Question

I don't know what to do with this question, please help!
A piece of string 10m long is cut into pieces, so that the lengths of the pieces form an arithmetic sequence.
(i) The lengths of the longest and shortes pieces are 1m and 25cm respectively; how many pieces are there?
Answer:16
(ii) If the same string had been cut into 20 pieces with lengths that formed an arithmetic sequence, and if the length of the second longest had been 92.5 cm, how long would the shortest piece have been?
(ii) Is the one I don't know what to do with. I have already completed (i)

Answer 1

which one !?

Answer 2

I need help with the last question please.

Answer 3

Arithmetic Series:
2nd term = a1 + d where a = first term and d = common difference
so here
92.5 = a1 + d

Answer 4

the sum of 10 terms = the length of the string = 10,000 cms

Answer 5

Um isn't 1000cm?

Answer 6

Now the sum of n terms is
(n/2)[2a1 + (n - 1)d]

so here n = 20 and Sum = 10,000 so we have

10,000 = 10[2a1 + 19d]

Answer 7

so we have the system of equations

a1 + d = 92.5
20a1 + 19d = 10,000

Answer 8

Ok, one little question- why is it 2a1 ?

Answer 9

2 * first term
a1 will be the first term = shortest piece

Answer 10

Oh ok

Answer 11

so you need to find the value of a1

Answer 12

I use a different form of that equation normally so was confused for a second

Answer 13

if you multiply the first equation by -19 and add you can find a1

Answer 14

yes a1 is usually used in the US
in UK we usually use just a

Answer 15

a1 + d = 92.5 sorry, could you just explain more about how you got this part? I dont really understand why the first term plus the common term equals the second last term? I know I am missing something

Answer 16

oh - maybe i was wrong there sorry - -i took that to be the second term
- its the 19th term

Answer 17

also the total length = 1000 (wrong again!)

Answer 18

Haha, it happens! No worries.

Answer 19

so the 2 equations are
a1 + 18d = 92.5
20a1 + 19d = 1000

Answer 20

Ok, I am not sure I understand the concept behind that second equation

Answer 21

that is the formula for the sum of n terms of an arithmetic series

Sn = (n/2)[2a1 + (n - 1)d]

here Sn = length of the string = 1000 cms, n = 20 (20 pieces) , a1 = first term, d = common difference

Answer 22

Ok. Then I am meant to simultaneously solve those two equations?

Answer 23

yes

Answer 24

i bit messy Have you got a graphical calculator?

Answer 25

No I don't. Oh well, I will try it quickly now without one

Answer 26

is our shortest piece still 25cm?

Answer 27

No I don't think so. That was for the first question

Answer 28

how long should shortest piece ... forgot to read thru it all the way

Answer 29

hmm the results don't seem to give rational values..

Answer 30

Yeah I am not getting anything that makes sense.

Answer 31

I got d=2.5 a=47.625 after rounding

Answer 32

i thnk the equations are correct..

Answer 33

I probably just did something wrong then.

Answer 34

yes the calc gives 47.631 and 2.4927

Answer 35

10m long

20(2a+19d)/2 = 1000
a+18d = 92.5

a is about 2.5, and d is about 5

Answer 36

yea
my second equation was wrong
10(2a + 19d) = 201 + 190d

Answer 37

2a+19d = 100
a+18d = 92.5

2a +19d = 100
-19/18a -19d = -19/18(92.5)

a = (100-(19/18(92+1/2)))/(2-(19/18))

Answer 38

2.5 for shortest length
and 5 for common difference

Answer 39

it was 190d not 19d

Answer 40

Ok..

Answer 41

do you see the mistake
I didnt distribute 10 over the (2a + 19) correctly

Answer 42

I thought someone would spot my deliberate mistake
- oh amistre did!

Answer 43

Oh! I see now!

Answer 44

good

there is another formula for sum of n terms of an arithmetic series

Sn = (n/2)[ a1 + L] where L is the last term

If you know the last term its a lot simpler

Answer 45

Yes, thats the form I normally use and I didn't remeber about the other one not needing the last term which is a big part of why I got stuck I think
Just trying to solve those two equations myself now.

Answer 46

probably best to multiply first one by -20 and add to get d first

Answer 47

Yup! Did it and got the two right answers! Thank you so much for your help!

Answer 48

yw