(Pṛthūdakasvāmī, commentary on Brahmagupta’s Brāhmasphuṭasiddhānta, c. 864 CE) A “hasta” (meaning “forearm”) refers to a unit of length measuring around 18 inches.
A pool 30 hastas long is dug to different depths along its length. It is divided into 5 sections having lengths 4, 5, 6, 7, and 8 hastas, and is dug to depths of 9, 7, 7, 3, and 2 hastas, respectively. Find the mean depth of the pool.

Step-by-step solution

Idea: Each depth applies to a different length of the pool, so the mean depth is the weighted mean of the depths with the section lengths as weights (Brahmagupta’s formula for the mean depth of a pit).

1. Weights are the lengths 4, 5, 6, 7, 8; values are the depths 9, 7, 7, 3, 2. Sum of weights = 30 hastas (the whole pool).½ mark
2. Σ(length × depth) = 4 × 9 + 5 × 7 + 6 × 7 + 7 × 3 + 8 × 2 = 36 + 35 + 42 + 21 + 16 = 150.1½ marks
3. Mean depth = 15030 = 5 hastas (about 5 × 18 = 90 inches).1 mark

Check: A pool 30 hastas long and 5 hastas deep everywhere would have the same side area: 30 × 5 = 150 = 36 + 35 + 42 + 21 + 16 ✓. The plain mean of the depths, 9 + 7 + 7 + 3 + 25 = 5.6, is too large because the shallow sections are the longest.

Answer to write in the exam

Common mistakes that cost marks

• Taking the plain mean of the depths (5.6 hastas), ignoring that the sections have different lengths.
• Dividing by 5 (the number of sections) instead of 30 (the total length).
• Pairing lengths and depths in the wrong order.

How this can come in the exam

Try one yourself

A canal has three parts: 10 m long and 4 m deep, 15 m long and 2 m deep, 25 m long and 3 m deep. Find its mean depth.

More questions like this

• Suvarna had purchased 1g gold at ₹15k (short for ₹15,000). This is shown by point O denoting the average price of gold that she possesses. Six different scenarios are given below for her next transaction. For each scenario estimate and mark the average price of gold she will have after the transaction.
• Given some data with corresponding weights, how would the weighted average change if all the weights are doubled? If required, experiment with some data. What do you observe? Justify your answer using algebra.
• Answer the following questions based on the graph.
• Observe the following infographic.
• Decision Dilemma:

All Data and statistics questions · All maths questions