need help with my homework?

Question

Michelle has more than 50 hairpins.
if she puts them in packets of 7,she has 5 hairpins left.
if she puts them in packets of 9,she has 6 hairpins left.
How many hairpins does michelle have?

erm.. can you show me the working?

Answer 1

96

Answer 2

I came up with 96. I did this by taking multiples of 7 greater than 50 plus 5, and comparing them to multiples of 9 greater than 50 plus 6. The first common number was 96.

7's:
x 7 = 49 + 5 = 54
x 8 = 56 + 5 = 61
x 9 = 63 + 5 = 63
x10= 70 + 5 = 75
x11= 77 +5 = 82
x12= 84 +5 = 89
x13= 91 +5 = 96

9's:
x5 = 45+6 = 51
x6 = 54+6 = 60
x7 = 63+6 = 69
x8 = 72+6 = 78
x9 = 81+6 = 87
x10=90+6 = 96

I'm sure there is a simpler, more algabreic way to figure it out...

Answer 3

Michelle has 33 + 63n hairpins, where n is some nonnegative integer.

To see why, notice that 33 divided by 7 is 4 remainder 5 and 33 divided by 9 is 3 remainder 6.

Now because both 7 and 9 evenly divide 63, you know that the remainders from dividing 33 + 63n by 7 and 9 are 5 and 6 respectively. So any number of the form 33 + 63n is a posible number of hairpins. However, since 96 is the smallest such number bigger than 50, I am guessing they just wanted 96 as an answer.

Answer 4

Let n be the number of hairpins.

n=6+9k, where k is some integer.

So 6+9k ≡ 5 mod 7
9k ≡ -1 mod 7
2k ≡ 6 mod 7 (since 9k≡2k and -1≡6)
k ≡ 3 mod 7
k=3+7j, where j is some integer.

n=6+9(3+7j)
n=6+27+63j
n≡33 mod 63

Now, obviously there are an infinite number of answers where Michelle has more than 50 hairpins, but the only solution where Michelle has less than 60 hairpins is 33 itself. Thus I'm guessing that the problem meant "Michelle has less than 50 hairpins" and the answer is 33, but it's possible that they meant more, in which case she might have 96 hairpins, 159 hairpins, 222 hairpins, etc.

Answer 5

x=number of hairpins
y=packets of 7
z=packets of 9

x=7*y+5
x=9*z+6

7*y+5=9*z+6
7*y=9*z+1 <--- (1)
z=(7*y-1)/9

We need to try and error of y's so that z would be a natural number. y should be above 8 because 7*7 would be less than 50.

so..
(7*8-1)/9 = 6.1111 (so.. y is not 8)
(7*9-1)/9 = 6.8888 (so.. y is not 9)
(7*10-1)/9= 7.6666 (so.. y is not 10)
(7*11-1)/9= 8.44444 (so.. y is not 11)
(7*12-1)/9= 9.22222 (so.. y is not 12)
(7*13-1)/9= 10 (it's a natural number.. so y=13).

so you would get z=10 packs of 9.. that would be 90. +6 remaining pins.. that would add up to 96 (number of pins). so.. try adding the z to the (1).

7*y=9*10+1
y=91/7=13.

so try putting y onto it.. to solve all the questions..

ps:// you should do your homework yourself after this .. it was easy.

Answer 6

I am only in yr 7 but im too tired to figure this out but ill tell u how to. U have to look for multiples of 7 over 50. then look for multiples of 9 over 50. look at the multiples over 50. if one of the multiples of 7 is one number more then a multiple of 9 u have your answer! simple.

Answer 7

(96-5) / 7 = 13

(96-6) / 9 =10

96

Answer 8

54 i the answer

Answer 9

she has 61.