Answer :
Sure! Let's solve this step-by-step.
First, let's introduce variables to represent the initial amounts of milk:
- Let [tex]\( M \)[/tex] be the amount of milk Mia had initially.
- Let [tex]\( D \)[/tex] be the amount of milk Dante had initially.
### Step 1: Express the Situation Mathematically
1. Mia gave [tex]\(\frac{1}{4}\)[/tex] of her milk to Dante.
- The amount of milk given to Dante is [tex]\(\frac{1}{4}M\)[/tex].
- After giving the milk away, Mia has [tex]\( M - \frac{1}{4}M \)[/tex] amount of milk left, which simplifies to [tex]\(\frac{3}{4}M\)[/tex].
2. Dante receives the milk from Mia.
- Dante's new amount of milk is [tex]\( D + \frac{1}{4}M \)[/tex].
### Step 2: Condition based on the Problem
Dante ends up having twice the amount of milk as Mia after receiving the milk. Mathematically, we can write this condition as:
[tex]\[ D + \frac{1}{4}M = 2 \left(\frac{3}{4}M\right) \][/tex]
### Step 3: Simplifying the Equation
Next, we simplify the equation:
[tex]\[ D + \frac{1}{4}M = 2 \cdot \frac{3}{4}M \][/tex]
[tex]\[ D + \frac{1}{4}M = \frac{6}{4}M \][/tex]
[tex]\[ D + \frac{1}{4}M = \frac{3}{2}M \][/tex]
[tex]\[ D = \frac{3}{2}M - \frac{1}{4}M \][/tex]
[tex]\[ D = \frac{6}{4}M - \frac{1}{4}M \][/tex]
[tex]\[ D = \frac{5}{4}M \][/tex]
### Step 4: Find the Ratio
We now have [tex]\( D = \frac{5}{4}M \)[/tex]. To find the ratio of the amount of milk Mia had to the amount of milk Dante had initially, we compare [tex]\( M \)[/tex] to [tex]\( \frac{5}{4}M \)[/tex].
Thus, we can express it as a ratio:
[tex]\[ M : D = M : \frac{5}{4}M \][/tex]
Dividing both parts of the ratio by [tex]\( M \)[/tex]:
[tex]\[ 1 : \frac{5}{4} \][/tex]
To get rid of the fraction, multiply both parts by 4:
[tex]\[ 4 : 5 \][/tex]
### Conclusion
The ratio of the amount of milk Mia had to the amount of milk Dante had at first is [tex]\( 4 : 5 \)[/tex].
First, let's introduce variables to represent the initial amounts of milk:
- Let [tex]\( M \)[/tex] be the amount of milk Mia had initially.
- Let [tex]\( D \)[/tex] be the amount of milk Dante had initially.
### Step 1: Express the Situation Mathematically
1. Mia gave [tex]\(\frac{1}{4}\)[/tex] of her milk to Dante.
- The amount of milk given to Dante is [tex]\(\frac{1}{4}M\)[/tex].
- After giving the milk away, Mia has [tex]\( M - \frac{1}{4}M \)[/tex] amount of milk left, which simplifies to [tex]\(\frac{3}{4}M\)[/tex].
2. Dante receives the milk from Mia.
- Dante's new amount of milk is [tex]\( D + \frac{1}{4}M \)[/tex].
### Step 2: Condition based on the Problem
Dante ends up having twice the amount of milk as Mia after receiving the milk. Mathematically, we can write this condition as:
[tex]\[ D + \frac{1}{4}M = 2 \left(\frac{3}{4}M\right) \][/tex]
### Step 3: Simplifying the Equation
Next, we simplify the equation:
[tex]\[ D + \frac{1}{4}M = 2 \cdot \frac{3}{4}M \][/tex]
[tex]\[ D + \frac{1}{4}M = \frac{6}{4}M \][/tex]
[tex]\[ D + \frac{1}{4}M = \frac{3}{2}M \][/tex]
[tex]\[ D = \frac{3}{2}M - \frac{1}{4}M \][/tex]
[tex]\[ D = \frac{6}{4}M - \frac{1}{4}M \][/tex]
[tex]\[ D = \frac{5}{4}M \][/tex]
### Step 4: Find the Ratio
We now have [tex]\( D = \frac{5}{4}M \)[/tex]. To find the ratio of the amount of milk Mia had to the amount of milk Dante had initially, we compare [tex]\( M \)[/tex] to [tex]\( \frac{5}{4}M \)[/tex].
Thus, we can express it as a ratio:
[tex]\[ M : D = M : \frac{5}{4}M \][/tex]
Dividing both parts of the ratio by [tex]\( M \)[/tex]:
[tex]\[ 1 : \frac{5}{4} \][/tex]
To get rid of the fraction, multiply both parts by 4:
[tex]\[ 4 : 5 \][/tex]
### Conclusion
The ratio of the amount of milk Mia had to the amount of milk Dante had at first is [tex]\( 4 : 5 \)[/tex].
