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NCERT Solutions Ganita Prakash (Part 1) Chapter 3 –76Section 3.7 Decimal Sequences / Estimating Sums and Differences — In-text Questions

Book page 75 Updated on2026-09-05

Q1.
Observe this sequence of decimal numbers and identify the change after each term: 4.4, 4.8. 5.2, 5.6, 6.0, … Continue this sequence and write the next 3 terms.
Answer
4.8 – 4.4 = 0.4
5.2 – 4.8 = 0.4
The change is + 0.4 each time.
6.0 + 0.4 = 6.4
6.4 + 0.4 = 6.8
6.8 + 0.4 = 7.2
Sequence: 4.4, 4.8, 5.2, 5.6, 6.0, 6.4, 6.8, 7.2
Why it happens: think in tenths — 44, 48, 52, 56, 60, 64, 68, 72. The numerator simply goes up by 4 each time, which makes the mental arithmetic easy.
Q2.
Similarly, identify the change and write the next 3 terms for each sequence given below. Try to do this computation mentally. (a) 4.4, 4.45, 4.5, … (b) 25.75, 26.25, 26.75, … (c) 10.56, 10.67, 10.78, … (d) 13.5, 16, 18.5, … (e) 8.5, 9.4, 10.3, … (f) 5, 4.95, 4.90, … (g) 12.45, 11.95, 11.45, … (h) 36.5, 33, 29.5, …
Answer
ChangeNext three terms
(a) 4.4, 4.45, 4.5+ 0.054.55, 4.6, 4.65
(b) 25.75, 26.25, 26.75+ 0.527.25, 27.75, 28.25
(c) 10.56, 10.67, 10.78+ 0.1110.89, 11.00, 11.11
(d) 13.5, 16, 18.5+ 2.521, 23.5, 26
(e) 8.5, 9.4, 10.3+ 0.911.2, 12.1, 13.0
(f) 5, 4.95, 4.90– 0.054.85, 4.80, 4.75
(g) 12.45, 11.95, 11.45– 0.510.95, 10.45, 9.95
(h) 36.5, 33, 29.5– 3.526, 22.5, 19
Why it happens: in (c) the jump of 0.11 makes 10.78 + 0.11 = 10.89, and then 10.89 + 0.11 = 11.00 — the hundredths carry into the tenths and the tenths into the units at the same moment. In (e) adding 0.9 is quickest as "add 1, then take away 0.1".
Tip: for mental work, count in the smallest unit present. Sequence (a) is 440, 445, 450, 455, 460, 465 hundredths.
Q3.
Make your own sequences and challenge your classmates to extend the pattern.
Answer

Pick a starting number and a fixed change, then write the first three terms only.

Your sequenceHidden ruleNext three terms
2.05, 2.3, 2.55, …+ 0.252.8, 3.05, 3.3
9.009, 9.018, 9.027, …+ 0.0099.036, 9.045, 9.054
100.4, 99.6, 98.8, …– 0.898.0, 97.2, 96.4
0.5, 1.5, 1.4, 2.4, 2.3, …+1 then –0.1, alternately3.3, 3.2, 4.2
Try This: the last row is a two-rule pattern like Question 10 of the exercise. Those are the hardest — and the most fun — to give a friend.
Q4.
Sonu says, “If we add two decimal numbers, then the sum will always be greater than the sum of their whole number parts. Also, the sum will always be less than 2 more than the sum of their whole number parts.” What do you think about this claim? Verify if this is true for these numbers. Will it work for any 2 decimal numbers? What about for the sum of 25.93603259 and 8.202?
Answer

Checking it for 25.936 and 8.202:

25.936 + 8.202 = 34.138
Sum of whole number parts = 25 + 8 = 33
33 + 2 = 35
33 < 34.138 < 35 ✓ The claim holds.

For 25.93603259 and 8.202:

25.93603259 + 8.202 = 34.13803259
Again 33 < 34.13803259 < 35 ✓

Does it work for any two decimal numbers? Almost.

Let the numbers be A + f and B + g, where A, B are the whole number parts
and f, g are the fractional parts, so 0 ≤ f < 1 and 0 ≤ g < 1.
Sum = (A + B) + (f + g)
Since 0 ≤ f + g < 2,
A + B ≤ Sum < A + B + 2
Why it happens: each fractional part is less than 1, so together they add less than 2 — the sum can never reach "2 more than" the whole parts. The upper half of Sonu's claim is always true.
Check it yourself: the lower half needs one small repair. If both numbers are whole, say 4.0 + 5.0 = 9, the sum equals 9 rather than being greater than it. So the correct statement is "greater than or equal to the sum of the whole number parts", with equality only when both fractional parts are zero.
Q5.
Similarly, come up with a way to narrow down the range of whole numbers within which the difference of two decimal numbers will lie.
Answer

Use the same idea with subtraction.

Let the numbers be A + f and B + g, with 0 ≤ f < 1 and 0 ≤ g < 1.
Difference = (A – B) + (f – g)
Since –1 < f – g < 1,
(A – B) – 1 < Difference < (A – B) + 1

Rule in words: the difference of two decimal numbers is always within 1 of the difference of their whole number parts.

ExampleWhole partsPredicted rangeActual difference
25.936 – 8.20225 – 8 = 17between 16 and 1817.734 ✓
34.505 – 18.134 – 18 = 16between 15 and 1716.405 ✓
17 – 16.19817 – 16 = 1between 0 and 20.802 ✓
9.9 – 9.099 – 9 = 0between –1 and 10.81 ✓
Why it happens: subtracting the fractional parts can pull the answer down by at most a little less than 1 (when f = 0 and g is nearly 1) or push it up by at most a little less than 1 (when f is nearly 1 and g = 0). So the answer never strays a full 1 from A – B.
Tip: estimating like this before you calculate is a quick way to catch a misplaced decimal point. If your answer to 34.505 – 18.1 came out as 1.6405, the estimate 16 would immediately warn you.
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