Thinking Skills: Critical Thinking and Problem Solving

6.1 Using other mathematical methods 233

answer is the LCM of 6 and 8. The prime

factors of 6 are 2 and 3; the prime factors of 8

are 2, 2 and 2. One of the 2s is common to

both so the LCM is 2 × 2 × 2 × 3 = 24, the

same answer as before.

In this case there is little to choose between

the two methods, but if the counting method

gave no coincidence for 30 or 40 values, the

LCM method would be much faster. There is

another lighthouse example in the end-of-

chapter assignments, but with a twist. Problem-

solving question-setters often use such twists to

take problems out of the straightforwardly

mathematical so that candidates must use their

ingenuity rather than just knowledge. Even so,

using the mathematics you do know can often

reduce the time necessary for a question.

Permutations and combinations

Another area where a little mathematics can

help is in problems involving permutations

and combinations. Here is another simple

example.

Three married couples and three single

people meet for a dinner. Everybody shakes

hands with everybody else, except that

nobody shakes hands with the person to

whom they are married.

How many handshakes are there?

Activity

Commentary

Without the twist of the married couples, this

would be very straightforward – the answer is

9 × 28 = 36. You have to divide by 2 because the

‘9 × 8’ calculation counts A shaking hands

with B and B shaking hands with A. The

married couples can be taken care of easily,

because they would represent three of the

handshakes, so the total is 33.

The alternative way to do this is to count:

AB, AC, AD... AI, BC, BD, etc. This is very

time-consuming.

Commentary
Once again, there is more than one way of
answering this question, but algebra can make
it much more straightforward. If Kara has
walked x metres when Betsy catches her, the
time taken in seconds from Kara leaving
Betsy’s house is 15.x. The time for Betsy to cycle
this distance is x 5. We know that Kara takes
7 minutes (420 seconds) longer than Betsy, so:
x
1.5 −

x

15 = 420

Multiplying both sides by 15:

10 x − 3x = 420 × 15 = 6300, so

x = 900 metres

900 metres takes Kara 600 seconds and takes
Betsy 180 seconds – a difference of 420 seconds
or 7 minutes as required. We could also calculate
that it takes Betsy 3 minutes to catch Kara.

Lowest common factors and

multiples
Another example follows of a problem that
can be solved using a simple mathematical
technique.

From a boat at sea, I can see two

lighthouses. The Sandy Head lighthouse

flashes every 6 seconds. The Dogwin

lighthouse flashes every 8 seconds. They

have just flashed together. When will they

flash together again?

Activity

Commentary
There is a straightforward way of solving this
with little mathematics; just list when the
flashes happen:

Sandy Head: 6, 12, 18, 24, 30 seconds later

Dogwin: 8, 16, 24 seconds later

So they coincide at 24 seconds. Those with a
little more mathematical knowledge will
spot that this is an example of a lowest
common multiple (LCM) problem. The