3 glasses A, B and C whose capacities are in the ratio
1:2:3 are filled with mixture of acid and water. The ratios of acid to water in A, B and C are 1:5, 3:5 and 5:7 respectively. The contents of these glasses are emptied into a single vessel. What is the ratio of acid to water in the vessel?
A:B:C = 1:2:3 = 12x : 24x : 36x
for A:
acid:water = 1:5 = (1/6)12x : (5/6)12x = 2x : 10x
for B:
acid:water = 3:5 = (3/8)24x : (5/8)24x = 9x : 15x
for C:
acid:water = 5:7 = (5/12)36x : (7/12)36x = 15x : 21x
now pouring this into one container in 1:2:3
from A we are getting 2x acid and 10x water
from B we are getting
9x acid and 15x water
from C we are getting 15x acid and 21x water
acid : water = (2x+9x+15x) : (10x+15x+21x)
= 26x : 46x
= 13 : 23
could have used a specific volume, e.g.
suppose A has 1200 ml, B has 2400 ml , C has 3600 ml, notice ratio of 1:2:3
in A we have 200 ml acid and 1000 ml water, 200:1000 = 1:5
in B we have 900 ml acid and 1500 ml water, 900:1500 = 3:5
in C we have 1500 acid and 2100 water, 1500:2100 = 5:7
total acid = 200+900+1500 = 2600
total water = 1000+1500+2100 = 4600
ratio of acid:water = 2600:4600 = 13:23
To find the ratio of acid to water in the vessel, we need to first calculate the amount of acid and water in each glass.
Let's assume the capacities of glasses A, B, and C are x, 2x, and 3x respectively.
First, let's calculate the amount of acid in each glass:
In glass A, the ratio of acid to water is 1:5. This means that out of every 6 units of the mixture, 1 unit is acid and 5 units are water. So, the amount of acid in glass A is (1/6) * x.
In glass B, the ratio of acid to water is 3:5. This means that out of every 8 units of the mixture, 3 units are acid and 5 units are water. So, the amount of acid in glass B is (3/8) * (2x).
In glass C, the ratio of acid to water is 5:7. This means that out of every 12 units of the mixture, 5 units are acid and 7 units are water. So, the amount of acid in glass C is (5/12) * (3x).
Now, let's calculate the amount of water in each glass:
In glass A, the amount of water is 5 units for every 6 units of the mixture. So, the amount of water in glass A is (5/6) * x.
In glass B, the amount of water is 5 units for every 8 units of the mixture. So, the amount of water in glass B is (5/8) * (2x).
In glass C, the amount of water is 7 units for every 12 units of the mixture. So, the amount of water in glass C is (7/12) * (3x).
Now, let's add up the amount of acid and water from each glass:
Total amount of acid = (1/6)x + (3/8)(2x) + (5/12)(3x)
Total amount of water = (5/6)x + (5/8)(2x) + (7/12)(3x)
Simplifying these expressions:
Total amount of acid = (8/48)x + (24/48)x + (30/48)x = (62/48)x
Total amount of water = (40/48)x + (20/48)x + (21/48)x = (81/48)x
Therefore, the ratio of acid to water in the vessel is (62/48)x : (81/48)x, which simplifies to 62:81.
To find the ratio of acid to water in the vessel, we need to determine the total amount of acid and water in each glass, and then combine them in the single vessel.
Let's start by finding the total amount of acid and water in each glass:
Glass A:
The capacity ratio of A is 1:2:3.
Let's assume the capacity of glass A is x units.
Therefore, the capacities of glasses A, B, and C would be x, 2x, and 3x units, respectively.
The ratio of acid to water in A is given as 1:5.
So, the amount of acid in A is (1/6) * x units, and the amount of water is (5/6) * x units.
Glass B:
The ratio of acid to water in B is given as 3:5.
So, the amount of acid in B is (3/8) * 2x units, and the amount of water is (5/8) * 2x units.
Glass C:
The ratio of acid to water in C is given as 5:7.
So, the amount of acid in C is (5/12) * 3x units, and the amount of water is (7/12) * 3x units.
Now, let's add up the amounts of acid and water from each glass and combine them in the single vessel:
Total amount of acid = (1/6)x + (3/8) * 2x + (5/12) * 3x
Total amount of water = (5/6)x + (5/8) * 2x + (7/12) * 3x
Simplifying the above expressions:
Total amount of acid = (7/12)x
Total amount of water = (41/24)x
Finally, the ratio of acid to water in the vessel is:
(7/12)x : (41/24)x
Simplifying the ratio further:
7/12 : 41/24
Thus, the ratio of acid to water in the vessel is 7:41.