Chapter 1 of 24

ICSE Class 10 Mathematics★ ICSE Class 10 Mathematics · ~30 marks · Board: ICSE

Commercial Mathematics, Quadratic Equations & Algebra (GST, Banking, Shares)

Mathematics for the REAL WORLD — GST, banking (recurring deposits), shares and dividends. Plus ALGEBRA: quadratic equations, ratio and proportion, linear inequations, and matrices. ICSE Class 10 Mathematics.

⏱ 18 min readMedium difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Calculate GST (CGST, SGST, IGST) for intra-state and inter-state transactions and apply the Input Tax Credit mechanism
2Find the interest and maturity value for a Recurring Deposit using the equivalent principal formula
3Compute dividend income from shares and determine yield, return on investment, and compare different share investments
4Solve quadratic equations by factorisation and the quadratic formula; interpret the discriminant and nature of roots
5Apply componendo and dividendo and properties of proportion in ratio problems
6Solve linear inequations in one variable and represent the solution on a number line
7Perform matrix addition, subtraction, and multiplication; understand order and conformability conditions

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Why this chapter matters

Commercial Mathematics (GST, banking, shares) and Algebra (quadratic equations, matrices) together account for roughly 30 marks in the ICSE board paper — the single largest chunk. GST and banking questions are GUARANTEED every year and are straightforward: learn the formulas, substitute, done. Quadratic equations underpin half of Class 10 algebra — word problems on speed, age, and area almost always set up a quadratic. Matrices are a unique ICSE topic not tested in CBSE and carry 6–8 marks. Students who master this file gain a reliable 25+ marks before touching geometry.

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Commercial Mathematics & Algebra

1. Goods and Services Tax (GST)

GST is a COMPREHENSIVE indirect tax on goods and services. It replaced multiple taxes (VAT, Service Tax, Excise, etc.). 'One Nation, One Tax.'

Calculating GST

GST amount = (Tax Rate / 100) × Selling Price
Final Price = Selling Price + GST
Input Tax Credit: Tax paid on PURCHASES can be offset against tax collected on SALES. Tax payable = Output GST — Input GST.

2. Banking — Recurring Deposits

A RECURRING DEPOSIT (RD) is when a fixed amount is deposited EVERY MONTH for a fixed period. Interest is calculated using the formula:

Interest on RD

Equivalent Principal for 1 month = P × n(n+1) / (2 × 12) where P = monthly instalment, n = number of months.
Interest = Equivalent Principal × Rate / 100.
Maturity Value = Total Deposited + Interest.

3. Shares and Dividends

A COMPANY raises capital by selling SHARES.
Face Value (Nominal Value) : The original value printed on the share.
Market Value: The price at which shares trade in the stock market. Can be above (at premium) or below (at discount) face value.
Dividend: Share of PROFITS paid to shareholders. Expressed as a PERCENTAGE of FACE VALUE.
Dividend Income = (Dividend % / 100) × Face Value × Number of shares.

4. Quadratic Equations

Standard Form: ax² + bx + c = 0 (a ≠ 0)

Solving Methods

Method	When to Use
Factorisation	When roots are rational. Splitting the middle term.
Quadratic Formula	ALWAYS works. x = [—b ± √(b² — 4ac)] / 2a.

Discriminant: Δ = b² — 4ac

Δ > 0: TWO DISTINCT real roots. Δ = 0: ONE REAL root (equal). Δ < 0: NO real roots.

Nature of Roots

Real and EQUAL: b² — 4ac = 0. Real and UNEQUAL: b² — 4ac > 0.
Rational roots: b² — 4ac is a PERFECT SQUARE.

Word Problems

Speed, distance, time. Ages. Numbers. Work. Area. 'Let the unknown be x. Frame the quadratic. SOLVE. Check: discard negative/unrealistic solutions.'

5. Ratio and Proportion

Ratio

Comparison of two quantities of the SAME kind. a:b = a/b. Must be in SIMPLEST form.

Proportion — Key Properties

a:b :: c:d means ad = bc (product of extremes = product of means).
Continued Proportion: a, b, c are in continued proportion if a:b = b:c → b² = ac. b = geometric mean.
Componendo and Dividendo: If a/b = c/d, then (a+b)/(a—b) = (c+d)/(c—d).

6. Linear Inequations

Solving

Like equations — but: MULTIPLYING/DIVIDING by a NEGATIVE REVERSES the inequality sign. Solution set represented on a NUMBER LINE. Open circle: strict (<, >). Closed circle: inclusive (≤, ≥).

7. Matrices

Basics

Order: m × n (rows × columns). Addition/subtraction: element-wise. SAME ORDER required.

Multiplication

A (m×n) × B (n×p) = C (m×p). Inner dimensions MUST MATCH. NOT COMMUTATIVE (AB ≠ BA generally).

Transpose

Rows ↔ Columns. (A′)′ = A. (AB)′ = B′A′.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

GST

TOTAL GST = Tax Rate% × Selling Price (before tax). For INTRA-STATE: CGST = SGST = Total GST / 2. For INTER-STATE: IGST = Total GST. INPUT TAX CREDIT: Tax payable = Output GST (on sales) − Input GST (on purchases). FINAL PRICE = Selling Price + Total GST. REVERSE CALCULATION: If final price (inclusive of GST) is given, Selling Price = Final Price × 100 / (100 + Tax Rate%).

ICSE GST questions are always multi-step: a manufacturer sells to dealer (GST charged), dealer sells to customer (GST charged), Input Tax Credit reduces tax. Read carefully who is the 'final consumer' and who gets the credit.

Recurring Deposit (RD)

EQUIVALENT PRINCIPAL for 1 month: P_eq = P × n(n+1) / (2 × 12), where P = monthly instalment, n = total number of months. INTEREST: I = P_eq × r / 100, where r = annual rate%. MATURITY VALUE = (P × n) + I. TOTAL DEPOSITED = P × n.

The formula P × n(n+1)/24 gives the PRINCIPAL EQUIVALENT for 1 month. The interest formula is then like simple interest: Principal × Rate / 100 (for 1 year). Since the equivalent principal is already for 1 month, multiply only by the rate for 1 year (not for n months). This is a common confusion — the n(n+1)/24 formula already accounts for the varying time of each instalment.

Shares and Dividends

DIVIDEND INCOME = (Dividend % / 100) × Face Value × Number of shares. RETURN % on investment = (Dividend Income / Amount Invested) × 100. Amount Invested = Market Price × Number of shares. RELATIONSHIP: If dividend rate d% on Face Value FV and market price MP, then Return % = (d × FV) / MP. To compare two investments: compare their Return %.

The DIVIDEND is always a percentage of the FACE VALUE (not market value). Students often multiply the dividend % by the market price — that is wrong. Example: '10% dividend on Rs 10 shares at Rs 15 market price' → dividend per share = 10% × 10 = Re 1, not 10% × 15.

Quadratic Equations

STANDARD FORM: ax² + bx + c = 0. QUADRATIC FORMULA: x = [−b ± √(b² − 4ac)] / 2a. DISCRIMINANT: Δ = b² − 4ac. If Δ > 0: two distinct real roots. If Δ = 0: one repeated real root. If Δ < 0: no real roots. SUM OF ROOTS: α + β = −b/a. PRODUCT OF ROOTS: α × β = c/a. FORMING a quadratic from roots: x² − (α+β)x + αβ = 0.

WORD PROBLEM STRATEGY: Identify the unknown (always take a positive quantity as x — e.g., speed, age, number). Write the equation from the given condition. Solve. REJECT negative or unrealistic roots with a reason. ICSE requires you to state why you reject a root.

Matrices

ORDER: A matrix of order m × n has m rows and n columns. ADDITION/SUBTRACTION: Only possible if matrices have the SAME order. Add/subtract corresponding elements. MULTIPLICATION: A(m×n) × B(n×p) = C(m×p). Number of COLUMNS of A must equal number of ROWS of B. Element C[i][j] = sum of (row i of A) × (column j of B). ZERO MATRIX: All elements zero. IDENTITY MATRIX (I): Square matrix with 1s on diagonal, 0s elsewhere. A × I = A.

Matrix multiplication is NOT commutative (A×B ≠ B×A in general). ICSE tests: 'If A = [[a,b],[c,d]], find A² or verify A satisfies a given equation.' Expand carefully — multiply each element properly. Common error: students add instead of multiply, or swap rows and columns.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Multiplying dividend % by market price instead of face value

✓ Dividend is ALWAYS calculated on FACE VALUE. If a company declares a 15% dividend on Rs 10 shares, dividend per share = 15% × 10 = Rs 1.50, regardless of whether the market price is Rs 8 or Rs 20. The market price only determines the COST OF INVESTMENT (used to calculate % return). Formula: Return % = (Dividend Income / Market Price × No. of Shares) × 100.

WATCH OUT

✗ Forgetting to reject negative roots in quadratic word problems

✓ If x represents a physical quantity (speed, age, number of articles, length), it CANNOT be negative. After solving the quadratic, you will get two roots. State explicitly: 'Since [speed/age/length] cannot be negative, x = [negative root] is rejected.' Then give the POSITIVE root as the answer. ICSE examiners deduct marks if no reason is given for rejection.

WATCH OUT

✗ Applying the RD interest formula with n as years instead of months

✓ In the RD formula, P_eq = P × n(n+1) / (2 × 12), the n is in MONTHS. If the period is given as '2 years', convert to months: n = 24. Then P_eq = P × 24 × 25 / 24 = P × 25/2. The rate r is the ANNUAL rate (do not divide by 12 again — the formula already accounts for this through the equivalent principal concept).

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· GST

A washing machine is sold for Rs 25,000 exclusive of GST. If the GST rate is 12%, find (a) the GST amount, (b) the final selling price. If the retailer had bought it for Rs 18,000 + GST, find his Input Tax Credit and the net GST he pays to the government.

▶ Show solution

GST on sale = 12% × 25,000 = Rs 3,000. Final price = 25,000 + 3,000 = Rs 28,000. GST paid on purchase (Input GST) = 12% × 18,000 = Rs 2,160. GST collected on sale (Output GST) = Rs 3,000. Net GST payable to government = Output GST − Input GST = 3,000 − 2,160 = Rs 840. (CGST = Rs 420, SGST = Rs 420 if intra-state.)

Q2MEDIUM· quadratic-word-problem

Two taps can fill a tank together in 6 minutes. The larger tap takes 5 minutes less than the smaller tap to fill the tank alone. Find the time each tap takes alone.

▶ Show solution

Let smaller tap take x minutes alone. Larger tap takes (x−5) minutes. In 1 minute: smaller does 1/x, larger does 1/(x−5), together: 1/6. So 1/x + 1/(x−5) = 1/6. Multiply through by 6x(x−5): 6(x−5) + 6x = x(x−5). 6x − 30 + 6x = x² − 5x. 12x − 30 = x² − 5x. x² − 17x + 30 = 0. Factoring: (x−15)(x−2) = 0. x = 15 or x = 2. If x = 2, larger tap = −3 min (impossible). REJECT x = 2. Answer: Smaller tap = 15 minutes, Larger tap = 10 minutes. Check: 1/15 + 1/10 = 2/30 + 3/30 = 5/30 = 1/6 ✓.

Q3HARD· shares-investment-comparison

A man has a choice of investing Rs 72,000 in either (i) 16% Rs 100 shares at Rs 120 or (ii) 14% Rs 100 shares at Rs 105. Find which investment gives a better return and by how much (as %).

▶ Show solution

Investment (i): 16% Rs 100 shares at Rs 120. No. of shares = 72,000 / 120 = 600. Annual dividend = 600 × 16% × 100 = Rs 9,600. Return % = (9,600 / 72,000) × 100 = 13.33%. Investment (ii): 14% Rs 100 shares at Rs 105. No. of shares = 72,000 / 105 = 685.71 (take 685 shares). Investment = 685 × 105 = Rs 71,925. Annual dividend = 685 × 14% × 100 = Rs 9,590. Return % = (9,590 / 71,925) × 100 ≈ 13.33%. [Or use formula: Return% = (d × FV) / MP. (i) = 16×100/120 = 13.33%. (ii) = 14×100/105 = 13.33%.] Both give equal returns. [Examiners often phrase the numbers so one is clearly better — use the formula Return% = (dividend% × FV) / MP for a fast comparison.]

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