Averages and Ratios — AAI ATC GK Lesson

Before you start

Learning objectives and outcomes

Objective 1Correctly combine averages across groups of different sizes.

Objective 2Avoid the average-speed trap by using total distance over total time.

Objective 3Combine two ratios sharing a common term without error.

Outcome AUse the pair-average technique to isolate a single unknown from two given pair-averages.

Subtopic 1

Average essentials

An average summarises a group of values into a single representative number — but combining two group averages correctly requires weighting by group size, not simply averaging the averages.

Core average rules

Average = (Sum of values) / (Number of values)
Combined average of two groups = (n₁a₁ + n₂a₂) / (n₁+n₂)
Average speed (equal distances) = Total distance / Total time, never the simple average of two speeds

The average-speed trap is one of the most common mistakes in this whole topic: for equal distances travelled at two different speeds, the correct average is never the arithmetic mean of the two speeds.

✈ AAI ATC relevance — combined-average and average-speed traps are tested almost every attempt, precisely because the intuitive shortcut is wrong

Memory trick — "Never average an average without weights"

Whenever two groups of different sizes are combined, or two speeds cover the same distance, resist averaging the two numbers directly — convert to totals (sum of values, or total time) first.

Easy

The average of 5, 10, 15, 20 and 25 is:

The total is 75, and there are five values, so the average is 15.

Moderate

A car travels 60 km at 30 km/h and returns the same 60 km at 60 km/h. The average speed for the whole trip is:

Average speed = total distance/total time = 120/(2+1) = 40 km/h, not the simple average of 45.

Subtopic 2

Ratio fundamentals

A ratio compares two or more quantities by division. Simplifying, dividing an amount, and combining two ratios that share a common term are the three core ratio skills tested.

Core ratio operations

Dividing an amount in ratio a:b:c — each part = total × (its ratio number)/(sum of all ratio numbers)
Combining a:b and b:c — scale both so the shared term (b) matches, then read off a:c

To combine a:b=3:4 with b:c=8:9, scale the first ratio so its b-value (4) matches the second ratio's b-value (8) — multiplying 3:4 by 2 gives 6:8, which now shares its second term with 8:9, giving a:b:c = 6:8:9.

Figure 2.1 — Combining two ratios via a shared term A simple diagram showing ratio a:b and ratio b:c side by side, with an arrow showing both being scaled until their shared 'b' terms match, producing a single combined a:b:c ratio. Upload image here

Memory trick — "Match the shared term before combining"

Two ratios can only be combined once the value they share is scaled to be identical in both — skipping this step is the most common ratio-combination mistake.

Moderate

Rs. 1,200 is divided among three people in the ratio 2:3:5. The largest share is:

The ratio has 10 parts total, so each part is 120. The largest share, 5 parts, is Rs. 600.

Hard

If a:b = 3:4 and b:c = 8:9, then a:c is:

Scaling a:b=3:4 to 6:8 matches b in b:c=8:9, giving a:b:c=6:8:9, so a:c = 6:9 = 2:3.

Subtopic 3 · highest yield

Mixed average-ratio problems

The hardest questions in this topic combine averages and ratios in the same problem — typically requiring an equation set up from pair-averages, or a proportional-scaling insight.

A key insight for pair-average problems

If pair-averages of A+B and B+C are known, then (A+B) + (B+C) = A + 2B + C
Subtracting the three-item total isolates B directly

When only pair-averages are given but a single value is asked for, adding the two pair-totals together deliberately double-counts the shared item — which is exactly what makes it possible to isolate that shared item.

✈ AAI ATC relevance — the pair-average technique, and recognising that weighted combinations of equal ratios keep the same ratio, are the two most valuable shortcuts in this whole topic

Memory trick — "Adding pair-totals double-counts the shared item on purpose"

The trick to these problems is realising the double-counting is useful, not a mistake — subtracting the known three-item total from the sum of the two pair-totals leaves exactly the shared item, isolated.

Hard

The average price of three stationery items A, B and C is Rs. 80. The average price of A and B is Rs. 77.5, and of B and C is Rs. 87.5. The price of B is:

Total of all three = 240. (A+B)+(B+C) = 155+175 = 330 = A+2B+C. Subtracting 240 leaves B = 90.

Moderate

If a:b = c:d = e:f = 2:3, then (xa+yc+ze):(xb+yd+zf) equals:

When several ratios are all equal, any weighted sum of the numerators bears the same ratio to the matching weighted sum of the denominators — the answer is simply 2:3.

Exam strategy

Facts worth having automatic

Combined averageWeight by group size — never average two averages directly.

Average speed (equal distances)Total distance ÷ total time, never the mean of the two speeds.

Combining ratiosScale both until the shared term matches exactly.

Pair-average trickAdd the two pair-totals; subtract the known combined total to isolate the shared item.

Weighted equal ratiosAny weighted combination of equal ratios keeps the same ratio.

Quick revision

Chapter summary sheet

AverageSum of values ÷ count.

Combined average(n₁a₁+n₂a₂)/(n₁+n₂).

Average speedTotal distance ÷ total time.

Dividing in ratioEach part = total × (its number)/(sum of ratio numbers).

Combining ratiosScale until the shared term matches.

Pair-average isolateSum pair-totals, subtract the known combined total.

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End of chapter

Comprehensive assessment

Full solutions are in the downloadable PDF above.

Multiple choice questions

1. Find the average of 5, 10, 15, 20 and 25.

2. A car travels 60 km at 30 km/h and returns at 60 km/h. Find the average speed.

3. Divide Rs. 1,200 among three people in the ratio 2:3:5 and find the largest share.

4. If a:b=3:4 and b:c=8:9, find a:c.

5. The average price of A, B, C is 80; A+B averages 77.5, B+C averages 87.5. Find B's price.

6. If a:b=c:d=e:f=2:3, find (xa+yc+ze):(xb+yd+zf) for any x, y, z.

7. Find the mean proportional between 4 and 16.

8. Two numbers are in ratio 3:5 and sum to 64. Find the smaller number.

9. The average of 10 numbers is 20. One number, 15, is removed. Find the new average.

10. A batsman scores 79 in his 18th innings, raising his average by 2. Find his new average.

Learning objectives and outcomes

Average essentials

Core average rules

Ratio fundamentals

Core ratio operations

Mixed average-ratio problems

A key insight for pair-average problems

Facts worth having automatic

Chapter summary sheet

↓ Free download — Averages and Ratios question PDF

Comprehensive assessment

Multiple choice questions

Averages and Ratios — one more topic locked in.