Hi
I came across two exam questions :

1) Mr Tan bought thrice as many books as calculators and spent $232 in total. He spent $8 more on books than on calculators. Given that a book cost $9 less than a calculator, what is the cost of the calculator ? Ans $14.

Not sure what method to adopt for such type of questions.. Try modelling but cannot get the answer

2) Peter has some money.
He spends $10 on 3 exercise books and 7 pencils.
He will be short of $0.40 fi he were to buy another exercise book.
He will have $0.60 left if he were to buy one more pencil.
(a) how much does each exercise book cost ?
(b) how much money does Peter have ?

My son could not solved this equation. I apply simultaneou equation concept and gap and difference concept. Not sure if there are alternative ways to solve them ?

Ans : 3e + 7 p = $10 (1)
1e - 1p = $0.4 + $0.6 = $1 (2) (applying gap and difference. My son cannot
write this sentence)

(2) X 7 : 7e - 7p = $7 (3)
(1) + (3) : 10 e = $7
1e = $1,70 (a)

(b) $10 + $1.70 - 40c
= $11.30

Any alternative solutioning ?

Thank You

For the first question, we have four unknowns: Cost of a book, Cost of a calculator and the number of books and the number of calculators. However, we do have relationships between the cost of books and calculators and the ratio of books to calculators. Given this, for me the first method that comes to mind is a combination of number x value and simultaneous equations.



For the second question I would follow the same approach that you have. I will post my solution shortly.

Update: I have posted another solution to this question below, which does not use equations and may be easier to understand.

I have done the same as you have using gap & difference followed by simultaneous equations. Perhaps with a model drawing for the gap and difference it may be more clear.

Which part is he having problems with?

Another solution not using the number x value format. This solution does not openly use equations. Instead, we use logic to work out the difference, in price, of 1 unit of books in comparison to 1 unit of calculators. We know this difference is a result of 1 calculator being $9 more than 1 book.



And finally, almost all model.

Hi Chris

Thanks very much.

For the Question 2 , it is much clearer now with the model drawn.

As for Question 1, a combination of number and unit method seems easy for us but tough for my son to comprehend.
Need to break this questions into various steps for easier understanding and explanation. Will digest your model drawing again.

Hi everyone! I am a newbie to onsponge! When I read the thread above, shivers went down my spine n my stomach dropped to the ground! R u guys telling me tt my P4 going to P5 girl r heading for such questions?! Simultaneous eqns, didn't we study tt in sec sch?! And stop the gap is a new math concept or something. Pls tell me tt such qns r only given to GEP kids! I think I am going sleepless tonite!;(

Jennifer Tan wrote:
R u guys telling me tt my P4 going to P5 girl r heading for such questions?! Simultaneous eqns, didn't we study tt in sec sch?! And stop the gap is a new math concept or something. Pls tell me tt such qns r only given to GEP kids! I think I am going sleepless tonite!;(

Yes Simultaneous Equations were taught in Secondary School in our days. Currently, it is included in the P6 syllabus, limited to a few steps only, as can be found in Chapter 5 of the +hinkingMath P6 Book and questions may appear on the PSLE that could use this approach.

That said, to my surprise, my daughter's school (she is also in P4 going into P5) put a Simultaneous Equations question on one of her SA2 practise papers. Basically, each school should teach the defined syllabus but may choose to add additional topics and/or teach topics at different timing. Another example, also from my daughters school - they didn't learn Tessellation.

With regards to the gap and difference question, some pupils are learning very basic gap and difference concepts as early as P3. The one in this thread is more complex as it requires more than one concept - gap and difference and simultaneous equations.

The firt question is a pretty standard one that appears in most P6 test papers and sometimes in P5 yearend assesments. It can be modelled quite easily as chris shows above.

It is made up of just two basic concepts.

The first is using the Total and Difference between 2 quantities (taught from P3) to find the two actual values.

hence all the books cost (232+8)/2 = 120
& all calcs cost 112

The second concept is that as long as the number or quantity of books and cals is adjusted to be equal, the quantity can be obtained taking the total diff in cost and dividing by the unit diff in cost.

You can therefore either reduce the book quantity (divide by 3) or increase the calc quantity (multiply by 3).

3u of books = 120 1u of calcs = 112
is adjusted to
1u of books = 40 and 1 u of cals = 112

1u = (112-40)total diff divided by 9 (unit diff)= 72/9=8

8 calcs cost 112
24 books cost 120, and you can find the unit price of either.

Rather than jump into simultaneous equations, I usually pose practical examples to get the kids to realise the relative values of the 2 quantities. Eg you need $4 more than what you have to buy a file and $10 more than what you have to buy a bag. OR You need $4 more than what you have to buy the file but save $3 if you buy a pen instead.OR you save $3 if you buy a pen but save $5 if you buy pencil instead. Posing the question "Which is more expensive and how much more?" for each scenario helps the kids to derive the answers quite easily.

Then teach the modelling for each scenario.

After that it is easy to use substitution and models since the kids are taught this from P2 or P3 at least.

Hence 1 pencil= 1u but 1 exercise book= 1u + $1
7 pencils =7u 3 exercise books= 3u+ $3
10 u + $3 = $10
and so forth.

Which is not to say that simultanueous euations or other algebraic shortcuts are unnecessary, but for the kids who are struggling to understand, the whole idea is to make math as concrete as possibly rather than more abstract.