Final answer:
Using the concept of average age, we calculate the future average age of the babies in the day care group. With the current ages, it will take 14 full months for the average age to surpass 20 months.
Step-by-step explanation:
To find out how many full months from now will the average age of the 21 babies in the day care group first surpass 20 months old, we can use the concept of the average age of the group.
Currently, there are 9 babies who are 5 months old and 12 babies who are 7 months old. To calculate their current combined age, we do the following:
- 9 babies × 5 months = 45 months
- 12 babies × 7 months = 84 months
Adding these together gives us a total of 129 months for the 21 babies.
The current average age is calculated by dividing the total combined age by the number of babies:
129 months ÷ 21 babies = 6.14 months (This is the current average age.)
We want to know when the average age will exceed 20 months. Let's call the number of months from now that we are looking for x. In x months, the total age of all babies will increase by x months for each baby:
21 babies × x months = 21x months
The combined age in x months will then be:
129 months + 21x months
The average age in x months is:
(129 + 21x) ÷ 21
We want this average to be just over 20 months:
(129 + 21x) ÷ 21 > 20
Multiplying both sides by 21 to get rid of the denominator gives us:
129 + 21x > 420
Subtracting 129 from both sides gives us:
21x > 291
Dividing both sides by 21:
x > 13.857
Since we cannot have a fraction of a month, we round up to the next full month.
Therefore, it will take 14 full months for the average age of the babies to surpass 20 months.