Linda mixed 1/4/9 liters of carrot juice with 2/5/9 liters of orange juice. She poured the mixture into some glasses. If there were 2/3 liters of the mixture in each glass , how many glasses did she use?

Answer 1

To find out how many glasses Linda used for the juice mixture, we first need to calculate the total volume of the mixture.

1. Combine the volumes of carrot juice and orange juice:

   Carrot juice: \( \frac{1}{4} \) liters
   Orange juice: \( \frac{2}{5} \) liters

   To add these two fractions together, we need a common denominator. The least common multiple (LCM) of 4 and 5 is 20.

   • Convert \( \frac{1}{4} \) to a fraction with a denominator of 20: \[ \frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20} \]

   • Convert \( \frac{2}{5} \) to a fraction with a denominator of 20: \[ \frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20} \]

   Now, we can add the two fractions: \[ \frac{5}{20} + \frac{8}{20} = \frac{5 + 8}{20} = \frac{13}{20} \]

   So, the total volume of the juice mixture is \( \frac{13}{20} \) liters.

2. Calculate the number of glasses used:

   Each glass holds \( \frac{2}{3} \) liters of the mixture. To find the number of glasses, we divide the total volume of the mixture by the volume of each glass: \[ \text{Number of glasses} = \frac{\text{Total volume}}{\text{Volume per glass}} = \frac{\frac{13}{20}}{\frac{2}{3}} \]

   Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{\frac{13}{20}}{\frac{2}{3}} = \frac{13}{20} \times \frac{3}{2} = \frac{13 \times 3}{20 \times 2} = \frac{39}{40} \]

   The division indicates that the number of glasses is \( \frac{39}{40} \). Since a complete glass of juice must be a whole number, we can infer that Linda did not fill a complete additional glass.

3. Conclusion:

   Since the answer does not yield a whole number, we interpret this to mean that Linda filled \( \lfloor \frac{39}{40} \rfloor = 0 \) additional glasses. Thus, Linda used \( 0 \) full glasses of juice.

However, it's more likely that this scenario indicates that Linda could fill just under one full glass, so if she attempted to pour it into glasses, she only managed to use one glass partially (if needed, but functions as 0 glasses filled completely).

Thus, Linda used 0 full glasses of the mixture, as each glass could not be filled under these conditions with the provided juice volumes.