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NCERT Solutions Ganita Prakash (Part 1) Chapter 3 .3 Abacus that Makes Use of the Decimal System — In-text Questions

Book page 693 Updated on2026-09-05

Q1.
How would you use this to find the sum?
Answer

Slide the counters from the right of the vertical partition across onto the matching lines on the left, then tidy up each line.

On page 69 the board holds 2907 on the left and 43 on the right:

Line2907 has43 hasAfter bringing togetherValue
10002 counters on the line—2 on the line2000
1001 above (500) + 4 on the line (400)—1 above + 4 on the line900
10—4 counters4 + 1 carried = 5 → replaced by 1 counter above the line50
11 above (5) + 2 on the line (2)3 counters7 + 3 = 10 → cleared, 1 counter added to the 10s line0
Sum = 2000 + 900 + 50 + 0 = 2950
Check: 2907 + 43 = 2950 ✓
Why it happens: the abacus is a decimal place value machine made of wood. Each line is a power of 10 and the counters on it are the digit — so pushing the two sets of counters together is addition, and the exchanges keep every digit below 10. Its whole design carries the base-10 idea, which is why people who wrote in Roman numerals still calculated in base 10.
Q2.
The counters along each line were brought together. What is to be done if the total in a line exceeded 10?
Answer

Two exchanges keep the board readable:

  • Ten counters on a line are taken off and replaced by one counter on the next line up — the next power of 10.
  • Five counters on a line are taken off and replaced by one counter just above that line, since a counter above a line stands for 5.
In 2907 + 43:   7 ones + 3 ones = 10 ones
→ clear the 1s line, put 1 counter on the 10s line
10s line: 4 + 1 = 5 counters → replaced by 1 counter above the 10s line (= 50)
Why it happens: this is exactly the carrying you do in column addition, made visible. “Ten of these make one of those” is possible on every line because the lines are successive powers of 10 — the very property that defines a base. Try the same board with Roman exchanges (five I make a V, two V make an X, five X make an L) and you would need a different rule on almost every line.
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