To find the number of days it will take for a family to empty a 22-liter water jar, we need to determine how much water the family drinks in a day and then divide the total capacity of the jar by this amount.

Given that the family drinks 2 1/4 liters of water in a day, we need to convert this mixed fraction into an improper fraction. To do this, we multiply the whole number (2) by the denominator (4) and add the numerator (1) to get the numerator of the improper fraction. The denominator remains the same. So, 2 1/4 can be written as (2 * 4 + 1) / 4 = 9/4 liters.

Now, let's calculate how many times the family drinks 9/4 liters in order to empty the 22-liter water jar. We can do this by dividing the total capacity of the jar (22 liters) by the amount the family drinks in a day (9/4 liters).

22 / (9/4) can be simplified by multiplying the numerator (22) by the reciprocal of the denominator (4/9). This gives us 22 * (4/9) = 88/9 liters.

Therefore, the family will empty the 22-liter water jar in (88/9) days.

To convert this improper fraction into a mixed fraction, we divide the numerator (88) by the denominator (9). The quotient is the whole number, and the remainder becomes the numerator of the fraction. So, 88 divided by 9 is 9 with a remainder of 7. This means that the family will empty the water jar in 9 days with 7/9 of a day remaining.

So, the final answer is that it will take the family 9 days and 7/9 of a day to empty the 22-liter water jar.

8 day