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This method is used in questions where there is a change resulting in a 'before' situation and an 'after' situation.
You will need to compare the two situations in order to understand the question fully and find a way to solve it.
This method is combined with the drawing models method.
Study the questions below.
In the 'before' situation, Susan has more money than Mark. In the 'after' situation, they had the same amount of money.
For this question, it is more useful to draw models for the 'after' situation because they had an equal amount in the end.
Since they had the same amount of money, we draw 2 equal boxes.
Now we work backwards to figure out the rest of the drawing.
Let's start with Susan. Since Susan spent half of her money, this means that the 'before' part should be twice of the 'after' part. It's like adding back the part that was spent.
I've drawn the part that was spent in red so it's easy to understand.
Now let's consider Mark's position. Since Mark has spent one-quarter of his money, this means that the 'after' portion should actually be 3 parts and the 'before' portion should be 4 parts.
Therefore we need to make some adjustments to Mark's model. This is what Mark's 'after' model should look like.
Keep in mind that Mark's 3 parts should still be the same as Susan's one part because they had the same amount of money after their spending.
Do you know what Mark's 'before' should look like?
Did you get it right? Again the amount that Mark spent is drawn in red.
Now we make more adjustments to Susan's model so that all the boxes are the same size. We also add in the $22 more. (This portion belongs to Susan's model only because Susan had $22 more than Mark.)
From the model, can you tell how to find the answer to the question?
Yes, we use division to find out the value of one box, then multiplication to find the value of 6 boxes (Susan's total).
So Susan has $66 at first.
Let's check if all the figures fit in with the information in the question.
If Susan has $66, then Mark has $44 ($22 less than Susan). After spending half of her money, Susan has $33 left. After spending a quarter of his money ($11), Mark has $33 left.
Yes, they both have the same amount of money at the end.
So $66 is the correct answer.
Barry has twice as many postcards as his sister June. June gave him 16 postcards. Now Barry has 88 more postcards than June. How many postcards do both of them have altogether?
This question is quite straightforward and easy to understand. At first glance, you might think that to find the answer you need to subtract 16 from 88 then multiply by 3.
Let's see if that's correct.
Before we continue, let's organize the information in the question:
Now we are ready to start drawing the 'before' model. Remember to draw the boxes the same size because each box represents 1 unit.
Now we highlight the 16 postcards that June is going to give to Barry. We just estimate where the number 16 is going to be. I've drawn it in blue so it's clearer to see and understand.
Before we continue, let's highlight a matching portion in Barry's model. This helps us to visualize the numbers instead of trying to guess where the numbers fit in.
We're using the concept that the same number should be represented by the same sized box.
Since June is giving the 16 cards to Barry, we need to draw this next:
Now we add in the '88 more' postcards. To make the models easier to see, we remove June's 16 postcards.
It's now possible to find the number of the middle white box (represented by the question mark). This box is 1 unit in the 'before' model.
We subtract 16 twice from 88.
Then we multiply the result by 3 because there are 3 units in total at the beginning (See the before model.)
88 - 16 - 16 = 56
56 x 3 = 168
Now we know that Barry and June has 168 postcards altogether.
Once you are familiar with this method, you can figure out the middle steps in your head and just draw the before and after models.
In the meantime, draw all the steps to help you understand the question. If you do it correctly, the operations needed to solve the question will become clear.
Kavita had 50% fewer erasers than Mark. After Mark gave 15 of his erasers to Kavita, Kavita had 40% fewer erasers than Mark. How many erasers did Kavita have at first?
For this question, you must be careful of the percentages. Percentages are comparisons between 2 numbers. In the 'before' situation, Kavita had 50% fewer erasers than Mark. So we know that Kavita's number is smaller than Mark's number.
Now we have to figure this out: Is '50% fewer' the same as '50% of'?
Yes it is. 50% changed to fractions is half. So Kavita's number is half of Mark's number. Half less than a number is the same as half of a number.
Let's draw that.
Now let's figure out the 'after' situation.
Again Kavita's number is smaller than Mark's number. Kavita had 40% fewer erasers than Mark.
In this case, '40% fewer than' is not the same as '40% of'.
We are comparing Kavita's number to Mark's number so Mark's number is 100%. This means that 40% fewer than 100% is 60%. (100 - 40 = 60)
We are not going to draw 100 boxes for Mark so we need to do a little adjustment. We use ratio to help us.
We start with 60 : 100 and then reduce the ratio to its simplest form.
This is how we change percentages into 'units'. In the 'after' situation, Kavita has 3 units while Mark has 5 units. Now it's easy to draw the model.
These are the important concepts to understand:
To understand the last point, I'm going to put the 'before' and 'after' models side by side:
The important thing to understand is that the basic box in the 'before' model is of a different size than the basic box in the 'after' model.
But each box is the same size in the respective models. This is because Mark's number of erasers in the 'before' situation is not the same number as the 'after' situation.
Our job now is to somehow make them equal in some way. Only then can we understand how to link the 'before' to the 'after'.
What we know is that the total number of erasers did not change. We make use of this fact to find the common link between the 2 situations.
In the 'before' we have a total of 3 units. In the 'after' we have a total of 8 units. So we need to find a common multiple of 3 and 8. That multiple is 24. This means that we need to change the units of both 'before' and 'after' to give a total of 24 units.
We do this by multiplying the 'before' units by 8. And multiplying the 'after' units by 3.
Kavita has 8 units at first and 9 units after. Mark has 16 units at first, and 15 units after.
Now it's easy to see how the 'before' became the 'after'.
You can draw it out if it helps you understand better.
Mark gave 1 unit to Kavita. When you read the question again, it says that Mark gave 15 erasers to Kavita.
This means that 1 unit represents 15 erasers.
To find out how many erasers Kavita had at first, we multiply 8 by 15 because Kavita had 8 units at first.
8 x 15 = 120
Kavita had 120 erasers at first.
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