AFCAT Numerical Ability · Average

Weighted Averages, Rates and Percentages

When values count unequally, multiply each by its weight, add, and divide by the total weight: for marks with weightage, percentage shares of a group, average speed, and members a percentage above or below the average.

All Average notes

Why this matters

About a fifth of this page is HARD, and most of those are percentage stories. Marks go when two rates are averaged directly (40 and 60 km/h do not average 50 unless the times are equal) and when '20% above the average' is worked out from the wrong average.

Concept 1 of 4: Weights given as numbers or ratios

A weight says how many times a value counts. A mark with weightage 2 counts as if it appeared twice.

Definition

Multiply each value by its weight, add, and divide by the sum of the weights.
Weights in a ratio such as $2 : 3 : 5$ can be used as 2, 3 and 5 directly.
Counts are weights: 40 people rating a product 4 stars puts a weight of 40 on the value 4.
The answer lies between the smallest and largest value, nearer the heavily weighted ones.

Weighted average

\bar x = \dfrac{w_1x_1 + w_2x_2 + \dots}{w_1 + w_2 + \dots}

Worked example

Marks of 70, 80 and 90 carry weights of 1, 2 and 2. Find the weighted average.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Average · Weighted Averages

In an examination, a student scored

85

marks in Mathematics (weightage

4

78

marks in Physics (weightage

3

), and

92

marks in Chemistry (weightage

3

). What is the weighted average score?

Divide by the weights, not the count

60 and 80 with weights 1 and 3 average

(60 + 240) \div 4 = 75

. Dividing 300 by 2 values gives 150, which is not even between them.

Drill 18 more on weights given as numbers or ratios

Concept 2 of 4: Averages of rates: speed, pass rate, return

A rate is one total per unit of another. To average rates, rebuild both totals: total distance over total time, total passes over total items.

Definition

Average speed = total distance ÷ total time. It equals the plain average of the speeds only when the times are equal.
Equal distances at speeds $u$ and $v$ : average speed $\dfrac{2uv}{u + v}$ .
A percentage rate (pass rate, efficiency, return) is weighted by its base: 500 items at a 96% pass rate give 480 passes.
Turn minutes into hours before dividing: 45 minutes is $\tfrac34$ hour.

Average speed

\text{average speed} = \dfrac{\text{total distance}}{\text{total time}}

Worked example

A bus goes 150 km at 50 km/h and then 120 km at 40 km/h. Find its average speed.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Average · Weighted Averages

The average speed of a car for the first

2

hours is

60

km/h and for the next

3

hours is

80

km/h. What is the average speed for the entire

5

hours?

Speeds do not average directly

Going out at 30 km/h and back the same way at 50 km/h averages 37.5 km/h, not 40, because more time is spent at the slower speed.

Drill 13 more on averages of rates: speed, pass rate, return

Concept 3 of 4: Percentage shares as weights, and the missing share

If 35% of a class are boys, picture 100 students: 35 boys and 65 girls. Percentage shares are simply weights that add up to 100.

Definition

Overall average = (share₁ × average₁ + share₂ × average₂ + ...) ÷ 100.
Reverse: overall × 100 − the known parts = the missing share × its average.
'The rest' is 100% minus the shares given.
An average stated relative to another ('Commerce averages 15% less than Science') is worked out first, then weighted.

Overall average from percentage shares

\bar x = \dfrac{p_1x_1 + p_2x_2 + \dots}{100}

Worked example

In a test, 45% of the students averaged 80 and the rest averaged 62. Find the overall average.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Average · Weighted Averages

In a class of

50

students,

60

% are boys. If the average marks of boys is

78

and the overall average is

75

, what is the average marks of girls?

Turn the percentage into people

'15% of 40 students failed' means 6 students. Use 6 as the count of that group, not 15.

Drill 32 more on percentage shares as weights, and the missing share

Concept 4 of 4: Values a percentage above or below the average

'20% above the average' is just a number. Work it out from the average it refers to, then treat it like any value that joins, leaves or replaces.

Definition

First find the value: $p\%$ above $A$ is $A(1 + \tfrac{p}{100})$ ; $p\%$ below is $A(1 - \tfrac{p}{100})$ . Then use totals as on the earlier pages.
Every value up by $p\%$ : the average goes up by $p\%$ too.
'The average rises by $p\%$ when one member joins': write the new average as $(1 + \tfrac{p}{100})$ times the old one and solve.
'The average fell by 8%' means the new average is $0.92$ times the old one.

A value relative to the average

\text{value} = A\left(1 \pm \dfrac{p}{100}\right)

Worked example

The average mark of 20 students is 70. Five students whose average is 30% below that leave. Find the new average.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 4 · Average · Weighted Averages

The average weight of

12

persons is

60

kg. If

4

persons leave the group and their average weight was

25

% more than the group average, what is the new average weight of the remaining persons?

Which average?

'20% above the group average' uses the average before anyone leaves or joins, unless the question says otherwise. Working it out from the new average gives a different number.

Drill 20 more on values a percentage above or below the average

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Weights given as numbers or ratios
Weighted average
$\bar x = \dfrac{w_1x_1 + w_2x_2 + \dots}{w_1 + w_2 + \dots}$
Averages of rates: speed, pass rate, return
Average speed
$\text{average speed} = \dfrac{\text{total distance}}{\text{total time}}$
Percentage shares as weights, and the missing share
Overall average from percentage shares
$\bar x = \dfrac{p_1x_1 + p_2x_2 + \dots}{100}$
Values a percentage above or below the average
A value relative to the average
$\text{value} = A\left(1 \pm \dfrac{p}{100}\right)$

Watch out for (4)

Divide by the weights, not the count
→ Weights given as numbers or ratios
Speeds do not average directly
→ Averages of rates: speed, pass rate, return
Turn the percentage into people
→ Percentage shares as weights, and the missing share
Which average?
→ Values a percentage above or below the average

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