could someone help me with some math?

Question

1. The ratio of the number of Alfred's paper clips to the number of Cindy's was 3:8. After Alfred gave six to Cindy, Cindy had 4 times as many as Alfred. How many paper clips did Cindy have in the end?

2. The ratio of the number of Sharon's hair clips to the total number of Sharon's and Britteny's hair clips is 3:8. If Britteny has 40 hair clips, how many more hair clips hoes she have than Sharon?

3. The ratio of Sally's age to Lucy's is 8:5. 30 years later, the ratio will be 6:5. How much older is Sally tham Lucy NOW?

Answer 1

1) A/C = 3/8
(A-6)/(C+6) = 1/4

Rewrite the first equation to isolate a variable (A or C)
A = 3*C/8
Plug into the second equation:
((3C/8) - 6)/(C+6) = 1/4
((3C/8) - 6) = (C+6)/4
(3C/8) - (C/4) = (3/2)+6
C/8 = 15/2
C = 60
Therefore in the end, Cindy had 60+6 = 66 paper clips.

2)
S/(S+B) = 3/8
B = 40 (given)
S = (3/8)*(40+S)
(5/8)S = 15
S = 24
40-24 = 16 more clips.

3)
S/L = 8/5
(S+30)/(L+30) = 6/5
S = 8L/5
5*(8L/5 + 30)= 6*(L+30)
8L - 6L = 180-150
2L = 30
L = 15 (Lucy's age now)
then S = 24 (Sally's age now)
S-L = 9

Answer 2

1. To express these as equations:

8*Alfred = 3*Cindy

4*(Alfred-6) = Cindy+6
4*Alfred - 24 = Cindy + 6
4*Alfred = Cindy + 30

Now, let's take the second equation and double it:

8*Alfred = 2*Cindy + 60

We can now compare the Cindy's ...

3*Cindy = 8*Alfred = 2*Cindy + 60

or

3* Cindy = 2*Cindy + 60
Cindy = 60

Cindy ended with 66 clips.


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2. The ratio of the number of Sharon's hair clips to the total number of Sharon's and Britteny's hair clips is 3:8. If Britteny has 40 hair clips, how many more hair clips hoes she have than Sharon?

8 * Sharon = 3 * (Sharon + Britteny)
8 * Sharon = 3 * (Sharon + 40)
8 * Sharon = 3 * Sharon + 120
5 * Sharon = 120

Sharon = 24

Brittney has 16 more hair clips than Sharon

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3. The ratio of Sally's age to Lucy's is 8:5. 30 years later, the ratio will be 6:5. How much older is Sally tham Lucy NOW?

5 * Sally = 8 * Lucy

5 * (Sally+30) = 6 * (Lucy+30)
5 * Sally + 150 = 6 * Lucy + 180
5 * Sally = 6 * Lucy + 30

8 * Lucy = 5 * Sally = 6 * Lucy + 30
8 * Lucy = 6 * Lucy + 30
2 * Lucy = 30

Lucy = 15

5 * Sally = 8 * Lucy = 8 * 15 = 120
Sally = 24

Sally is 9 years older than Lucy.

Answer 3

Here's the first one...

Let the number of Alfred's paper clips be A, and the number of Cindy's be C.

A = 3C/8
and
(A-6) = (C+6)/4

Looks like a pair of simultaneous equations to me.

8A = 3C
8A - 3C = 0 for the first one, and

4A - 24 = C + 6
4A - C - 24 = 6
4A - C = 30 for the second one.

Hmm . To make this one a little easier to solve, multiply the second equation by -2:
-8A +2C = -60

Now, add them together:

8A -3C = 0
-8A +2C = -60

Which gives:

0A -C = -60
-C = -60
C = 60

That's at the beginning, at the end Cindy has C + 6 = 60 + 6 = 66 paperclips. Let's check:

A = (3C)/8
A = (3 * 60)/8
A = 180/8
A = 22.5

Wow, that's a little weird. At the start, Alfred has 22 1/2 paperclips. I've never seen a 1/2 paperclip! But let's keep checking...

If Alfred gives 6 paperclips to Cindy, he has 22.5 - 6 = 16.5 left. Cindy has 66, which is supposed to be 4 times as many as Alfred. 16.5 * 4 = 66...so yeah, it's a little strange, but it's right.

Answer 4

1) let Alfred has 3x and Cindy has 8x
then after exchange Alfred has 3x-6 and Cindy has 8x+6
so , 8x+6 = 4(3x-6)
i.e. 8x+4 = 12x-24
i.e. 4x = 28 or x= 7
so Cindy has 8*7+6 = 62

Answer 5

1)6 more than when she started
2)25
3)2