The woman is going to make a phone call costing any multiple of 10p. I am going to investigate the number of different ways she could put the 10p and 20p coins into the payphone.

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Introduction

A pay phone will take only 10p, 20p, 50p, and £1 coins. A woman has plenty of 10p and 20p coins. She has no other coins. She can put the coins into the pay phone in any order.

To make a call costing 50p, she could put in the coins in any order;

20p, 20p, 10p

or 10p, 20p, 20p

or 10p, 10p, 20p, 10p

There are more ways of making 50p with only 10p and 20p coins.

. The woman is going to make a phone call costing any multiple of 10p. I am going to investigate the number of different ways she could put the 10p and 20p coins into the payphone.

A man also wants to use the pay phone. He has plenty of 10p and 50p coins. He has no other coins. He wishes to make a telephone call costing any multiple of 10p.

2. I am going to investigate the number of different ways he has of entering the 10p and 50p coins into the telephone.

3. I will then investigate the more general cases leading into special cases.

There can be made 8 different combinations using only 10p and 20p coins to make a call costing 50p.

These are;

. 10,10,10,10,10,10

2. 10,10,10,20

3. 20,10,10,10

4. 10,20,10,10

5. 10,10,20,10

6. 10,20,20

7. 20,10,20

8. 20,20,10

. The woman is going to make a phone call costing any multiple of 10p. Investigate the number of different ways she could put the 10p and 20p coins into the pay phone.

Multiple of 10p Combinations for 10p, 20p Total no of combinations (Fibonacci sequence) Algebra/Terms

0p 10 1 T1

20p 10,1020 2 T2

30p 10,10,1010,2020,10 3 T3 = T2+T1

40p 10,10,10,10 20,2010,10,2010,20,1020,10,10 5 T4=T3+T2

50p 10,10,10,10,1020,20,1020,10,2010,20,2010,10,10,2020,10,10,1010,20,10,1010,10,20,10 8 T5=T4+T3

Fibonacci sequence

This sequence in which each term is the sum of the two preceding terms is very well known. It is called the Fibonacci sequence after the Italian mathematician who investigated it. The nth term Un, is very complicated and you probably don´t need to know how to find before your A-levels.

Prediction

Looking at the table above, we know that;

30p = T3 = T2+T1

40p = T4 = T3 +T2

50p = T5 = T4+T3

So, 60p´s prediction will be

60p = T6 = T5+T4

Explanation

As every "total no of combinations" of putting the coins into the pay phone are found by adding the 2 previous terms, the result of Tx can be solved by this formula;

Tx = T(x-1)+T(x-2)

Prediction & Explanation

So, for 60p we know it is the equivalent of T6. Let us put this into the formula, to find the "number of ways".

Tx = T(x-1)+T(x-2)

T6 = T(6-1)+T(6-2)

T6 = T5+T4

T6 = 8+5

T6 = 13

Testing

The prediction will be tested as to see whether it will work out.

Multiple of 10p Combinations for 10p, 20p Total no of combinations Algebra/Terms

60p 10,10,10,10,10,1020,20,2010,10,10,10,2020,10,10,10,1010,20,10,10,1010,10,20,10,1010,10,10,20,1010,10,20,2020,10,10,2020,20,10,1010,20,20,1010,20,10,2020,10,2010, 13 T6 = T5+T4

The pattern of the different number of ways a multiple of 10p can be used, using 10p and 20p coins only, is recorded below;

0p- 1

20p- 2

30p- 3

40p- 5

50p- 8

60p- 13

It can also be shown in a dot pattern

0p- .

20p- ..

30p-...

40p- .....

50p- ........

60p- .............

I am now going to use some "imaginary coins". Starting of with 10p and 30p, I will investigate the number of different ways you could put the 10p and 30p coins into the pay phone. The phone call will cost any multiple of 10p.

Multiple of 10p Combinations for 10, 30p Total no of combinations Algebra/Terms

0p 10 1 T1

20p 10,10 1 T2

30p 10,10,10,30 2 T3

40p 10,10,10,1010,3030,10 3 T4 = T3+T1

50p 10,10,10,10,1010,10,3030,10,1010,30,10 4 T5 = T4+T2

60p 10,10,10,10,10,1030,3010,10,10,3030,10,10,1010,30,10,1010,10,30,10 6 T6 = T5+T3

Prediction

Looking at the table at the table above, we know that;

40p = T4 = T3+T1

50p = T5 = T4+T2

60p = T6 = T5 +T3

So, 70p´s prediction will be

70p = T7 =T6+T4

Explanation

As every "total no of combinations" of putting the coins into the pay phone are found by adding the previous term together with the third previous term, the result of Tx can be solved by this formula;

Tx = T(x-1)+T(x-3)

Prediction & Explanation

So for 70p, we know it is the equivalent of T7. Let us put this into the formula, to find the "number of ways".

Tx = T(x-1)+T(x-3)

T7 = T(7-1)+T(7-3)

T7 = T6+T4

T7 = 6+3

T7 = 9

Testing

The prediction will be tested as to see whether it will work out.
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Multiple of 10p Combinations for 10p, 30p Total no of combinations Algebra/Terms

70p 10,10,10,10,10,10,1030,30,1030,10,3010,30,3010,10,10,10,3030,10,10,10,1010,30,10,10,1010,10,30,10,1010,10,10,30,10 9 T7 = T6+T4

The pattern of the different number of ways a multiple of 10p can be used, using 10p and 30p coins only, is recorded below;

0p- 1

20p- 1

30p- 2

40p- 3

50p- 4

60p- 6

70p- 9

It can also be shown in a dot pattern

0p- .

20p- .

30p- ..

40p- ...

50p- ....

60p- ......

70p- .........

The ...

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