The Constant Quantity Concept

The Constant Quantity Concept is derived from a combination of the

Part-Whole Concept , the Comparison Concept and/or the Change Concept . The Constant Quantity Concept is applicable when the problems deal with quantities being transferred in or transferred out of one of the two variables concerned. This leaves the other variable with its quantity unchanged, hence we refer to it as the "Constant Quantity Concept". The unique feature in this concept lies in the fact that after the transfer in or transfer out of quantities, the value of the second variable remains unchanged.

To illustrate this concept, consider the following problems:


(A) Constant Quantity after a Transfer Out


Pamela had thrice as many hats as Christine. After Pamela gave away 10 hats, she had half as many hats as Christine. How many hats did Christine and Pamela each had at first?


Answer:

Step 1: Draw 1 box to represent the number of hats Christine had and 3 boxes to represent the number of hats Pamela had at first.

constant-quantity-concept-001


Step 2: Since Christine did not receive or give away any hats, the model (1 box) representing her hats remains constant.

We also know that after Pamela gave away 10 hats, she had only half the number of hats that Christine had, i.e., Christine would have twice the number of hats that Pamela had.

Thus, Christine would have 2 units and Pamela 1 unit. We can then divide the 1 unit that Christine had into 2 equal smaller units and mark out 1 equivalent small unit for Pamela.

constant-quantity-concept-002


Step 3: Next, we also know that Pamela had given away 10 hats. The parts to the right of the 1 small unit in Pamela's model bar must be equal to 10 hats.

constant-quantity-concept-003


Step 4: After we had put all the information into the model, we need to check if the unknown units of the model are equal. If not, can we sub-divide them to make them equal.

In the question, we can sub-divide the part Pamela gave away into 5 equal parts as follow:

constant-quantity-concept-004


Hence, from the model,

5 units ----------> 10 hats

1 unit -----------> 10 hats / 5 = 2 hats

2 units ----------> 2 units X 2 hats = 4 hats

6 units ----------> 6 units X 2 hats = 12 hats

Therefore, Pamela had 12 hats and Christine had 4 hats at first.


(B) Constant Quantity after a Transfer In


Sean had thrice as many marbles as Irwyn. After Irwyn bought another 10 marbles, he had twice as many marbles as Sean. How many marbles did they each have at first?


Answer

Step 1: Draw 1 box to represent the number of marbles Irwyn had and 3 boxes to represent the number of marbles Sean had at first.

constant-quantity-concept-005


Step 2: Since Sean did not buy or give away any marbles, the model (3 boxes) representing his marbles remains constant.

We also know that after Irwyn bought another 10 marbles, he had twice as many marbles as Sean. Since Sean had 3 units, Irwyn would have twice as many units as Sean, i.e., 6 units. So, we add another 5 boxes to Irwyn's model bar to make it 6 units.

constant-quantity-concept-006
constant-quantity-concept-007


Step 3: Next, since Irwyn bought another 10 marbles, these 10 marbles must be represented by the extra 5 units added to his model bar.

constant-quantity-concept-008


Thus,

5 units ----------> 10 marbles

1 unit -----------> 10 marbles / 5 = 2 marbles

3 units ----------> 3 units X 2 marbles = 6 marbles

Therefore, Sean had 6 marbles and Irwyn had 2 marbles at first.




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