The average daily production of a manufacturing plant over a 16-day period is 65 units. The average production for the first 9 days is 60 units, and the average production for the last 5 days is 70 units. If the ratio of the output on the 10th day to the 11th day is 3 : 2, what was the production output on the 10th day?

The correct option is 90 units.

Step 1: Calculate the total production for all 16 days

The total production over the 16-day period is calculated by multiplying the total number of days by the overall average daily production:

$\text{Total production (16 days)}=16\times 65=1040\text{units}$

Step 2: Calculate the total production for the first 9 days and the last 5 days

Production for the first 9 days:

$\text{Total production (first 9 days)}=9\times 60=540\text{units}$

Production for the last 5 days (from day 12 to day 16):

$\text{Total production (last 5 days)}=5\times 70=350\text{units}$

Step 3: Calculate the combined production for the 10th and 11th days

The combined total production for the first 9 days and the last 5 days is:

$540+350=890\text{units}$

Subtracting this combined total from the overall 16-day total gives the sum of the production outputs on the 10th and 11th days:

$\text{Production on (10th + 11th day)}=1040-890=150\text{units}$

Step 4: Determine the production output on the 10th day using the given ratio

The ratio of the output on the 10th day to the 11th day is 3 : 2.

Let the production output on the 10th day be:

$3x$

And let the production output on the 11th day be:

$2x$

Summing both outputs:

$3x+2x=150$

$5x=150$

$x=\frac{150}{5}=30$

Now, calculate the production output on the 10th day:

$\text{Production on 10th day}=3x=3\times 30=90\text{units}$

Hence, the production output on the 10th day was 90 units.