Interesting word problem - Remainder Concept

I saw this interesting word problem posted in one of the thread in Kiasu parents forum (http://www.kiasuparents.com) the question look something like this :-

Question :

Kumar spent $1729 on a set of encyclopedia. He spend 1/4 of the remainder on a a camera and still had 2/5 of his money left. Find the total amount of money he had at first.

This is a typical question based on remainder concept. While it is relatively easily for experienced teacher to solve the question. However, in teaching, it is not how much you know, but rather how easy you make it for the children to understand. You have to approach it from the mentality of your students. There are a few points which you have to specially take note as below :-

1. The amount given $1729 is a relatively odd number and some students will perceive it as tough to calculate.

2. 3/4 of the remainder which makeup the Final fraction of 2/5 is not very clear cut to some students who are not fast learner. (as you need to further divide the 2/5 to 6/15 so that the 3/4 of the remainder fit nicely into the subunit. You could split the 1/4 & 3/4 of the remainder properly into the subunits nicely.)

3. After further divide the final fraction into 15 equal parts, it will not be easy for your students or your kids to work on as they have to be careful visually as well as when they carry out the computing.

4. For students that you have not introduce the remainder concept, you will need to guide and train them to identify why this question belong to the remainder concept. This question also give you to opportunity to train them how to think.

Let approach it from the model method

From the question we can draw the model below

We know that Kumar had spent the money on the encyclopedia and camera are left with 2/5 of the total amount. We know the amount spent on the encyclopedia but not the final fraction (We could immediately calculate the total amount if we know the final fraction of the encyclopedia. We know the final fraction left but not the actual money left. The great clue is the money left is 2/5 of the total and it is also 3/4 of the remainder after buying the encyclopedia and camera. So trying to relate the amount left which is 3/4 of the remainder to the final fraction we have to compare as below

We had to split the each 1/5 to 3 equal units so each 1/4 will make up 2/15 of the final fraction and 3/4 will make up 6/15 of the final fraction as indicated below

Hence we can find out the final fraction of the encyclopedia by substract 2 units (camera) and 6 units (left) from the total 15 units. Hence the encyclopedia is 7 units as shown below:-

Therefore the total amount is as work out as below

                            7 units  1729

                            15 units1729 / 7 * 15 = 3705