Then you can compare this method to the method you're used to.

Before we start drawing we organise the information given:

- we know the unit cost of both the pen and the book
- we know the number of pens bought
- we know the total cost of all the pens and books
- we need to find out the number of books bought

Let's start by drawing the smallest parts: 1 pen and 1 book.

(When you write the numbers it is optional whether to include the $ sign or not. I prefer not to as it makes the relationship of the numbers easier to understand.)

Now we add in the other 5 pens.

There is a problem with the books because we don't know how many there are. So we draw it this way with dotted lines:

Remember that the total cost of everything ($74) should be written at the side so there's no confusion.

The next step is to work out the total cost of all the pens using multiplication.

The total cost of all the pens is $18.

The cost of all the pens and all the books add up to $74. (Think of it as 2 numbers adding up to 74).

Next step is to work out the total cost of all the books using subtraction.

The total cost of all the books is $56.

If one book costs $7, then to find the total number of books we use division:

The final answer to the question is 8 books.

*If you know someone who needs help in Maths please share this page with them.*

Marla finished reading her book in 3 days. On the first day, she read 40% of the total number of pages. The ratio of the number of pages read on the 2nd day to the 3rd day is 3:5.

If she read 18 more pages on the 3rd day than on the 2nd, how many pages does the book have altogether?

This question involves percentage and ratio. You will learn to convert these into 'units' so we can draw out the models using boxes or rectangles.

**Watch the video:**

Let's organize the information in the question:

- 1st day: 40% of the total pages in the book
- This means that the pages read on 2nd and 3rd day combined comprise 60% of the total
- The ratio of 2nd day to 3rd day can be changed to 'units' (3 units : 5 units)
- The number of pages read on the 3rd day is 18 more than the number of pages read on the 2nd day

Let's draw the easiest part first: the ratio of the 2nd day to the 3rd day is 3:5. This means we draw 3 boxes for the 2nd day and 5 boxes for the 3rd day.

We add in the 18 pages below the 'extra' boxes.

From the model drawn, we can see that we need to use division to find out the number represented by one box.

After that, we use multiplication to find out the total number of pages read on the second and third days.

Since the total of 2 boxes is 18, we divide 18 by 2 to find out what is the number represented by one box.

Similarly, there are 8 boxes altogether, therefore, we multiply 9 by 8 to get the total number of pages for the two days.

(*Alternatively, we can multiply 9 and 3 to find out the pages on the second day. Then multiply 9 by 5 to find out the pages on the third day, and then add both numbers together.*)

So now we know that Marla read 72 pages altogether on the second and third days. Wow! She is a real bookworm!

Now we are ready to deal with the first day. Remember that first day is 40% of the whole book and the 2nd and 3rd days together is 60% of the whole book.

For convenience in drawing, we convert the percentage into 4 units and 6 units respectively.

(*We reduce 40:60 into 4:6 to make it easier to draw. You can also use 2 units and 3 units, which is the simplest form of 40 and 60. Try drawing the model using 2 units and 3 units yourself and see if your final answer is the same as mine.*)

Then we add in the numbers.

*
In the first set of models, the total number of units for Day 2 and Day
3 is 8 units. But in the second set of
models, the total number of units for Day 2 and Day 3 is only 6 units. *

*
The reason for this is that in the first set of models we are comparing
Day 2 to Day 3 whereas in the second set, we are comparing Day 2 and Day 3
combined to Day 1.*

*
This is an important distinction to make when drawing models.*

By looking at the models, can you tell which operations are needed to work out the total number of pages?

Yes, it's division to work out the number represented by one box. Then multiplication to work out the total number. Since there are 10 boxes altogether, we multiply by 10.

The book has 120 pages.

(Of course this is a Math question so the numbers are not very realistic. Can anyone actually finish reading that many pages in just 3 days??)

From all the questions you've seen so far, we've used model drawing to help us visualize how numbers relate to each other; how they are compared to each other.

This is the main strength of Drawing Models method in Singapore Math.