Using models to solve maths problems part 3

More maths problems using models, at the request of Lilian!

These two are found in the P4 CASCO Challenging Maths assessment book. Neither Lesley-Anne nor I could solve them last year (when she was in p4). Only when I re-visited them this year could I solve them, probably due to more practice. Lesley-Anne could only get Q1 right with some guidance.

Question 1:

Jack and Ali are given a certain number of maths problems to solve. If Jack solves 3 problems and Ali solves 1 problem every minute, Jack will have 12 problems unsolved when Ali has finished solving all his problems. If Jack solves 1 problem and Ali solves 2 problems every minute, Jack will have 42 problems unsolved when Ali has finished solving all his problems.

a) How many problems were given to Ali?
b) To finish solving the problems at the same time as Ali, how many problems must Jack solve every minute if Ali solves 4 problems every minute?

I'm guessing that some of us have a mental block with this problem because it deals with time. Since time is linear and not an object, we don't know how to graphically capture it in a model. It also appears to add another variable to the problem, making it harder to pin down. But notice that the question never asks how long Jack or Ali takes to solve the problems. In other words, time is not an issue here.

In cases where you have two different models, it's often important to find the constant that is valid for both models. In this question, the constant for both scenarios is 1 minute.

So I started by drawing the basic model for the first part of the question (right). The unknown shows the number of problems solved by Jack and Ali per minute. 12 is the number of problems unsolved by Jack. Easy so far?

Now we need to figure out the model for the second part of the question. If Jack solves 1 problem and Ali solves 2 problems every minute, Jack will have 42 problems unsolved when Ali has finished solving all his problems. Some people might draw the model this way (right). Add an unknown part to Ali and the portion after Jack's one unknown part is 42. Sounds logical, right? WRONG. (I marked a big X in case it escaped you, DON'T FOLLOW THIS!!)

Here's why - if you add another part to Ali, you will need to add 3 more parts to Jack, otherwise the first model doesn't hold true anymore (ie If Jack solves 3 problems and Ali solves 1 problem every minute, Jack will have 12 problems unsolved when Ali has finished solving all his problems).

Instead of having to redraw a model by adding parts, it's easier to just cut Ali's unknown part in the original model into 2 (below). Remember one of my basic assumptions of models (mentioned in my previous post), ie every unknown part should be equal. So I also cut each of the other unknown parts into 2. Since Jack solves 1 problem when Ali solves 2 problems, the entire part after Jack's one part is 42.

5 parts + 12 = 42
5 parts = 42-12 = 30
Therefore, 1 part = 30÷5 = 6

Ali has 2 parts, so 6 x 2 = 12

Answer for a): Ali was given 12 problem sums.

Next part should be quite straightforward. If Ali solves 4 problem sums per minute, he would take (12÷4) 3 minutes to solve all his sums. Jack has 48 (42 + 1 part = 42 + 6) problem sums to solve. To finish solving all of them in 3 minutes, he needs to solve (48÷3) 16 per minute.

Answer for b): Jack has to solve 16 problem sums every minute, ie Jack's mother is a sadist.

Question 2:

There are 500 male and 200 female employees in Company A. There are 400 male and 600 female employees in Company B. Some employees are transferred from Company A to Company B. After the transfer, the number of male employees is the same as the number of female employees in Company B and the number of male employees is twice the number of female employees in Company A. How many female employees were transferred from Company A to Company B?

This is what I call a "transference" question, ie there is a change in value from one scenario to another. (By the way, all the lingo is coined by me, it's by no means the standard in the maths community! All the tips and assumptions are also mine, based on my own experience. If they don't help you, by all means chuck them.)

First, I need to draw the model depicting the number of employees in both companies before the transfer. When you need to compare models, it's usually easier to draw in comparable parts than in one block with a value. Looking at the values given, it's clear that it would be easiest to draw in 100 employee parts. So this is the initial model (right). Each part represents 100.

Now we need to do the transfer. This is an interesting problem because it's an example of a case where the PROCESS of drawing the model (and not the final model itself) helps you arrive at the answer. Sometimes, we tend to attempt to find the answer first then try to draw the model to fit the answer (come on, admit it! I know I do!) but this really defeats the purpose of the model.

We know that after the transfer, Company B had the same number of male and female employees. This means that at least 200 (2 parts) male employees were transfered from Company A. So first we move 2 parts male from A to B (right, indicated by shading). Looking at the remaining parts in Company A, we can see that the number of male employees (3 parts) is not twice that of female employees (2 parts), in other words, we need to move more people.

Now, we know that from this point, we have to move an equal number of male and female employees from Company A to B, in order for Company B to have an equal number of male and female employees. Again by looking at the model (below), we can see that if we move 100 (1 part) male and 100 (1 part) female employees from Company A, we will be left with 2 parts male and 1 part female, ie male employees twice the number of female employees. Voila! Let the picture do the talking.

Answer: 100 female employees were moved from Company A to Company B.