NCERT Solutions Ganita Prakash (Part 1) Chapter 3 .3 Advantages of a Base-n System — Figure it Out

Book page 653 Updated on2026-09-05

Q1.

Add the following Egyptian numerals: (i) … and … (ii) … and …

Answer

(i) Read the two numerals on page 65 first.

First: 9 lotus, 6 coils, 8 strokes = 9000 + 600 + 8 = 9608
Second: 5 coils, 7 strokes = 500 + 7 = 507

Pool the symbols and regroup from the smallest upwards — ten of any symbol make one of the next.

Symbol	Total	Regrouping	Kept
strokes (1)	8 + 7 = 15	10 strokes → 1 arch	5 strokes, carry 1 arch
arches (10)	0 + 0 + 1 = 1	—	1 arch
coils (100)	6 + 5 = 11	10 coils → 1 lotus	1 coil, carry 1 lotus
lotus (1000)	9 + 1 = 10	10 lotus → 1 finger	0 lotus, carry 1 finger
finger (10000)	1	—	1 finger

Sum = 1 finger + 1 coil + 1 arch + 5 strokes
= 10000 + 100 + 10 + 5 = 10115
Check: 9608 + 507 = 10115 ✓

(ii)

First: 1 lotus, 8 arches = 1000 + 80 = 1080
Second: 4 arches, 6 strokes = 40 + 6 = 46

strokes: 0 + 6 = 6
arches: 8 + 4 = 12 → 1 coil and 2 arches
coils: 0 + 1 (carried) = 1
lotus: 1

Sum = 1 lotus + 1 coil + 2 arches + 6 strokes = 1000 + 100 + 20 + 6 = 1126
Check: 1080 + 46 = 1126 ✓

Why it happens: this is column addition, and the girl on page 66 is right to notice the resemblance. “Ten strokes make an arch, ten arches make a coil” is exactly “carry 1 from the ones column to the tens column”. It works because every landmark is 10 times the one below, so the same carrying rule serves at every level — which is precisely what fails in the Roman system.

Q2.

Add the following numerals that are in the base-5 system that we created. Remember that in this system, 5 times a landmark number gives the next one!

Answer

Read the two numerals from page 65:

○ ⬡⬡ □ △△ = 125 + 25 + 25 + 5 + 1 + 1 = 182
○○○ ⬡ □□ △△ = 375 + 25 + 5 + 5 + 1 + 1 = 412

Now pool them, symbol by symbol:

Symbol	Value	First + Second	Total	Regroup?
△	1	2 + 2	4	No, 4 < 5
□	5	1 + 2	3	No
⬡	25	2 + 1	3	No
○	125	1 + 3	4	No

Sum = ○○○○ ⬡⬡⬡ □□□ △△△△
= 4 × 125 + 3 × 25 + 3 × 5 + 4 × 1
= 500 + 75 + 15 + 4 = 594
Check: 182 + 412 = 594 ✓

Why it happens: this sum happens to need no regrouping at all — every column came to 4 or less. Had any column reached 5, five of that symbol would have been swapped for one of the next symbol up, exactly as ten strokes became an arch in Q1. The reminder in the book is a warning to check every column, not a hint that a carry is hiding here.

Try This: add ○○○ □□□ △△△ (375 + 15 + 3 = 393) to ○○ □□ △△△ (250 + 10 + 3 = 263). Now the triangles come to 6, so five of them become a □. The answer is 656; check that your symbols give 5×125 + 0×25 + 6×5 + 1, regrouped to ∿ ⬡ □ △ = 625 + 25 + 5 + 1 = 656.

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