Combined Average: Formula, Methods and Solved Examples

Combined Average is the average of two or more groups considered together. It is calculated by dividing the total value of all groups by their total number of items. The combined average formula uses each group’s size and average, so groups with different numbers of items must be weighted accordingly.

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What Is Combined Average?

Combined average is the overall average obtained by joining two or more groups. It equals the sum of the total values of all groups divided by the total number of items in those groups.

If a group has n items and average A, its total value is nA. For multiple groups, add these total values and divide by the sum of their sizes. Thus, the average of each group is multiplied by its number of items before combining.

Combined Average Formula & Tricks

Important Formulas

Combined average of two groups
Combined Average = (n₁A₁ + n₂A₂) / (n₁ + n₂)

Here, n₁ and n₂ are the group sizes, while A₁ and A₂ are their respective averages.

Combined average of multiple groups
Combined Average = (n₁A₁ + n₂A₂ + ... + nₖAₖ) / (n₁ + n₂ + ... + nₖ)

Multiply every group average by its size, add the products, and divide by the total number of items.

Total value from average
Total value = Number of items × Average

This relation converts a group average into the total sum needed for combination.

Quick Tricks

Use the weighted-average form

When group sizes are unequal, do not take the simple average of the group averages. Multiply each average by its group size first.

Example: For 20 students averaging 60 and 30 students averaging 70, the combined average is (20×60 + 30×70)/50 = 63.
Check the range

The combined average must lie between the smallest and largest group averages when all group sizes are positive.

Example: If two group averages are 50 and 80, their combined average must be between 50 and 80; 90 cannot be the answer.
Use deviation from a convenient value

Choose a base value and combine deviations with their group sizes to reduce calculation.

Example: For groups of 25 and 15 with averages 48 and 52, take 50 as the base: combined average = 50 + [25(−2) + 15(2)]/40 = 49.25.

Combined Average Concepts

Combining Two Groups with Different Sizes

For two groups, multiply each group’s average by its size and divide the sum by the total size.

If the first group has n₁ items with average A₁ and the second has n₂ items with average A₂, first find their totals as n₁A₁ and n₂A₂. Then use the combined average formula. The larger group has greater influence on the result.

Example: A class of 24 students has an average score of 55, and another class of 16 students has an average score of 65. Combined average = (24×55 + 16×65)/(24+16) = 2360/40 = 59.

Combining More Than Two Groups

For three or more groups, add the total values of all groups and divide by the total number of items.

For groups with sizes n₁, n₂, n₃ and averages A₁, A₂, A₃, the combined average is (n₁A₁+n₂A₂+n₃A₃)/(n₁+n₂+n₃). The same method extends to any number of groups.

Example: Three groups contain 10, 20 and 30 items with averages 40, 50 and 60. Combined average = (10×40 + 20×50 + 30×60)/60 = 52.

Finding a Missing Group Average

A missing group average can be found by subtracting the known group total from the combined total and dividing by the missing group size.

If the combined group has N items and average A, its total is NA. If a known group has n₁ items and average A₁, then the missing group average for n₂ items is [(NA) − (n₁A₁)]/n₂.

Example: A combined group of 50 items has average 62. One group has 20 items with average 55. The other group’s average is (50×62 − 20×55)/30 = 66⅔.

Relation Between Group Averages and Combined Average

The combined average is a weighted average, so it lies between the group averages and is closer to the average of the larger group.

For two positive group sizes, if A₁ < A₂, then A₁ < combined average < A₂, unless one group has zero size. If both groups have equal sizes, the combined average is the simple average (A₁+A₂)/2.

Example: Groups of equal size with averages 46 and 64 have combined average (46+64)/2 = 55.

Combined Average Video Lessons

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3 Lessons
Lesson 1 of 3 Quick Revision

Average of Two Groups

Understand how to calculate and compare the average of two groups using their group sizes and individual averages in quantitative aptitude problems.

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Practice Combined Average Questions

Practise published questions related to this topic.

1A bag was sold for ₹880 at a 12% loss. What was the cost price?→ 2An item is marked at Rs. 2000 and sold after two successive discounts of 20% and 10%. What is the selling price?→ 3What is the largest 2 digit number which is exactly divisible by both 2 and 7?→ 4Three bells ring together at 3:00 PM. They ring at intervals of 18 minutes, 24 minutes, and 32 minutes. After how much time will they ring together again?→ 5The HCF of two numbers is 6, their LCM is 36, and their sum is 30. Find the numbers.→ 63 different pieces of iron with lengths 81 cm, 54 cm and 36 cm. Rods are to be made by cutting these pieces. Each rod should be of same length. No iron is to be wasted. What is the length (in cm) of longest such rod?→ 7By selling an article for ₹117, a man loses 35%. At what price (in ₹) should he sell the same article to gain 50%?→ 8Simplify: √(144 × 49) + √(64 × 25) - √(100 × 36)→ 9A shopkeeper is offering a 20% discount on a Mixer that originally costs Rs 400. What is the sale price of the Mixer?→ 10If 4^x - 4^(x-1) = 48 , then the value of (2x-1)/(2x+3) is:→

Combined Average Quick Quiz

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Quick Revision Notes

Combined Average: Quick Revision

Use group size as the weight for every group average.

  • Total value of a group = Number of items × Group average.
  • Combined average = Sum of group totals ÷ Sum of group sizes.
  • For unequal group sizes, never take the simple average of the group averages.
  • The combined average lies between the smallest and largest group averages.
  • For equal-sized groups, combined average = Sum of group averages ÷ Number of groups.
  • To find a missing average, subtract the known total from the combined total and divide by the missing group size.

Combined Average FAQs

What is the combined average formula for two groups?

Combined Average = (n₁A₁ + n₂A₂)/(n₁+n₂), where n₁ and n₂ are group sizes and A₁ and A₂ are their averages.

Can the averages of two groups be added and divided by 2?

Only when the two groups have equal sizes. For unequal sizes, use the weighted formula: (n₁A₁+n₂A₂)/(n₁+n₂).

What is the combined average of 15 items with average 40 and 25 items with average 52?

Combined average = (15×40 + 25×52)/40 = 1900/40 = 47.5.

How is a group average converted into its total value?

Multiply the number of items by the group average. For example, 18 items with average 35 have a total value of 18×35 = 630.

If two groups have averages 45 and 75, can their combined average be 80?

No. With positive group sizes, the combined average must lie between 45 and 75.

What is the combined average when three equal-sized groups have averages 20, 30 and 40?

Since the groups have equal sizes, combined average = (20+30+40)/3 = 30.

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