Question

Is there a combination of 15 cent and 33 cent stamps that can make $1.77? If so, find all possible ways of doing this. If not, explain clearly why not.

Answer 1

you want to solve
\[15a + 33b = 177\]

in integers

Answer 2

would i have to cancel one out by doing 15a(-15a)?

Answer 3

oh! would i have to do them from 1 all the way up until i get to 177 and stop when i get there? like 15(1)+33(1)=177?

Answer 4

That works! but there are more efficinet ways...
familiar with finding gcd using euclid algorithm ?

Answer 5

no, i haven't learned the euclid algorithm :o

Answer 6

Okay, lets not use it then.
First of all notice that this equation has solutions in integers because both sides are multiples of \(3\). May be lets divide that

Answer 7

\[15a + 33b = 177\]

divide 3 through out

\[5a + 11b = 59\]

Answer 8

Clearly \(b\) cannot be more than \(5\) (why ?)

consider five cases \(b = 1,2,3,4,5\) and see which value(s) of \(b\) give you integer values for \(a\)

Answer 9

why can't b be more than 5?

Answer 10

isn't 11 already greater than 5?

Answer 11

you want 5a+11b to equal 59
when b is greater than 5, the left hand side will be greater than 59. so NO solutions.

Answer 12

oh okay! so then would my equation be: 5(1)+11(1)=59, then all the way up to 5?

Answer 13

consider 5 cases

Answer 14

case 1 : b=1
\(5a + 11(1) = 59\)
solve \(a\)

Answer 15

i got 9.6...

Answer 16

which is not an integer, so case1 wont work

Answer 17

work out case2

Answer 18

case 2 : b=2
\(5a + 11(2) = 59\)
solve \(a\)

Answer 19

7.4, so that doesnt work..

Answer 20

Right, work remaining 3 cases also

Answer 21

only case 4 works, a=3

Answer 22

does that mean the only solution is a=3, b=4 ?

Answer 23

yes, because they are equal right?

Answer 24

or do i need to create another equation?

Answer 25

Yep!

Answer 26

No, we're done

Answer 27

awesome, thank you so much!!

Answer 28

so now if i wanted to make $1.77 using only 15 and 35 cent stamps how could i find all the possible ways for that?

Answer 29

could i still use a = 3 and b = 4?

Answer 30

yes we figured out that the only possible combination is by taking three 15 cent stamps and four 35 cent stamps

Answer 31

lets double check maybe..

Answer 32

3(0.15) + 4(0.35) = ?

Answer 33

wait i got 1.85!

Answer 34

3(0.15) + 4(0.33) = ?

Answer 35

let me check my math again!

Answer 36

there was a typo, its 33 cent stamps right ?

Answer 37

yes!

Answer 38

3(0.15) + 4(0.33) = ?

Answer 39

okay so thats 1.77! awesome, thank you so much!! you definitely made it a lot easier to understand a problem that i was over thinking! how did you know to use the integers and create the equation 15a+33b=177 though?

Answer 40

we want to find a, b such that :

0.15a + 0.33b = 1.77

Answer 41

since working with integers is easy, i have simply multiplied 100 both sides :)

Answer 42

oh i see! thanks!! :)

Answer 43

yw:)