NCERT Solutions for Class 12 Economics Chapter 5: Government Budget and the Economy
These Class 12 Economics Chapter 5 solutions cover Government Budget and the Economy from the NCERT textbook Introductory Macroeconomics (Reprint 2026–27). The chapter explains the meaning and components of the government budget, the three budget objectives (allocation, redistribution and stabilisation), the classification of receipts and expenditure, the measures of government deficit — revenue deficit, fiscal deficit and primary deficit — fiscal policy with the government expenditure, tax, transfer and balanced-budget multipliers, and the debt issue. Below you get the complete, step-by-step NCERT exercise solutions (every theory question answered and every numerical solved with full working), plus key concepts, formulas, extra practice, MCQs, Assertion–Reason and FAQs.
Class: 12Subject: EconomicsBook: Introductory MacroeconomicsChapter: 5Topic: Government Budget and the EconomySession: 2026–27
Chapter 5, Government Budget and the Economy, looks at how the government influences economic life through its budget — the Annual Financial Statement of estimated receipts and expenditures placed before Parliament under Article 112 for each financial year (1 April to 31 March). The budget is split into a revenue budget and a capital budget, and serves three objectives: allocation (providing public goods that are non-rivalrous and non-excludable), redistribution (changing the distribution of income through taxes and transfers) and stabilisation (correcting fluctuations in income, output and prices). Receipts are classified as revenue or capital, and expenditure as revenue or capital. The chapter then defines the key measures of deficit — revenue, fiscal and primary — and uses simple fiscal-policy algebra (the government expenditure, tax, transfer and balanced-budget multipliers, plus the automatic stabiliser of proportional taxes). It ends with the debt issue: how deficits add to debt, the burden of debt, Ricardian equivalence, the ‘crowding-out’ debate and ways to reduce the deficit, including the FRBMA, 2003.
Key Terms & Concepts
Government budget: a statement of the estimated receipts and expenditure of the government for a financial year, presented to Parliament under Article 112 (the Annual Financial Statement).
Public goods: goods that are non-rivalrous (one person’s use does not reduce the amount available to others) and non-excludable (no one can be kept from enjoying the benefit, giving rise to ‘free-riders’), e.g. national defence and roads. The market cannot supply them, so the government must.
Revenue receipts: receipts that neither create a liability nor reduce assets of the government (non-redeemable) — tax revenue (direct and indirect) and non-tax revenue (interest, dividends, fees).
Capital receipts: receipts that either create a liability (borrowing) or reduce financial assets (disinvestment, recovery of loans).
Revenue expenditure: expenditure not creating physical or financial assets — the normal running of government, interest payments, subsidies and grants.
Capital expenditure: expenditure that creates physical or financial assets or reduces liabilities — acquisition of land, buildings, machinery, investment in shares, and loans by the government.
Revenue deficit: the excess of revenue expenditure over revenue receipts; it means the government is dissaving.
Fiscal deficit: the excess of total expenditure over total receipts excluding borrowing; it measures the government’s total borrowing requirement.
Primary deficit: fiscal deficit minus net interest payments; it shows the current year’s fiscal imbalance, free of past interest obligations.
Automatic stabiliser: a feature such as proportional income tax that, without any deliberate action, lowers the multiplier and cushions the economy against booms and slumps.
Ricardian equivalence: the view that, because consumers are forward-looking, financing spending by borrowing or by taxes has the same effect, since people save more today to meet the future taxes that the debt implies.
Proportional tax (T = tY): Y* = A/(1 − c(1 − t)); multiplier = 1/(1 − c(1 − t)), where A = C̄ + cTR̄ + I + G.
NCERT Exercises — Full Solutions
All questions below are reproduced verbatim from the NCERT Introductory Macroeconomics Chapter 5 exercise. Answers are original; every numerical is solved step by step and verified.
1. Explain why public goods must be provided by the government.
ANSWERPublic goods such as national defence, roads and government administration must be provided by the government because of two features that the market cannot handle:(i) Non-rivalrous in consumption: one person’s consumption of a public good does not reduce the amount available to others (a public park or clean air can be enjoyed by many at once), so the usual link between paying and consuming breaks down.(ii) Non-excludable: there is no feasible way to keep a non-payer from enjoying the benefit. This gives rise to ‘free-riders’ — people who consume the good without paying. Since consumers will not voluntarily pay for what they can get free, private firms cannot earn revenue and will not provide such goods.Because the market mechanism fails to supply public goods, the government must step in and finance them through the budget (public provision), whether it produces them itself or buys them from the private sector.
2. Distinguish between revenue expenditure and capital expenditure.
ANSWERThe two differ mainly in whether the spending creates assets or reduces liabilities:
Basis
Revenue expenditure
Capital expenditure
Assets/liabilities
Does not create physical or financial assets, nor reduce liabilities
Creates physical or financial assets or reduces liabilities
Acquisition of land, buildings, machinery; investment in shares; loans by the government to states/PSUs
3. ‘The fiscal deficit gives the borrowing requirement of the government’. Elucidate.
ANSWERFiscal deficit = Total expenditure − (Revenue receipts + Non-debt creating capital receipts). The receipts subtracted are all the government’s receipts other than borrowing. So the deficit is precisely the gap that must be filled by borrowing.From the financing side, Gross fiscal deficit = Net borrowing at home + Borrowing from the RBI + Borrowing from abroad. Therefore the fiscal deficit indicates the total borrowing requirement of the government from all sources in a year.It is a key indicator of the financial health of the public sector: a large fiscal deficit means heavy borrowing, a growing stock of debt and rising future interest payments.
4. Give the relationship between the revenue deficit and the fiscal deficit.
ANSWERThe two are linked by the identity: Fiscal deficit = Revenue deficit + Capital expenditure − Non-debt creating capital receipts.Thus the revenue deficit is a part of the fiscal deficit. A large share of revenue deficit within the fiscal deficit signals that a large part of the borrowing is being used to meet the government’s consumption (revenue) needs rather than investment, which points to a deterioration in the quality of government expenditure and lower future capital formation.
5. Suppose that for a particular economy, investment is equal to 200, government purchases are 150, net taxes (that is lump-sum taxes minus transfers) is 100 and consumption is given by C = 100 + 0.75Y (a) What is the level of equilibrium income? (b) Calculate the value of the government expenditure multiplier and the tax multiplier. (c) If government expenditure increases by 200, find the change in equilibrium income.
SOLUTIONGiven I = 200, G = 150, net taxes T = 100 (lump-sum taxes minus transfers), C = 100 + 0.75Y where here Y is disposable income, so C = 100 + 0.75(Y − 100). Here c = 0.75.(a) Equilibrium income. Y = C + I + G = 100 + 0.75(Y − 100) + 200 + 150.Y = 100 + 0.75Y − 75 + 200 + 150 = 375 + 0.75Y.Y − 0.75Y = 375 ⇒ 0.25Y = 375 ⇒ Y* = 1500.(b) Multipliers. Government expenditure multiplier = 1/(1 − c) = 1/(1 − 0.75) = 1/0.25 = 4.Tax multiplier = −c/(1 − c) = −0.75/0.25 = −3.(c) Effect of ΔG = 200. ΔY = (1/(1 − c)) × ΔG = 4 × 200 = 800. Equilibrium income rises by 800 (new Y = 1500 + 800 = 2300).
6. Consider an economy described by the following functions: C = 20 + 0.80Y, I = 30, G = 50, TR = 100 (a) Find the equilibrium level of income and the autonomous expenditure multiplier in the model. (b) If government expenditure increases by 30, what is the impact on equilibrium income? (c) If a lump-sum tax of 30 is added to pay for the increase in government purchases, how will equilibrium income change?
SOLUTIONGiven C = 20 + 0.80YD, I = 30, G = 50, TR = 100, with c = 0.80. (No tax is mentioned initially, so T = 0 and YD = Y + TR.)(a) Equilibrium income. Y = C + I + G = 20 + 0.80(Y + 100) + 30 + 50.Y = 20 + 0.80Y + 80 + 30 + 50 = 180 + 0.80Y.0.20Y = 180 ⇒ Y* = 900. Autonomous expenditure multiplier = 1/(1 − 0.80) = 1/0.20 = 5.(b) ΔG = 30. ΔY = 5 × 30 = 150. New equilibrium income = 900 + 150 = 1050.(c) A lump-sum tax of 30 added (T = 30) along with ΔG = 30. The tax reduces income through the tax multiplier −c/(1 − c) = −0.80/0.20 = −4. Tax effect = −4 × 30 = −120.Net change = (G effect) + (tax effect) = +150 − 120 = +30. This is the balanced-budget multiplier of unity: an equal rise in G and T (30 each) raises income by exactly 30. New equilibrium income = 1050 − 120 = 930.
7. In the above question, calculate the effect on output of a 10 per cent increase in transfers, and a 10 per cent increase in lump-sum taxes. Compare the effects of the two.
SOLUTIONFrom Q6, c = 0.80, TR = 100. A 10% increase in transfers means ΔTR = 10; a 10% increase in lump-sum taxes (on the tax of 30 introduced in part c) means ΔT = 3.Transfer increase. Transfer multiplier = c/(1 − c) = 0.80/0.20 = 4. ΔY = 4 × 10 = +40.Tax increase. Tax multiplier = −c/(1 − c) = −4. ΔY = −4 × 3 = −12.Comparison. A 10% rise in transfers raises output by 40, whereas a 10% rise in lump-sum taxes lowers output by 12. The two work in opposite directions: transfers add to disposable income and demand, while taxes reduce them. (The numbers differ because the base of transfers, 100, is larger than the tax base of 30; per unit, the transfer multiplier 4 and the tax multiplier −4 are equal in size but opposite in sign.)
8. We suppose that C = 70 + 0.70YD, I = 90, G = 100, T = 0.10Y (a) Find the equilibrium income. (b) What are tax revenues at equilibrium income? Does the government have a balanced budget?
SOLUTIONGiven C = 70 + 0.70YD, I = 90, G = 100, proportional tax T = 0.10Y, so YD = Y − T = Y − 0.10Y = 0.90Y. Here c = 0.70, t = 0.10.(a) Equilibrium income. Y = C + I + G = 70 + 0.70(0.90Y) + 90 + 100.Y = 70 + 0.63Y + 90 + 100 = 260 + 0.63Y.Y − 0.63Y = 260 ⇒ 0.37Y = 260 ⇒ Y* = 260/0.37 = 702.70 (approx. 702.7).(b) Tax revenue. T = 0.10Y = 0.10 × 702.70 = 70.27.Balanced budget? Government spending G = 100, tax revenue T = 70.27. Since T (70.27) < G (100), the government does not have a balanced budget — it runs a budget deficit of 100 − 70.27 = 29.73.
9. Suppose marginal propensity to consume is 0.75 and there is a 20 per cent proportional income tax. Find the change in equilibrium income for the following (a) Government purchases increase by 20 (b) Transfers decrease by 20.
SOLUTIONGiven c = 0.75 and proportional tax rate t = 0.20. With proportional taxes the multiplier for autonomous changes is 1/(1 − c(1 − t)).c(1 − t) = 0.75 × (1 − 0.20) = 0.75 × 0.80 = 0.60. So multiplier = 1/(1 − 0.60) = 1/0.40 = 2.5.(a) ΔG = 20. ΔY = 2.5 × 20 = +50.(b) ΔTR = −20. A change in transfers enters multiplied first by c (only c times any transfer is spent), so the transfer multiplier with proportional taxes is c/(1 − c(1 − t)) = 0.75/0.40 = 1.875. ΔY = 1.875 × (−20) = −37.5.So a rise in government purchases of 20 raises income by 50, while a fall in transfers of 20 lowers income by 37.5.
10. Explain why the tax multiplier is smaller in absolute value than the government expenditure multiplier.
ANSWERThe government expenditure multiplier is 1/(1 − c) and the tax multiplier is −c/(1 − c). Since 0 < c < 1, the absolute value of the tax multiplier, c/(1 − c), is smaller than 1/(1 − c) by exactly one.The reason is that an increase in government spending adds to total demand directly — the whole of ΔG enters aggregate demand in the first round. A tax change, however, affects demand only indirectly, through disposable income: when taxes fall by ΔT, consumption rises only by c × ΔT in the first round (the fraction (1 − c) of the tax cut is saved, not spent).Because the first-round impulse from a tax change (cΔT) is smaller than that from a spending change (ΔG), the full multiplier effect of taxes is smaller in absolute value — always one less than the spending multiplier.
11. Explain the relation between government deficit and government debt.
ANSWERA government deficit is a flow — the shortfall of receipts below expenditure in a single year, financed mainly by borrowing. Government debt is a stock — the total accumulated amount the government owes at a point in time.The two are closely linked: each year’s deficit adds to the stock of debt. If the government keeps running deficits year after year, debt keeps accumulating, and it has to pay more and more by way of interest.These interest payments themselves enlarge future expenditure and so feed back into still larger deficits and debt — a self-reinforcing process. Thus, persistent deficits are the chief cause of a rising public debt.
12. Does public debt impose a burden? Explain.
ANSWERPublic debt may impose a burden, but the issue must be judged for the economy as a whole, not like a single trader’s debt — the government can also raise resources through taxation and printing money.Arguments that it is a burden: (i) By borrowing today and repaying through higher taxes later, the government shifts the burden of reduced consumption onto future generations, lowering their disposable income and consumption. (ii) Government borrowing reduces the savings available to the private sector, which can crowd out private investment and slow capital formation and growth. (iii) Debt owed to foreigners is a real burden, since goods must be sent abroad to pay the interest.Arguments that it need not be a burden: (i) Debt owed to ourselves merely transfers purchasing power within the nation. (ii) Under Ricardian equivalence, forward-looking consumers save more to offset future taxes, so national saving need not fall. (iii) If the borrowed funds are invested in infrastructure whose return exceeds the rate of interest, future generations are better off and growth can pay off the debt.Hence debt is burdensome only if it reduces future growth in output; if it raises productive capacity, the growth of the economy can outpace the debt.
13. Are fiscal deficits inflationary?
ANSWERFiscal deficits are not always inflationary; it depends on the state of the economy.When they can be inflationary: a deficit raises aggregate demand (through higher spending or tax cuts). If the economy is near full employment and firms cannot raise output at existing prices, the extra demand pushes prices up.When they need not be: if there are unutilised resources and output is being held back by a lack of demand, a higher fiscal deficit raises demand, output and employment without raising prices. Here the deficit increases real output rather than the price level.Therefore a high fiscal deficit accompanied by higher output need not be inflationary — the inflationary impact depends on whether the economy has spare capacity.
14. Discuss the issue of deficit reduction.
ANSWERA government deficit can be reduced either by raising receipts or by cutting expenditure.Raising receipts: increase tax revenue, with greater reliance on direct taxes (indirect taxes are regressive, hitting all income groups equally), and raise non-debt receipts through disinvestment (sale of shares in PSUs).Cutting expenditure: the major thrust has been here. This can be done by making government activities more efficient through better planning and administration (e.g. cash transfers instead of food subsidy, since the Planning Commission found the government spends Rs 3.65 to transfer Re 1 to the poor), or by narrowing the scope of government. But cutting back vital areas like agriculture, education, health and poverty alleviation would harm the economy.Many countries adopt self-imposed limits such as India’s FRBMA, 2003. However, larger deficits do not always mean a more expansionary policy — the same fiscal measures can yield a large or small deficit depending on the state of the economy (in a recession, falling tax revenue automatically widens the deficit). So deficit reduction must be pursued with care, keeping growth and welfare in view.
15. What do you understand by G.S.T? How good is the system of G.S.T as compared to the old tax system? State its categories.
ANSWERMeaning: The Goods and Services Tax (GST), operational from 1 July 2017, is a single comprehensive indirect tax on the supply of goods and services, levied right from the manufacturer/service provider to the consumer. It is a destination-based consumption tax with the facility of Input Tax Credit (ITC) throughout the supply chain, applied at one rate for one type of good/service across the country.How it is better than the old system: (i) It removes the cascading (tax-on-tax) of the old regime by taxing only the value added at each stage and allowing credit of tax paid at the previous stage. (ii) It subsumes many Central and State taxes (Central Excise, Service Tax, CST, VAT, Entry Tax, Octroi, Luxury and Entertainment Tax, etc.), creating ‘One Nation, One Tax, One Market’. (iii) It establishes parity in taxation across states, eases the movement of goods and services, widens the tax base, raises transparency, reduces the cost of business and is expected to raise GDP. (iv) Compliance is online through a common portal, reducing human interface and easing the doing of business.Categories (rate slabs): there are six standard GST rates — 0%, 3%, 5%, 12%, 18% and 28% — applied on the supply of goods and/or services across the country. (Five petroleum products are kept out of GST for the time being, States continue to levy VAT on alcoholic liquor for human consumption, and tobacco attracts both GST and Central Excise Duty.)
Extra Practice Questions
Short Answer Type Questions
Q1. Define a balanced budget, a surplus budget and a deficit budget.
ANSWERA balanced budget is one in which estimated government expenditure equals estimated receipts. A surplus budget is one in which receipts exceed expenditure (tax collection exceeds required spending). A deficit budget is one in which expenditure exceeds receipts — the most common situation, financed by borrowing.
Q2. Distinguish between direct and indirect taxes with one example each.
ANSWERA direct tax is levied on income or wealth and its burden cannot be shifted to another person, e.g. personal income tax and corporation tax. An indirect tax is levied on goods and services and its burden can be shifted to the final consumer, e.g. GST and customs duty. Indirect taxes are regressive, while progressive direct taxes help redistribution.
Q3. What is the primary deficit and why is it calculated?
ANSWERPrimary deficit = Fiscal deficit − Net interest payments. It is calculated to focus on the current year’s fiscal imbalance by removing the interest burden inherited from past borrowing. A zero primary deficit means the government is borrowing only to pay interest on existing debt, not for fresh expenditure.
Q4. How does a proportional income tax act as an automatic stabiliser?
ANSWERA proportional tax (T = tY) lowers the marginal propensity to consume out of income to c(1 − t), making disposable income and consumption less sensitive to changes in GDP. When GDP rises, more is siphoned off as tax, limiting the rise in consumption; when GDP falls, disposable income falls less sharply. This works automatically, without any policy decision, to cushion the economy.
Q5. State two reasons why public goods cannot be supplied by the market.
ANSWER(i) Public goods are non-rivalrous — one person’s consumption does not reduce the amount available to others, so there is no exclusive title to sell. (ii) They are non-excludable — non-payers cannot be kept out, creating free-riders, so private firms cannot collect fees and earn revenue. Hence the government must provide them.
Long Answer Type Questions
Q1. Explain the three objectives of a government budget.
ANSWERThe government budget serves three objectives. Allocation: the government provides public goods (defence, roads, administration) that the market cannot supply because they are non-rivalrous and non-excludable; it finances these through the budget (public provision) and may also produce them itself (public production). Redistribution: through progressive taxes and transfer payments the government changes the distribution of income, affecting households’ personal disposable income so as to bring about a distribution considered fair by society. Stabilisation: the government uses fiscal policy to correct fluctuations in income, output, employment and prices — raising aggregate demand when there is unemployment and restraining it when demand exceeds output and threatens inflation. Together these functions let the budget promote efficiency, equity and stability.
Q2. Derive and explain the balanced-budget multiplier.
ANSWERThe balanced-budget multiplier measures the effect on income when government spending and taxes rise by the same amount (ΔG = ΔT), keeping the budget balanced. The government expenditure multiplier is 1/(1 − c) and the tax multiplier is −c/(1 − c). Adding the two policy multipliers gives 1/(1 − c) + (−c/(1 − c)) = (1 − c)/(1 − c) = 1. So a balanced-budget multiplier of unity means a 100 increase in G financed by a 100 increase in taxes raises income by exactly 100. This happens because the whole of ΔG enters demand directly, while the tax increase reduces consumption only by c times ΔT (the rest comes out of saving); the net first-round effect is ΔG − cΔT = ΔG(1 − c), and after the multiplier process income rises by exactly ΔG. Thus an equal rise in spending and taxes is still mildly expansionary.
Q3. Examine whether public debt is a burden on the economy.
ANSWERWhether public debt is a burden must be judged for the whole economy, since the government — unlike a single trader — can also tax and print money. The case for a burden: borrowing today shifts reduced consumption onto future generations who must pay higher taxes; government borrowing reduces private savings and may crowd out investment, lowering capital formation and growth; and external debt is a genuine burden because goods must be exported to pay the interest. The case against: internal debt is owed ‘to ourselves’, so purchasing power stays within the nation; under Ricardian equivalence forward-looking households save more to meet future taxes, leaving national saving unchanged; and the flow of savings is not fixed — if deficits succeed in raising output, income and saving both rise, so both government and industry can borrow more. Crucially, if borrowing finances infrastructure whose return exceeds the interest rate, growth can repay the debt and future generations gain. Conclusion: debt is burdensome only if it reduces future growth in output; productive debt judged against the growth of the economy as a whole need not be a burden.
MCQs & Assertion–Reason
1. The government budget is presented to Parliament under which Article of the Constitution?
2. Which of the following is a non-tax revenue receipt?
(a) Corporation tax (b) Customs duty (c) Dividends and profits from PSUs (d) GST
3. A feature of a public good is that it is:
(a) rivalrous and excludable (b) non-rivalrous and non-excludable (c) rivalrous and non-excludable (d) non-rivalrous and excludable
4. Revenue deficit is the excess of:
(a) total expenditure over total receipts (b) revenue expenditure over revenue receipts (c) capital expenditure over capital receipts (d) fiscal deficit over interest payments
5. Fiscal deficit equals:
(a) total expenditure − total receipts including borrowing (b) total expenditure − (revenue receipts + non-debt creating capital receipts) (c) revenue receipts − revenue expenditure (d) interest payments − fiscal deficit
For each Assertion–Reason question, choose: (A) Both true and the Reason correctly explains the Assertion; (B) Both true but the Reason is not the correct explanation; (C) Assertion true, Reason false; (D) Assertion false, Reason true.
A-R 1. Assertion: The fiscal deficit indicates the total borrowing requirement of the government.
Reason: Fiscal deficit equals total expenditure minus all receipts other than borrowing.
A-R 2. Assertion: The tax multiplier is larger in absolute value than the government expenditure multiplier.
Reason: A tax change affects demand only indirectly through disposable income, while a spending change affects demand directly.
A-R 3. Assertion: A proportional income tax acts as an automatic stabiliser.
Reason: It makes disposable income and consumption less sensitive to fluctuations in GDP.
A-R 4. Assertion: The revenue deficit is a part of the fiscal deficit.
Reason: Fiscal deficit = revenue deficit + capital expenditure − non-debt creating capital receipts.
A-R 5. Assertion: A fiscal deficit is always inflationary.
Reason: When there are unutilised resources, a higher deficit raises output rather than prices.
Answer key: 1-(A), 2-(D), 3-(A), 4-(A), 5-(D).
Exam Tips & Common Mistakes
How to score full marks in this chapter
Memorise the three deficit formulas exactly and show the substitution clearly in numericals. For multiplier sums, first state the value of c (and t if proportional taxes apply), compute the multiplier, then multiply by the change in the autonomous variable. Remember the sign conventions: the tax multiplier is negative, the transfer multiplier positive. Quote the precise definitions (non-rivalrous, non-excludable, revenue vs capital) and use the book’s own examples — FRBMA 2003, Article 112, Ricardian equivalence, GST slabs — to show you have studied the chapter. Always end a numerical with a one-line conclusion (e.g. “hence the budget is in deficit”).
Common mistakes to avoid
Confusing the tax multiplier (−c/(1 − c)) with the government expenditure multiplier (1/(1 − c)) — the tax one is negative and one smaller in absolute value.
Forgetting that disposable income YD = Y − T (+ TR), so substitute YD correctly before solving for Y.
Treating a renewable receipt like a tax as creating a liability — revenue receipts are non-redeemable; only borrowing creates a liability.
Saying fiscal deficits are “always” inflationary — with spare capacity they raise output, not prices.
Mixing up deficit (a yearly flow) with debt (an accumulated stock).
Stating the balanced-budget multiplier as 0 — it is exactly 1.
Frequently Asked Questions
What is Chapter 5 of Class 12 Economics (Introductory Macroeconomics) about?
Chapter 5, Government Budget and the Economy, explains the meaning and components of the government budget, its three objectives (allocation, redistribution, stabilisation), the classification of receipts and expenditure, the measures of deficit (revenue, fiscal, primary), fiscal-policy multipliers, and the issue of government debt.
What is the difference between fiscal deficit and primary deficit?
Fiscal deficit is total expenditure minus all receipts other than borrowing, and shows the government’s total borrowing requirement. Primary deficit is the fiscal deficit minus net interest payments, and shows the current year’s fiscal imbalance free of the interest burden inherited from past debt.
Are these Class 12 Economics Chapter 5 solutions free?
Yes. All solutions are free and follow the official NCERT Introductory Macroeconomics textbook for the 2026–27 session, with every theory question answered and every numerical solved and verified step by step.
