The frozen yogurt machine dispenses yogurt at a constant rate. The machine needs [tex]\(\frac{1}{4}\)[/tex] hour to dispense [tex]\(\frac{3}{4}\)[/tex] of the total amount of yogurt used for each lunch period.

How long will it take the frozen yogurt machine to dispense the total amount of frozen yogurt used for each lunch period? Show or explain your reasoning.



Answer :

Let's determine how long it will take the frozen yogurt machine to dispense the total amount of frozen yogurt used for each lunch period.

We know the following details:
- The machine needs [tex]\(\frac{1}{4}\)[/tex] hour (or 15 minutes) to dispense [tex]\(\frac{3}{4}\)[/tex] of the total amount of yogurt.

First, we need to find the rate at which the machine dispenses the yogurt.

1. Dispensing Rate Calculation:
The dispensing rate can be found by dividing the portion of yogurt dispensed by the time taken.
[tex]\[ \text{Dispensing Rate} = \frac{\frac{3}{4} \text{ (fraction of yogurt)}}{\frac{1}{4} \text{ (hours)}} = \frac{3/4}{1/4} = 3 \text{ (total yogurt per hour)} \][/tex]

2. Time to Dispense Total Yogurt:
The total amount of yogurt is equivalent to 1 (since it represents the whole amount of yogurt for a lunch period). To find out the total time required to dispense all the yogurt, we use the previously calculated dispensing rate.
[tex]\[ \text{Time to dispense total yogurt} = \frac{\text{Total yogurt}}{\text{Dispensing rate}} = \frac{1 \text{ (total yogurt)}}{3 \text{ (total yogurt per hour)}} = \frac{1}{3} \text{ hour} \][/tex]

Thus, it will take the frozen yogurt machine [tex]\(\frac{1}{3}\)[/tex] hour (or 20 minutes) to dispense the total amount of frozen yogurt used for each lunch period.