3.10
Another type of problem involves identifying whether there is enough data to solve the problem and, if not, which data is missing.
This is a useful building block in problem-solving. It highlights one of the key elements of problem solving, which is to find a way to solve a problem without, in this case, having to do any arithmetic.
The words ‘necessity’ and ‘sufficiency’ are used in mathematics but have exactly the same meaning as they do in normal language.
An individual piece of data is necessary to solve a problem if we cannot solve the problem without it. A set of data is sufficient to solve a problem if it contains all the information we need.
Identifying which data is needed to solve a problem can save effort in finding unnecessary data or in making unnecessary calculations.
Such questions are approached in a manner similar to those described in earlier chapters.
To illustrate the type of question described here, we start with a very simple example.
Suppose someone is taking a car journey. We know their leaving time and we know the average speed they will do. We want to know their arrival time. Which other piece of information is necessary for us to calculate this?
The solution is very straightforward: we need the distance of the journey. We can then calculate the journey time (distance divided by speed) and thus the arrival time. All of the three pieces of data we now have are necessary to do this calculation. The three pieces taken together are sufficient.
Here is a slightly more complex example.
I use the trip meter on my car to measure the distance driven since I last had the car serviced, so that I know when the next service is due. The trip meter can be set to zero by the press of a button and records the kilometres driven since it was last reset.
I set the trip meter to zero after my last service. The next service is due after 20,000 km have been driven. Some time later, I lent the car to my brother. I forgot to tell him about the trip meter; he pressed the button to zero it and drove 575 km. I then started driving again without realising what he had done.
What should the trip meter read when the next service is due?
The above problem cannot be solved with the information given. What additional piece of information is needed to solve it?
Activity
Commentary
This question is actually rather easier than it may at first seem. The distance driven from the last service when my brother returned the car was the distance I had driven plus the distance he had driven. I know how far he had driven, so what I need to know was the distance on the trip meter when I gave the car to my brother.
In this case, like the previous example, we were not asked to solve the problem, merely to identify what pieces of information were needed to solve it. In real-life problem solving, the data is not generally given; it has to be
found. Having the skill to know which pieces of data are needed can save considerable time and effort. Solving this type of problem does not need particular mathematical skills – just some clear and logical thinking.
I have a small collection of three types of old coin. The collection contains 15 coins in total. There are more pennies than half-crowns and more half-half-crowns than guineas.
Which one of the following single pieces of information would enable you to know exactly how many of each type of coin there was?
A There are 4 more half-crowns than guineas.
B There are 5 more pennies than guineas.
C There are 3 more pennies than half-crowns.
D There is one fewer penny than guineas and half-crowns together.
Activity
Commentary
In this problem, we are being asked to find which of the options is sufficient (along with the information we have already been given) to solve the problem.
There are 12 ways that 15 can be partitioned into three different numbers:
Guineas Half-crowns Pennies
1 2 12
1 3 11
1 4 10
1 5 9
1 6 8
Guineas Half-crowns Pennies
2 3 10
2 4 9
2 5 8
2 6 7
3 4 8
3 5 7
4 5 6
Of the options given, only C gives a unique set. If there are 3 more pennies than half-crowns, there must be 8 pennies, 5 half-crowns and 2 guineas.
Why do the other options not work?
• We have met a new type of problem where, rather than being asked to find a solution, we are asked to find what pieces of information are necessary or sufficient to solve it.
• We have also encountered problems where we have to find a solution, but need to identify an additional piece of information which is necessary either to help us with the method of solution or to choose between different possible solutions.
• We have learned the meaning, in this context, of the words ‘necessary’ and
‘sufficient’.
• We have seen various types of problem which require extra data: some needing mathematical solutions; some only requiring us to establish a logical method of solution.
Summary
assistant to count the bags. However, the assistant, not being very bright, counted the total number of pieces of fruit instead.
George was about to send him back to repeat it when he realised that the number that the assistant had given him was not only sufficient information for him to work out how many bags there were of each, but was also the maximum such number. How many bags of pears and bananas were there?
3 Kuldip told me she had 12 coins in her pocket, all either 1¢, 2¢ or 5¢, with a different number of each denomination.
There were more 2¢ than 1¢ coins and more 5¢ than 2¢ coins. She asked me how much money she had in her pocket altogether. I told her that I did not have enough information to answer.
Which of the following additional pieces of information would enable me to know how much money she had in his pocket?
A She had three 2¢ coins.
B The total amount of money was a multiple of 10¢.
C 5¢ coins amounted to 34 of the total value.
D She had two more 5¢ coins than 1¢
and 2¢ together.
Answers and comments are on pages 321–2.
1 I have made a dice out of a sheet of cardboard in the form of an octahedron, which has eight faces as shown below.
I now want to number the faces from 1 to 8. The numbers on opposite faces must add up to 9, so when I number a face 1, the opposite face must be 8.
If I start with number 1 and work up, how many faces can I number before I am left with no choice about where to put the numbers?
2 (Harder task) George stocks bags of pears and bananas in his shop. Each bag contains either five pears or three bananas. He wanted to know how many to order to keep his stocks up, so he sent his
End-of-chapter assignments
The current structure of income tax collection in Bolandia is that the first $2000 of annual earnings are tax-free (this is called the tax threshold), then 20¢ of tax is charged on every dollar earned over this (this could also be described as a 20% tax rate).
The most obvious and familiar use of the word
‘model’ is that of a replica of an object, for example a car, at a smaller scale. In this book the word is used in a wider sense. Models can be pictures, graphs, descriptions, equations, word formulae or computer programs, which are used to represent objects or processes.
These are sometimes called ‘mathematical models’; they help us to understand how things work and give simplified
representations that can enable us to do ‘what if?’ type calculations.
Architects, for example, use a wide range of models. They may build a scale model of a building to let the client see it and to give a better impression of how it will look. Their drawings are also models of the structure of the building. In modern practice, these drawings are made on a computer, which will contain a three-dimensional model of the building in digital form. This may be used to estimate material costs and carry out
structural calculations as well as producing a three-dimensional ‘walk-through’ picture on the screen.
This chapter deals with the recognition and use of appropriate models. A simple example of a model is a word formula used to calculate cost. The amount of a quarterly electricity bill can be described as ‘A standing charge of $35 plus 10¢ per unit of electricity used’. This may equally be shown algebraically as:
c = 35 + 0.1u
where c is the amount to pay (in dollars) and u is the number of units used.
A more complicated example of a model would be the type that governments set up to simulate their economies. These usually