Question

tax returns filed manually have a 20% chance of containing errors, while tax returns filed electronically have a 0.05% chance of containing the same. if 2.7 million tax returns are filed each way, how many more erroneous manually filed returns will there be than erroneous electronically filed returns?
a. 270,675
b. 538,650
c. 541,350
d. 269,325
please select the best answer from the choices provided
o a
o b
o c
o d

Explanation:

Step1: Find number of manual returns

Total returns: 2.7 million = 2,700,000. Since half are manual, manual returns = $\frac{2,700,000}{2}$ = 1,350,000.

Step2: Calculate manual errors

Manual error rate: 20% = 0.2. Manual errors = $1,350,000 \times 0.2$ = 270,000. Wait, no—wait, electronic returns: also half, 1,350,000. Electronic error rate: 0.05% = 0.0005. Electronic errors = $1,350,000 \times 0.0005$ = 675. Wait, no, the question is "how many more erroneous manual than electronic". So manual errors: 20% of 1.35M = 0.21,350,000 = 270,000. Electronic errors: 0.05% of 1.35M = 0.00051,350,000 = 675. Wait, no, that can't be. Wait, maybe I misread. Wait, 2.7 million total, each way (manual and electronic) so each is 1.35 million. Manual error rate: 20% (0.2), electronic: 0.05% (0.0005). So manual errors: 1,350,000 0.2 = 270,000. Electronic errors: 1,350,000 0.0005 = 675. Then difference: 270,000 - 675 = 269,325? Wait, no, wait the options: d is 269,325. Wait, maybe I miscalculated manual. Wait, 20% of 1.35 million: 1,350,000 0.2 = 270,000. Electronic: 1,350,000 0.05% = 1,350,000 0.0005 = 675. Then 270,000 - 675 = 269,325. Wait, but option a is 270,675. Wait, maybe I messed up the error rates. Wait, manual: 20% (0.2), electronic: 0.05% (0.0005). Wait, 20% is 0.2, 0.05% is 0.0005. So manual errors: 1.35M 0.2 = 270,000. Electronic: 1.35M 0.0005 = 675. Then 270,000 - 675 = 269,325, which is option d. Wait, but let's recalculate: 1,350,000 0.2 = 270,000. 1,350,000 * 0.0005 = 675. 270,000 - 675 = 269,325. Yes, that's option d. Wait, but maybe the error rates are 20% and 0.05% (not 0.05% as 0.0005, but 0.05% is 0.0005). Wait, 0.05% is 0.05 per 100, so 0.05/100 = 0.0005. Correct. So difference is 270,000 - 675 = 269,325. So option d.

Step1: Calculate manual errors

Manual returns: $2,700,000 \div 2 = 1,350,000$.
Manual errors: $1,350,000 \times 0.2 = 270,000$.

Step2: Calculate electronic errors

Electronic returns: $1,350,000$ (same as manual, since total is split evenly).
Electronic error rate: $0.05\% = 0.0005$.
Electronic errors: $1,350,000 \times 0.0005 = 675$.

Step3: Find the difference

Difference: $270,000 - 675 = 269,325$.

Answer:

d. 269,325