Question

In a shipment of \[120\] machine parts \[5\] percent were defective. In a shipment of \[80\] machine parts \[10\] percent were defective. For the two shipments combined what percent of the machine parts were defective?
A. \[6.5\% \]
B. \[7.0\% \]
C. \[7.5\% \]
D. \[8.0\% \]

Answer 1

Hint: Calculate the number of defective pieces in each shipment separately using the given information and then take the sum of defective pieces from both the shipments and use that sum to find the percentage of defective pieces of combined shipments.
* Percentage refers to the portion or part of a whole. Calculate percentage by the formula
\[p\% \] of \[x = \dfrac{p}{{100}} \times x\]
* Also, convert any fraction into percentage by multiplying it to \[100\].

Complete step-by-step answer:
Given, a shipment of \[120\] machine parts \[5\] percent were defective.
Therefore number of defective pieces in shipment of \[120\] pieces \[ = \] \[5\% \] of \[120\]
Using the formula to calculate percentage of a quantity
\[5\% \] of \[120 = \dfrac{5}{{100}} \times 120\]
                     \[ = \dfrac{{5 \times 12 \times 10}}{{100}} = \dfrac{{60}}{{10}} = 6\]
Therefore, number of defective pieces in shipment of \[120\] pieces is \[6\] \[...(i)\]
Similarly, calculate for the second shipment.
Given, a shipment of \[80\] machine parts \[10\] percent were defective.
Therefore number of defective pieces in shipment of \[80\] pieces \[ = \] \[10\% \] of \[80\]
Using the formula to calculate percentage of a quantity
\[10\% \] of \[80 = \dfrac{{10}}{{100}} \times 80\]
                     \[ = \dfrac{{10 \times 80}}{{100}} = \dfrac{{800}}{{100}} = 8\]
Therefore, number of defective pieces in shipment of \[80\] pieces is \[8\] \[...(ii)\]
Now, to calculate the combined percentage of defective pieces from the total of two shipments.
Total number of defective pieces \[ = \] number of defective pieces from first shipment \[ + \]number of defective pieces from second shipment.
From equations \[(i)\] and \[(ii)\]
Total number of defective pieces \[ = 6 + 8 = 14\]
Also, total number of machine parts \[ = \] number of machine parts in first shipment \[ + \]number of machine parts in second shipment.
Total number of machine parts \[ = 120 + 80 = 200\]
Percentage of defective pieces from total shipment \[ = \] total defective pieces/machine parts in total shipment \[ \times 100\]
Percentage of defective pieces from total shipment \[ = \dfrac{{14}}{{200}} \times 100 = \dfrac{{14}}{2} = 7\]

Therefore, the percentage of defective pieces from total shipment is \[7\% \]. So, option B is correct.

Note: Students are likely to make mistakes while converting a percentage into a fraction, always try to cancel out multiples of \[10\] from the denominator and numerator to make calculations easier. Always remember the percentage part of the whole will always be less than the whole because the percentage gives a part of the whole, so it can never be greater than the whole, this concept comes in handy to verify the answers.