Average Questions and Answers

Answer: B

The combined average is \(75\frac{5}{7}\). The total marks are the sum of the two sections' totals, divided by the total number of students.

Since 70 times 75 is 5250 and the remaining 50 gives \(\frac{50}{70} = \frac{5}{7}\), the options with 74, 76 or 77 as the whole part do not equal 5300 divided by 70.

A weighted average is used because the sections have different sizes. A simple average of 70 and 80 would give 75, but section B has more students (40 against 30), so the combined mean is pulled above 75, to \(75\frac{5}{7}\), which is the first check that option A, 74 5/7, cannot be right.

Answer: C

The average monthly expenditure is Rs.5,500. Total income for the year is \[ 2 \times 5000 + 4 \times 6000 + 6 \times 7000 = 76000 \] Savings are 10,000, so expenditure is 66,000 over 12 months, which is 5,500 a month.

Rs.5,000 would leave savings of 16,000, Rs.6,000 would leave 4,000, and Rs.6,500 would leave a deficit of 2,000.

Check: monthly average income is 76000 / 12, about 6,333, and monthly savings are 10000 / 12, about 833, which leaves 5,500 spent.

Answer: B

The sixth student weighs 48 kg. All eleven weigh 11 × 54 = 594 kg. The first six weigh 6 × 55 = 330 kg and the last six weigh 6 × 52 = 312 kg. These groups overlap only at the sixth student, so \[ 330 + 312 - 594 = 48 \]

So the sixth student is 48 kg, not 49, 50 or 51.

Answer: C

The average is 55. The sum of 40, 50, 60 and 70 is 220, and dividing by 4 gives 55. 50 is the average of only the first three, 45 would be too low for these numbers, and 65 is above the middle of the set. The four numbers are evenly spaced with a difference of 10, so their average is also the midpoint of the first and the last, (40 + 70)/2 = 55.

Answer: C

The new average weight is 67.675 kg. The total weight of the 35 students is 35 x 66 = 2310 kg. When six students of average 58.5 kg leave, 351 kg goes out, and when eleven students of average 68 kg join, 748 kg comes in. The class then has 35 - 6 + 11 = 40 students.

\[ \text{Total} = 2310 - 351 + 748 = 2707 \text{ kg} \]

\[ \text{Average} = \frac{2707}{40} = 67.675 \text{ kg} \]

The options 66.375, 66.5 and 68.5 do not equal 2707 divided by 40, so the third option is the answer. As a check, 40 students at 67.675 kg each weigh 2707 kg, which is the total found above.

Answer: A

The average is 68.

Ten students average 80, giving a total of 800. The other 15 students average 60, giving 900. The total for 25 students is 1700.

\[ \frac{800+900}{25}=68 \]

78, 65 and 64 do not come out of this calculation.

This is a weighted average. Ten students bring 800 marks and the other fifteen bring 900, so the total is 1700 for 25 students. Since 80 and 60 are the two group averages, the class average must lie between them and nearer 60 because there are more students in that group, which gives 68.

Answer: A

The average speed is 2.17 km/h.

The walking times for the five kilometres are 1/2, 1/3, 1/5, 1/6 and 1/10 hours, which add up to 39/30 = 1.3 hours. He rests 15 minutes after each kilometre except at the end, that is four rests, which add 1 hour. The total time is 2.3 hours.

\[ \frac{5}{2.3}\approx2.17 \]

Counting a fifth rest would give 1.96, which is not an option.

Average speed is total distance divided by total time, never the mean of the speeds. The time on the road is 1.3 hours and the rests add 1 hour, so the total is 2.3 hours. The mean of the five speeds, 5.2 km/h, would be wrong.

Answer: C

The average is 5.5.

The first 10 natural numbers add up to 55, and 55 divided by 10 is 5.5.

55 is the sum, not the mean, 0.5 and 5 do not equal the mean.

The mean of the first n natural numbers is (n + 1)/2. For n = 10 this is 11/2 = 5.5. It is also the middle of the list between the fifth and sixth numbers, 5 and 6, which is another way to see that it is not 5.

Answer: C

The largest possible value is 50.

Five numbers with mean 12 add up to \(5 \times 12 = 60\). To make one number as large as possible, make the other four as small as possible. They must be different natural numbers, so the smallest choice is 1, 2, 3 and 4, which add to 10.

\[ \text{largest} = 60 - (1 + 2 + 3 + 4) = 50 \]

60 would need the other four numbers to be zero, and zero is not a natural number, nor could four zeros be different. 12 is just the mean and 40 is well below what is achievable.

Answer: B

Let the sale in the sixth month \(=x\)
Then, \(\dfrac{6435+6927+6855+7230+6562+x}{6}=6500\)
\(\Rightarrow 6435+6927+6855+7230+6562+x=6 \times 6500\\ \Rightarrow 34009 + x = 39000\\ \Rightarrow x = 39000 - 34009 = 4991\)