Another Problem from Singapore Grade 6B Placement Test

Note: As usual, the comments section contains insightful contributions from Denise, mathercize et al. I expounded briefly on a ratio approach and provided a link to some other bar diagram solutions. By the way, I've been using the misnomer, 'fraction bars' instead of the correct phrase bar diagram. I have a lot to learn here...

Lots of traffic from people viewing this placement test. The following is Question #11 from this test and I think it can lead to fruitful discussion of methods and strategies for middle school and secondary students. Older students will usually use an algebraic approach, prealgebra students might use 'Guess-Test-Revise' with a calculator. However, Singapore students apparently use a 'fraction bar' model for many of these, which I see as a ratio approach. How would you solve it? How would you present this to your students? What methods would you expect your students to use? What percent of your students would feel they don't even know where to begin and give up quickly? Would group work help here?

My first inclination was to set up an equation in one variable, however, because it was on the Singapore test, I tried a ratio approach which enabled me to solve it mentally. Have fun! By the way, to personalize it for your students, you may want to change the names to two students in your class and place it in the context of spending money at the mall!

Also, consider that if one were to use 'Guess-Test', it would make sense to start with a number divisible by $48 (since $240 is a multiple of $48, an intuitive student with strong quantitative skills might easily guess the answer!).

The Question:

Peter and Paul each had an equal amount of money. Each day Peter spent $36 and Paul spent $48. When Paul used up all his money, Peter still had $240 left. How much money did each of them have at first?

Singapore Math - Primary Math 6B Placement Test...

[Note: As promised, I have now posted a lengthy comment to this post, which only scratches the surface. I welcome your comments.]

Although Singapore Math is no secret to many homeschooling parents and districts around the country, I thought that the sample placement test for Grade 6B, available for free downloading, might lead to some fruitful discussion. Pls try all 14 questions before commenting. I did and I will have much to say later. There are many other placement tests at the same site so have fun! Go to the Singapore Math home page for background if needed.