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NCERT Solutions for Class 9th Maths Chapter 2 Exercise Set 2.3 — Exploring linear patterns

Book page 23–24 Updated on2026-09-08

Q1.
A student has ₹500 in her savings bank account. She gets ₹150 every month as pocket money. How much money will she have at the end of every month from the second month onwards? Find a linear expression to represent the amount she will have in the nth month.
Answer

She begins with ₹500 and adds ₹150 at the end of each month.

End of month n12345…
Amount (₹)65080095011001250…

From the second month onwards the balances are ₹800, ₹950, ₹1100, ₹1250, … — each ₹150 more than the one before.

Amount in the nth month = ₹(500 + 150n)

Check: n = 2 gives 500 + 300 = ₹800. ✓

Why the expression splits this way: ₹500 is there before any pocket money arrives, so it is the constant term — the value at n = 0. The ₹150 arrives once per month, so it is multiplied by n. Since the difference between consecutive balances is the fixed ₹150, this is a linear pattern of degree 1, and 150 is its slope.
Try This: when will she first have more than ₹3000? Solve 500 + 150n > 3000, giving n > 16.67, so in the 17th month.
Q2.
A rally starts with 120 members. Each hour, 9 members drop out of the group. How many members will remain after 1, 2, 3, … hours? Find a linear expression to represent the number of members at the end of the nth hour.
Answer
After n hours12345…
Members left111102938475…
Members at the end of the nth hour = 120 − 9n

Check: n = 3 gives 120 − 27 = 93. ✓

Why the coefficient is negative: members leave, so each hour subtracts a fixed 9 — this is linear decay, and its slope is −9. As in Bela’s pocket money, the model has a natural end: 120 − 9n ≥ 0 requires n ≤ 13⅓. At n = 13 there are 3 members left, and within the next hour the rally is over. Beyond that the expression would report a negative number of people, which is meaningless.
Check it yourself: the count never reaches exactly 0, since 120 is not a multiple of 9. Solving 120 − 9n = 0 gives n = 13⅓ — useful as an estimate of when the rally empties, but not an hour number.
Q3.
Suppose the length of a rectangle is 13 cm. Find the area if the breadth is (i) 12 cm, (ii) 10 cm, (iii) 8 cm. Find the linear pattern representing the area of the rectangle.
Answer

With the length fixed at 13 cm, area = 13 × breadth.

(i)   13 × 12 = 156 cm²
(ii)   13 × 10 = 130 cm²
(iii)   13 × 8 = 104 cm²
If the breadth is b cm,   Area A = 13b cm²
Breadth b (cm)81012
Area A = 13b (cm²)104130156
Step for ∆b = 2—+26+26
Why area is linear here although area is usually not: area lb involves two variables and is not linear in general — that was the point of the garden in Example 2. But here the length is held fixed at 13, so only one variable is free, and 13b has degree 1 in b. Every extra centimetre of breadth adds a strip 13 cm long, i.e. 13 cm² — a constant, which is exactly what makes the pattern linear.
Tip: the constant term is 0: zero breadth, zero area. Its graph is the line A = 13b through the origin.
Q4.
Suppose the length of a rectangular box is 7 cm and breadth is 11 cm. Find the volume if the height is (i) 5 cm, (ii) 9 cm, (iii) 13 cm. Find the linear pattern representing the volume of the rectangular box.
Answer

Length × breadth = 7 × 11 = 77 cm², and that base is fixed. So volume = 77 × height.

(i)   77 × 5 = 385 cm³
(ii)   77 × 9 = 693 cm³
(iii)   77 × 13 = 1001 cm³
If the height is h cm,   Volume V = 77h cm³
Height h (cm)5913
Volume V = 77h (cm³)3856931001
Step for ∆h = 4—+308+308
Why a volume can behave linearly: volume is a product of three lengths, so in general it is a degree-3 quantity. Fixing two of them turns 7 × 11 into the single number 77, and the box grows by stacking identical slabs of 77 cm³ — one slab per centimetre of height. Equal steps in h therefore give equal steps in V, and the step is 4 × 77 = 308 cm³ for every 4 cm.
Check it yourself: 1001 − 693 = 308 and 693 − 385 = 308 — a good arithmetic check on all three volumes at once.
Q5.
Sarita is reading a book of 500 pages. She reads 20 pages every day. How many pages will be left after 15 days? Express this as a linear pattern.
Answer
Pages read in 15 days = 20 × 15 = 300
Pages left = 500 − 300 = 200 pages
Pages left after n days = 500 − 20n
Day n0510152025
Pages left5004003002001000
Why it is a linear pattern: the number left drops by the same 20 pages every day, so consecutive terms differ by the constant −20. That constant difference is the definition of a linear pattern, and it is the coefficient of n in 500 − 20n. Because the decrease never speeds up or slows down, we can jump straight to day 15 instead of subtracting 20 fifteen times.
Tip: Solve 500 − 20n = 0 to find that she finishes the book on the 25th day — and 500 is exactly 25 × 20, so the last day is a full day’s reading.
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