CDS Mathematics · Averages

Weighted and Combined Averages

The average of two groups together is weighted by their sizes, so it always lies between the two group averages, nearer the bigger group.

All Averages notes CDS Mathematics strategy

Why this matters

Nineteen PYQs, one of them HARD. Two questions keep coming back: given the group sizes, find the combined average; or given all three averages, find the ratio of the group sizes. The second has a shortcut — the sizes are in the inverse ratio of the distances to the combined average.

Concept 1 of 2: The combined average

Pour two groups together and the total is the sum of their totals. Dividing by the combined count gives an average that sits between the two, pulled toward the larger group.

Definition

$\bar x = \dfrac{n_1\bar x_1 + n_2\bar x_2}{n_1 + n_2}$ ; for more groups, sum every $n_i\bar x_i$ .
It always lies between the smallest and largest group averages.
Averaging a RATE (price per pencil, kg per rupee) weights each rate by the quantity bought, not by the number of purchases.
A month's daily average: count each kind of day (five Sundays in a month starting on a Saturday).

Combined average

\bar x = \dfrac{n_1\bar x_1 + n_2\bar x_2}{n_1 + n_2}

Worked example

One class of

30

averages

64

marks; another of

20

averages

74

. Find the combined average.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (II) 2017 — Elementary Mathematics · Q100Easy

Example 1 · Averages · Weighted and Combined Averages

The average height of 22 students of a class is 140 cm and the average height of 28 students of another class is 152 cm. What is the average height of students of both the classes ?

Not the plain average of the averages

Averages of

140

and

152

combine to

146

only when the groups are equal. With more students in the taller class, the answer lies above

146

Drill 11 more on the combined average

Concept 2 of 2: Ratio of group sizes

The combined average balances the two groups like a see-saw. The group further from the balance point must be the lighter one, so the sizes are in the inverse ratio of the distances.

Definition

$n_1 : n_2 = (\bar x_2 - \bar x) : (\bar x - \bar x_1)$ .
Combined average exactly midway: the groups are equal.
Given the ratio and one group's average, the other follows from the same balance.
Three groups known in pairs: find each pair's ratio, then chain them.

Alligation

\dfrac{n_1}{n_2} = \dfrac{\bar x_2 - \bar x}{\bar x - \bar x_1}

Worked example

Boys average

62

kg, girls

47

kg, the whole class

56

kg. Find the ratio of boys to girls.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (II) 2018 — Elementary Mathematics · Q56Easy

Example 2 · Averages · Weighted and Combined Averages

The average marks of section A are 65 and that of section B are 70. If the average marks of both the sections combined are 67, then the ratio of number of students of section A to that of section B is

The ratio is inverted

The group CLOSER to the combined average is the BIGGER one. With means

65

and

70

and a combined

67

, the distances are

2

and

3

, so the sizes are

3 : 2

, not

2 : 3

Drill 6 more on ratio of group sizes

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 (2)

The combined average
Combined average
$\bar x = \dfrac{n_1\bar x_1 + n_2\bar x_2}{n_1 + n_2}$
Ratio of group sizes
Alligation
$\dfrac{n_1}{n_2} = \dfrac{\bar x_2 - \bar x}{\bar x - \bar x_1}$

Watch out for (2)

Not the plain average of the averages
→ The combined average
The ratio is inverted
→ Ratio of group sizes

Test yourself on Averages

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Averages chapter test Drill all 19 questions