Question 1006346: A bag of batteries in the ratio of 3 size D batteries to 7 size C batteries is combined with a bag having a ratio of 5 size D to 1 size C.
If the two bags together now contain an equal number of size D and C batteries, how many batteries are in the second bag?
My attempt:
Step 1: I can see that the first bag in the ratio of 3/7 has 18 size D batteries and 42 size C batteries, given that the total number in the bag is 60.
Step 2: I can see that the second bag is in the ratio of 5/1.
Step 3: Through trial and error, I can see that I need to get to a combined bag with a ratio of 8/8 for a total of 96. But I don't know how to do this mathematically.
This is often my problem. Can anybody help me set this up?
Thank you.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you are correct in that both bags will contain 96 batteries.
there will be 48 D and 48 C.
you are given that the first bag contains 60 batteries total and that the ratio of D to C is 3/7
let x = the common multiplier.
you get 3x + 7x = 60
combine like terms to get 10x = 60
solve for x to get x = 6
the common multiplier is 6.
3*6 + 7*6 = 18 + 42 = 60
the total number of batteries is 60.
the ratio of D to C is 18 / 42 = 3 / 7.
the batteries in the second bag are in the ratio of D / C = 5 / 1
let y be the common multiplier for the second bag.
you get 5y + 1y = z
z is the total for the second bag.
when you add the D batteries together and you add the C batteries together, you will get a ratio of 1/1.
you get:
(18 + 5y) / (42 + y) = 1/1 which becomes:
(18 + 5y) / (42 + y) = 1
multiply both sides of this equation by (42 + y) to get
18 + 5y = 42 + y
subtract y from both sides of this equation and subtract 18 from both sides of this equation to get:
5y - y = 42 - 18
simplify to get:
4y = 24
solve for y = get y = 6
when y = 6, the equation for your second bag of 5y + y = z becomes 30 + 6 = z which results in 36 = z
your second bag contains 36 batteries.
30 of them are D
6 of them are C.
the ratio of D/C = 30/6 = 5/1.
your combined bag contains 60 + 36 = 96 batteries.
your total D are 18 + 30 = 48
your total C are 42 + 6 = 48
the ratio of D to C is 1/1.
|
|
|
