Answer :
Sure, let’s break down the problem step-by-step to find the value of [tex]\(A + B\)[/tex]:
### Step 1: Initial Setup
- You have an initial volume of solution = 200 ml.
- The solution contains 25% milk.
- The initial volume of milk in the solution = 25% of 200 ml = [tex]\( \frac{25}{100} \times 200 = 50 \)[/tex] ml.
- The initial volume of water in the solution = 200 ml - 50 ml = 150 ml.
### Step 2: Adding Milk to Reverse the Ratio
- Initially, the milk to water ratio is 25:75 (or 1:3).
- We need to reverse this ratio, so we need to make the solution 75% milk.
Let [tex]\( A \)[/tex] ml of milk be added to reach 75% milk.
- New total volume of the solution = [tex]\( 200 + A \)[/tex] ml.
- New volume of milk = [tex]\( 50 + A \)[/tex] ml.
For the solution to be 75% milk:
[tex]\[ \frac{50 + A}{200 + A} = 0.75 \][/tex]
Solve for [tex]\( A \)[/tex]:
[tex]\[ 50 + A = 0.75 (200 + A) \][/tex]
[tex]\[ 50 + A = 150 + 0.75A \][/tex]
[tex]\[ A - 0.75A = 150 - 50 \][/tex]
[tex]\[ 0.25A = 100 \][/tex]
[tex]\[ A = \frac{100}{0.25} \][/tex]
[tex]\[ A = 400 \text{ ml} \][/tex]
### Step 3: Adding Water to Reverse the Ratio Again
- After adding 400 ml of milk, the solution has 75% milk (or 3:1 ratio of milk to water).
- The total volume after adding milk = 200 ml + 400 ml = 600 ml.
- The volume of milk = 50 ml + 400 ml = 450 ml.
- The volume of water = 600 ml - 450 ml = 150 ml.
Next, add [tex]\( B \)[/tex] ml of water to make the milk to water ratio 25:75 (1:3) again.
- New total volume of the solution = [tex]\( 600 + B \)[/tex] ml.
- The total volume of milk remains 450 ml.
For the solution to be 25% milk:
[tex]\[ \frac{450}{600 + B} = 0.25 \][/tex]
Solve for [tex]\( B \)[/tex]:
[tex]\[ 450 = 0.25 (600 + B) \][/tex]
[tex]\[ 450 = 150 + 0.25B \][/tex]
[tex]\[ 450 - 150 = 0.25B \][/tex]
[tex]\[ 300 = 0.25B \][/tex]
[tex]\[ B = \frac{300}{0.25} \][/tex]
[tex]\[ B = 1200 \text{ ml} \][/tex]
### Step 4: Calculate [tex]\( A + B \)[/tex]
[tex]\[ A = 400 \text{ ml} \][/tex]
[tex]\[ B = 1200 \text{ ml} \][/tex]
[tex]\[ A + B = 400 + 1200 = 1600 \text{ ml} \][/tex]
So, the value of [tex]\( A + B \)[/tex] is [tex]\( \boxed{1600} \)[/tex].
### Step 1: Initial Setup
- You have an initial volume of solution = 200 ml.
- The solution contains 25% milk.
- The initial volume of milk in the solution = 25% of 200 ml = [tex]\( \frac{25}{100} \times 200 = 50 \)[/tex] ml.
- The initial volume of water in the solution = 200 ml - 50 ml = 150 ml.
### Step 2: Adding Milk to Reverse the Ratio
- Initially, the milk to water ratio is 25:75 (or 1:3).
- We need to reverse this ratio, so we need to make the solution 75% milk.
Let [tex]\( A \)[/tex] ml of milk be added to reach 75% milk.
- New total volume of the solution = [tex]\( 200 + A \)[/tex] ml.
- New volume of milk = [tex]\( 50 + A \)[/tex] ml.
For the solution to be 75% milk:
[tex]\[ \frac{50 + A}{200 + A} = 0.75 \][/tex]
Solve for [tex]\( A \)[/tex]:
[tex]\[ 50 + A = 0.75 (200 + A) \][/tex]
[tex]\[ 50 + A = 150 + 0.75A \][/tex]
[tex]\[ A - 0.75A = 150 - 50 \][/tex]
[tex]\[ 0.25A = 100 \][/tex]
[tex]\[ A = \frac{100}{0.25} \][/tex]
[tex]\[ A = 400 \text{ ml} \][/tex]
### Step 3: Adding Water to Reverse the Ratio Again
- After adding 400 ml of milk, the solution has 75% milk (or 3:1 ratio of milk to water).
- The total volume after adding milk = 200 ml + 400 ml = 600 ml.
- The volume of milk = 50 ml + 400 ml = 450 ml.
- The volume of water = 600 ml - 450 ml = 150 ml.
Next, add [tex]\( B \)[/tex] ml of water to make the milk to water ratio 25:75 (1:3) again.
- New total volume of the solution = [tex]\( 600 + B \)[/tex] ml.
- The total volume of milk remains 450 ml.
For the solution to be 25% milk:
[tex]\[ \frac{450}{600 + B} = 0.25 \][/tex]
Solve for [tex]\( B \)[/tex]:
[tex]\[ 450 = 0.25 (600 + B) \][/tex]
[tex]\[ 450 = 150 + 0.25B \][/tex]
[tex]\[ 450 - 150 = 0.25B \][/tex]
[tex]\[ 300 = 0.25B \][/tex]
[tex]\[ B = \frac{300}{0.25} \][/tex]
[tex]\[ B = 1200 \text{ ml} \][/tex]
### Step 4: Calculate [tex]\( A + B \)[/tex]
[tex]\[ A = 400 \text{ ml} \][/tex]
[tex]\[ B = 1200 \text{ ml} \][/tex]
[tex]\[ A + B = 400 + 1200 = 1600 \text{ ml} \][/tex]
So, the value of [tex]\( A + B \)[/tex] is [tex]\( \boxed{1600} \)[/tex].
