8 min readPathway sheet3-mark and 5-mark word problems; often the 4-mark case-based question

Daily-life problems on APs

Turn the story into numbers: find a, d and n. Then decide if the question wants one term (aₙ) or a total (Sₙ).

One value in some positionUse aₙ = a + (n − 1)d

A total, 'in all', 'altogether'Use Sₙ = n/2 [2a + (n − 1)d] or n/2 (a + l)

Unknown nSolve aₙ = value (a simple equation) or Sₙ = value (a quadratic in n)

Always end withA sentence answer with the unit (₹, rows, poles, months)

The four-step method

1
Read the story and write the first value as a.
2
Find the fixed change each time. This is d (negative if the value falls).
3
Decide what is asked: one value (aₙ), a total (Sₙ), or how many steps (n).
4
Write the formula, substitute, solve, and give the answer as a sentence with its unit.

Example 1: monthly savings

Problem

Meena saves ₹ 100 in January, ₹ 120 in February, ₹ 140 in March, and so on, saving ₹ 20 more each month. (i) How much does she save in December? (ii) What does she save in the whole year?

Working

a = 100, d = 20, n = 12 (January to December)
(i) Formula: aₙ = a + (n − 1)d
a₁₂ = 100 + 11 × 20 = 100 + 220 = 320
(ii) Formula: Sₙ = n/2 (a + l)
S₁₂ = 12/2 × (100 + 320) = 6 × 420 = 2520

Answer

(i) She saves ₹ 320 in December. (ii) She saves ₹ 2520 in the year.

Example 2: a stack of bamboo poles

Problem

Bamboo poles are stacked with 25 in the bottom row, 23 in the next, 21 in the next, and so on. How many rows hold 160 poles, and how many poles are in the top row?

Working

a = 25, d = −2, Sₙ = 160
Formula: Sₙ = n/2 [2a + (n − 1)d] = n/2 [50 − 2(n − 1)] = n/2 (52 − 2n) = n(26 − n)
n(26 − n) = 160, so n² − 26n + 160 = 0
(n − 10)(n − 16) = 0, so n = 10 or n = 16
For n = 16, a₁₆ = 25 + 15 × (−2) = −5. A row cannot hold −5 poles, so reject n = 16.
For n = 10, a₁₀ = 25 + 9 × (−2) = 7

Answer

The poles are in 10 rows, and the top row has 7 poles.

Example 3: when is a value reached?

Problem

Sita's monthly salary starts at ₹ 18,000 and rises by ₹ 1,500 every year (example figures). In which year will her monthly salary be ₹ 30,000?

Working

a = 18000, d = 1500, aₙ = 30000
Formula: aₙ = a + (n − 1)d
18000 + (n − 1) × 1500 = 30000
(n − 1) × 1500 = 12000, so n − 1 = 8
n = 9

Answer

Her monthly salary will be ₹ 30,000 in the 9th year.

Words that tell you what to find

'in the 8th week', 'the 15th row', 'the top row': find one term aₙ.
'in all', 'total', 'altogether', 'in the whole year': find a sum Sₙ.
'how many rows', 'after how many days', 'in which year': find n.
If two values of n come out, test each in the story and reject the one that makes no sense.

In real life

Planning Diwali or Pongal savings that grow each week is an AP problem. So are loans paid in rising instalments and bricks stacked at a building site.

Exam angle

2 marks: “A skating class charges ₹ 200 for the first month and ₹ 25 more each month after that (example amounts). Find the fee for the 9th month.” a = 200, d = 25. a₉ = 200 + 8 × 25 = 200 + 200 = ₹ 400.

3 marks: “A loan of ₹ 2400 is repaid by paying ₹ 25 in the first month and ₹ 10 more in each later month. How many months does it take?” a = 25, d = 10. Sₙ = n/2 [50 + 10(n − 1)] = 5n² + 20n = 2400, so n² + 4n − 480 = 0. (n + 24)(n − 20) = 0, so n = 20 months.

5 marks: “Ramesh saves ₹ 50 in week 1 and ₹ 10 more each week. Suresh saves ₹ 100 every week. After how many weeks are their total savings equal?” Ramesh: Sₙ = 5n² + 45n. Suresh: 100n. 5n² + 45n = 100n gives 5n(n − 11) = 0, so n = 11 (n = 0 rejected). Each has ₹ 1100.

Remember it

a, d, n, then ask: one value or the total? One value is aₙ; a total is Sₙ.