See-Saw Method for Weighted Averages in Placement Tests

The see-saw method solves weighted average problems without computing the full formula, useful when group sizes arrive as a ratio rather than raw counts.

Placement aptitude tests from TCS NQT to AMCAT include weighted average questions in every full-length test. They appear in three patterns: find the combined average, find what ratio the groups are in, or find a missing group value. One rule handles all three.

The alligation cross that powers the see-saw is derived fully in the mixtures and alligations complete guide. This article is the applied playbook: the rule stated once, then drilled on four placement-exam-style problems.

Where Placement Tests Use Weighted Averages

Weighted average problems appear across the assessment stack that most IT and analytics fresher hiring runs through:

• TCS NQT Numerical Ability — weighted averages appear alongside percentages and ratios in the quantitative section
• AMCAT Quantitative Ability — alligation and average problems are part of the standard question bank
• eLitmus pH Test Quantitative — mixture-and-average problems are a regular fixture
• Mu Sigma MuApt — the Mu Sigma aptitude test emphasises analytical reasoning that includes weighted average reasoning

Time per question in these tests runs from 45 to 75 seconds. A formula approach (multiply, add, divide) uses all of that on a multi-step problem. The see-saw cuts the same problem to 15 to 20 seconds once you practise it.

To understand what other topic areas these tests cover, the campus placement evaluation test guide maps the full syllabus structure. For Mu Sigma specifically, the Mu Sigma MuApt preparation guide covers the test pattern in detail.

The See-Saw Rule

Two groups: Group 1 has n1 members with average A1, Group 2 has n2 members with average A2. The combined average is M. The see-saw rule states:

Think of a physical balance. M is the pivot point. A group with a higher average is like a lighter weight placed farther from the centre, and a larger group pulls the pivot closer to itself. That image keeps the rule stable when exam nerves are high.

• n1 : n2 = (A2 − M) : (M − A1)

Read it as a balance: M is the pivot. The distance from M to each group’s average is inversely proportional to that group’s size. The larger group pulls the combined average closer to itself.

Three things follow from this one rule:

• Given n1, n2, A1, A2 — compute M by solving the proportion
• Given A1, A2, M — read off n1:n2 directly from the two distances
• Given A1, A2, M and one absolute group size — compute the other group size

The full algebraic derivation of why n1:n2 = (A2 − M):(M − A1) is in the mixtures and alligations guide. Do not re-memorise the derivation for exam day. Memorise the identity and practise plugging numbers into it until it is automatic.

Three Question Types in Placement Tests

Type 1: Find the Combined Average

A batch of 25 students scored an average of 72 in the quantitative section. A second batch of 15 scored an average of 88. What is the combined average for all 40 students?

See-saw approach:

• Step 1: Write n1:n2 = 25:15 = 5:3
• Step 2: The gap between the two averages is 88 − 72 = 16
• Step 3: The inverse ratio is 3:5 (swap the digits). M sits 3 parts from A1 and 5 parts from A2, out of a total of 8 parts
• Step 4: M = 72 + (3/8) × 16 = 72 + 6 = 78

Verification (formula):

• (25 × 72 + 15 × 88) / 40 = (1800 + 1320) / 40 = 3120 / 40 = 78 ✓

Type 2: Find the Group Ratio

In a test, Section A candidates averaged 65 and Section B averaged 80. The combined average was 71. What is the ratio of Section A to Section B candidates?

See-saw approach:

• Step 1: Draw the lever: A1 = 65, M = 71, A2 = 80
• Step 2: Distance from M to A1 = 71 − 65 = 6
• Step 3: Distance from M to A2 = 80 − 71 = 9
• Step 4: n_A : n_B = (A2 − M) : (M − A1) = 9 : 6 = 3 : 2

Verification:

• If A = 3 parts and B = 2 parts: (3 × 65 + 2 × 80) / 5 = (195 + 160) / 5 = 355 / 5 = 71 ✓

Type 3: Find a Group Size

In a campus drive, Group X averaged 62 and Group Y averaged 80 on the aptitude test. The combined average was 68 and Group Y had 30 candidates. How many candidates were in Group X?

See-saw approach:

• Step 1: n_X : n_Y = (A_Y − M) : (M − A_X) = (80 − 68) : (68 − 62) = 12 : 6 = 2 : 1
• Step 2: If n_Y = 30 and the ratio is 2:1, then n_X = 2 × 30 = 60

Verification:

• (60 × 62 + 30 × 80) / 90 = (3720 + 2400) / 90 = 6120 / 90 = 68 ✓

The Averaging-the-Averages Trap

The most common error on weighted average questions: taking a simple average when the groups are different sizes.

Two classes, Class A with 40 students averaging 30 kg and Class B with 60 students averaging 40 kg. The wrong approach:

• Simple average = (30 + 40) / 2 = 35 kg (WRONG — treats both classes as equal-sized)

The correct combined average:

• (40 × 30 + 60 × 40) / 100 = (1200 + 2400) / 100 = 36 kg

See-saw check: n1:n2 = 40:60 = 2:3. Distance from M to each average must be in ratio 3:2. Check: (36 − 30) : (40 − 36) = 6 : 4 = 3 : 2. Confirmed.

The simple-average error is off by exactly 1 kg here. On a placement test it’s worth 1 mark. The see-saw eliminates the error by making the group sizes part of the calculation from the first step.

For additional practice on quantitative aptitude topics that appear alongside weighted averages in placement tests, the time and work aptitude guide covers the work-rate shortcuts that follow the same ratio-reasoning structure.

Speed Check: Mastering the Shortcut

To time yourself: all four examples above should take under 20 seconds each once the see-saw is automatic. The order of practice that works:

1. Drill Type 2 first (find ratio — the purest application of the rule)
2. Then Type 1 with ratio inputs (not raw counts)
3. Then Type 3 (reverse-engineer group size)
4. Type 1 with raw counts last (the formula is equally fast here, so the see-saw adds less value)

Placement test quantitative sections reward this kind of method-switching: know two approaches to the same problem type, pick the faster one based on how the numbers are given.

The same speed-over-formula thinking applies in AI-era hiring contexts. Recruiters at analytics firms now score candidates on structured problem-solving alongside aptitude scores. If that interests you, TinkerLLM starts at ₹299 and puts practical AI problem-solving in your hands. The ratio-and-proportion reasoning you just drilled maps directly onto how LLM output evaluation works.