Averages, Ratio & Percentage

Overview

Averages, Ratios, and Proportions form one of the most tested arithmetic clusters on both CAT and GMAT. Average problems blend with mixtures and data interpretation; ratio problems connect to partnerships, ages, and work. Together they appear in 2–4 CAT questions per exam and are staple 500–650 difficulty GMAT Problem Solving questions.

Averages

Average = Sum of items / Number of items

Sum = Average × Number of items — this form is more useful when solving for missing elements.

Adding / Removing an element

If the average of n items is A, and a new item with value x is added:

• New average = (n×A + x) / (n+1)
• If new average rises to A', then x = (n+1)A' − nA

If an element with value x is removed:

• New average = (n×A − x) / (n−1)

Example: Average of 8 numbers is 40. A 9th number is added; average becomes 42. The new number = 9×42 − 8×40 = 378 − 320 = 58.

Weighted Average

When two groups (sizes n₁ and n₂, averages A₁ and A₂) are combined:

Combined average = (n₁A₁ + n₂A₂) / (n₁ + n₂)

This is identical to the mixtures weighted average formula — the same logic applies.

Average Speed

When the same distance is covered at two different speeds s₁ and s₂:

Average speed = 2s₁s₂ / (s₁ + s₂) (harmonic mean)

This is NOT the arithmetic mean (s₁+s₂)/2. The formula gives a result closer to the smaller speed.

Example: 60 km at 40 km/h and 60 km at 60 km/h. Average speed = 2×40×60 / (40+60) = 4800/100 = 48 km/h (not 50).

Consecutive Integers

Average of n consecutive integers starting from a = a + (n−1)/2.

For consecutive integers: if the count is odd, the average equals the middle number. If even, the average is the mean of the two middle numbers.

Ratios

Ratio a:b means for every a parts of the first quantity, there are b parts of the second.

Scaling ratios

Multiply or divide both parts by the same factor — ratio unchanged.

• a:b = ka:kb for any k ≠ 0
• To compare ratios, convert to fractions: 3:5 = 3/5 = 0.6 vs 5:8 = 0.625 → 5:8 > 3:5

Combining ratios

If A:B = 2:3 and B:C = 4:5, find A:C:

• Make B common: A:B = 8:12, B:C = 12:15 → A:B:C = 8:12:15 → A:C = 8:15

Dividing a quantity in a given ratio

Total quantity Q in ratio a:b:c → parts are Qa/(a+b+c), Qb/(a+b+c), Qc/(a+b+c).

Example: Rs.2100 split in 2:3:2 → parts = 600 : 900 : 600.

Proportion

Direct proportion: y ∝ x → y = kx. As x increases, y increases. k = y/x (constant).

Inverse proportion: y ∝ 1/x → xy = k. As x increases, y decreases.

Example (inverse): 6 men take 8 days; 4 men take 6×8/4 = 12 days.

Continued proportion

a, b, c are in continued proportion if a/b = b/c → b² = ac (b is the geometric mean).

Partnership

Partners share profit proportional to capital × time.

If A invests C₁ for t₁ months and B invests C₂ for t₂ months: Profit ratio = C₁t₁ : C₂t₂

Example: A invests Rs.10000 for 6 months, B invests Rs.12000 for 8 months. Ratio = 10000×6 : 12000×8 = 60000 : 96000 = 5:8.

Common Mistakes

• Using arithmetic mean for average speed — always use harmonic mean when equal distances are covered
• In ratios, treating a:b as a percentage (3:5 ≠ 3/5 of total; 3/5 only if ratio is part-to-whole)
• Weighted average: forgetting that group sizes determine the weights, not just the values
• Partnership: ignoring the time dimension — profit share requires capital × time, not capital alone

Exam Tips

• For average problems, always convert to sums (Sum = Avg × Count) immediately
• When a new member joins: use Sum + new value = new average × new count; solve for the unknown
• For combined average: use the deviation method (how far each group average is from combined average); the ratio of deviations equals the inverse ratio of sizes
• Ratio chain (A:B, B:C → A:B:C): equalize the linking term; write all three in one ratio
• For CAT, ratio problems often embed an unknown — assign the ratio multiplier as k and solve for k from a given total or difference