All practice questions

For the gaseous reaction represented by the equation:

2\text{A}(g) + \text{B}(g) \rightarrow \text{C}(g) + 3\text{D}(g)

the quantity of substance

\text{A}

present in a

2.0\text{ dm}^3

reaction vessel decreases from

0.80\text{ mol}

to

0.32\text{ mol}

in

40\text{ s}

. What is the average rate of formation of product

\text{D}

in

\text{mol dm}^{-3}\text{ s}^{-1}

?

Show answer & explanation

Answer: 0.009

Answer

The average rate of formation of product D is

0.009\text{ mol dm}^{-3}\text{ s}^{-1}

.

The change in concentration of reactant A over

40\text{ s}

is

\frac{0.48\text{ mol}}{2.0\text{ dm}^3} = 0.24\text{ mol dm}^{-3}

. The rate of consumption of A is

\frac{0.24}{40} = 0.006\text{ mol dm}^{-3}\text{ s}^{-1}

. Because

2\text{ moles}

of A produce

3\text{ moles}

of D, the rate of formation of D is

\frac{3}{2} \times 0.006 = 0.009\text{ mol dm}^{-3}\text{ s}^{-1}

.

Step-by-Step Solution

1

Calculate the change in concentration of reactant A during the time interval.

\Delta [A] = \frac{0.80\text{ mol} - 0.32\text{ mol}}{2.0\text{ dm}^3} = 0.24\text{ mol dm}^{-3}

Concentration is moles per unit volume.

2

Calculate the rate of consumption of reactant A per unit time.

\text{Rate}_A = -\frac{\Delta [A]}{\Delta t} = \frac{0.24\text{ mol dm}^{-3}}{40\text{ s}} = 0.006\text{ mol dm}^{-3}\text{ s}^{-1}

Reaction rate is defined as the change in concentration over elapsed time.

3

Use the stoichiometric coefficients from the balanced equation to calculate the rate of formation of D.

\text{Rate}_D = \frac{3}{2} \times \text{Rate}_A = \frac{3}{2} \times 0.006 = 0.009\text{ mol dm}^{-3}\text{ s}^{-1}

According to the balanced chemical equation,

2

moles of A are consumed for every

3

moles of D formed, so

\frac{1}{3}\text{Rate}_D = \frac{1}{2}\text{Rate}_A

.

Key Concept

Stoichiometric relationship between rates of consumption of reactants and rates of formation of products

Question 282Question →

A profit-maximizing monopolist produces at an equilibrium output level of $100$ units. At this output, the product is sold at a price of $\$60$ per unit and the average total cost is $\$45$ per unit. What is the total profit earned by the monopolist in dollars?

Show answer & explanation

Answer: 1500

Answer

The total profit earned by the monopolist is

1500

dollars.

Total profit is calculated as

(P - ATC) \times Q = (60 - 45) \times 100 = 15 \times 100 = 1500

dollars.

Step-by-Step Solution

1

Calculate Total Revenue (

TR

)

TR = \$6,000

Total revenue is obtained by multiplying price per unit by total output (

P \times Q = 60 \times 100

).

2

Calculate Total Cost (

TC

)

TC = \$4,500

Total cost is obtained by multiplying average total cost per unit by total output (

ATC \times Q = 45 \times 100

).

3

Calculate Profit (

\pi

)

\pi = \$1,500

Economic profit is the excess of total revenue over total cost (

\pi = TR - TC = 6,000 - 4,500

).

Key Concept

Short-run monopoly profit calculation using revenue and cost figures.

Estimated Time:1m 0s

Question 283Question →

A consumer spends their entire monthly budget of $\text{₦}2,400$ on two commodities, Good $X$ and Good $Y$ . The market price of Good $Y$ ( $P_y$ ) is $\text{₦}30$ per unit. At utility-maximizing equilibrium under ordinal utility analysis, the consumer purchases $20$ units of Good $X$ and $40$ units of Good $Y$ . What is the magnitude of the Marginal Rate of Substitution of Good $X$ for Good $Y$ ( $MRS_{xy}$ ) at this equilibrium point?

Show answer & explanation

Answer: 2

Answer

The Marginal Rate of Substitution of Good X for Good Y (MRS_xy) at equilibrium is 2.

Under ordinal utility theory, consumer equilibrium occurs at the point of tangency between the budget line and the highest attainable indifference curve. At this point, the slope of the indifference curve (

MRS_{xy}

) equals the absolute slope of the budget line (

\frac{P_x}{P_y}

). First, calculating

P_x

from the budget equation

2400 = 20 P_x + 30(40)

yields

P_x = \text{₦}60

. Then, substituting

P_x

and

P_y

into the equilibrium condition gives

MRS_{xy} = \frac{60}{30} = 2

.

Step-by-Step Solution

1

Formulate the budget line equation using total income and expenditures.

2400 = 20 P_x + 30(40)

Total expenditure on both goods must equal total income at budget exhaustion.

2

Calculate the price of Good X (

P_x

).

P_x = \text{₦}60

Finding the price of Good X is necessary to establish the price ratio.

3

Calculate the Marginal Rate of Substitution at consumer equilibrium.

MRS_{xy} = \frac{P_x}{P_y} = \frac{60}{30} = 2

At consumer equilibrium under ordinal utility, the indifference curve is tangent to the budget line, meaning

MRS_{xy} = \frac{P_x}{P_y}

.

Key Concept

Consumer Equilibrium Condition under Ordinal Utility Analysis

Estimated Time:2m 0s

Question 284Question →

The table below shows the distribution of weekly expenditures (in thousands of Naira, ₦'000) of a group of market traders:

Expenditure (₦'000)	Frequency ( $f$ )
10	5
20	8
30	$x$
40	4
50	3

If the mean weekly expenditure of the traders is ₦28,000 (represented as $28$ in the table units), find the value of $x$ .

Show answer & explanation

Answer: 20

Answer

The value of

x

is 20.

Using the arithmetic mean formula

\bar{x} = \frac{\sum fx}{\sum f}

, we set up the equation with the given mean of 28:

\frac{520 + 30x}{20 + x} = 28

. Cross-multiplying gives

520 + 30x = 560 + 28x

. Rearranging terms yields

2x = 40

, which gives

x = 20

.

Step-by-Step Solution

1

Sum all given frequencies including the unknown

x

to get the total frequency expression.

\sum f = 20 + x

The total number of observations is needed for the denominator of the arithmetic mean formula.

2

Multiply each expenditure value by its respective frequency and aggregate the terms.

\sum f x = 520 + 30x

The total weighted value of all expenditures is needed for the numerator of the mean formula.

3

Substitute the known mean value of 28 into the equation

\bar{x} = \frac{\sum f x}{\sum f}

and solve for

x

.

x = 20

Isolating

x

yields the exact missing frequency.

Key Concept

Finding a missing frequency from a frequency distribution given the arithmetic mean.

Question 285Question →

The table below presents the monthly Production Possibility Schedule for a manufacturing firm in Aba producing Leather Shoes and Leather Bags using a fixed quantity of labor and raw materials:

Production Combination	Leather Shoes (hundreds of pairs)	Leather Bags (hundreds of units)
P	$0$	$150$
Q	$20$	$140$
R	$40$	$120$
S	$60$	$90$
T	$80$	$50$
U	$100$	$0$

If the plant is currently producing at Combination Q ( $20$ hundred pairs of shoes and $140$ hundred bags) and reallocates its resources to increase shoe production to Combination T ( $80$ hundred pairs of shoes), what is the average opportunity cost of producing ONE additional pair of Leather Shoes over this range? Express your answer in units of Leather Bags foregone.

Show answer & explanation

Answer: 1.5

Answer

1.5 bags foregone per additional pair of shoes

Opportunity cost measures the sacrifice of one good required to obtain an additional quantity of another good. Moving from Combination Q to Combination T increases shoe production by 60 hundred pairs (from 20 to 80 hundred pairs) while reducing bag production by 90 hundred units (from 140 to 50 hundred units). The opportunity cost per additional pair of shoes is calculated as 90 hundred bags divided by 60 hundred pairs of shoes, yielding exactly 1.5 bags foregone per pair of shoes.

Step-by-Step Solution

1

Determine initial production at Combination Q

Shoes =

20

hundred pairs (

2,000

pairs), Bags =

140

hundred units (

14,000

bags)

Establishes the baseline output before resource reallocation.

2

Determine final production at Combination T

Shoes =

80

hundred pairs (

8,000

pairs), Bags =

50

hundred units (

5,000

bags)

Identifies the new production level after expanding shoe production.

3

Calculate the net gain in shoe production and net loss in bag production

Additional shoes =

80 - 20 = 60

hundred pairs; Bags foregone =

140 - 50 = 90

hundred bags

Opportunity cost evaluates the sacrifice required to gain additional units of the preferred good.

4

Calculate the unit opportunity cost

Opportunity cost per shoe pair =

\frac{90\text{ hundred bags}}{60\text{ hundred pairs}} = 1.5\text{ bags}

Divides the total quantity of the sacrificed good by the total gain of the produced good to get per-unit real cost.

Key Concept

Opportunity Cost / Marginal Rate of Transformation

Question 286Question →

A monopolist faces a market demand function given by $P = 140 - 2Q$ , where $P$ is the price in Naira and $Q$ is the output level. The total cost function of the firm is $TC = 20Q + Q^2 + 200$ . What is the maximum economic profit, in Naira, earned by the monopolist at equilibrium?

Show answer & explanation

Answer: 1000

Answer

The maximum economic profit earned by the monopolist at profit-maximizing equilibrium is 1000 Naira.

To maximize economic profit, a monopolist sets marginal revenue equal to marginal cost (

MR = MC

). From

P = 140 - 2Q

,

TR = 140Q - 2Q^2

, giving

MR = 140 - 4Q

. Differentiating

TC = 20Q + Q^2 + 200

gives

MC = 20 + 2Q

. Equating

MR = MC

yields

140 - 4Q = 20 + 2Q \implies Q = 20

units. Substituting

Q = 20

into the demand equation gives price

P = 100

Naira. Total revenue is

2000

Naira (

100 \times 20

) and total cost is

1000

Naira (

20(20) + 20^2 + 200

). The resulting maximum economic profit is

2000 - 1000 = 1000

Naira.

Step-by-Step Solution

1

Derive Total Revenue (TR) and Marginal Revenue (MR) functions

TR = 140Q - 2Q^2

and

MR = 140 - 4Q

Marginal revenue is the first derivative of total revenue with respect to quantity.

2

Derive Marginal Cost (MC) function

MC = 20 + 2Q

Marginal cost is the first derivative of total cost with respect to quantity.

3

Equate MR to MC to solve for the profit-maximizing output level (Q)

140 - 4Q = 20 + 2Q \implies 6Q = 120 \implies Q = 20

units

The necessary condition for profit maximization in all market structures is

MR = MC

.

4

Determine the equilibrium price (P) from the demand curve

P = 140 - 2(20) = 100

Naira

Monopolists set price based on consumer willingness to pay at the profit-maximizing output level.

5

Calculate Total Revenue (TR), Total Cost (TC), and Economic Profit (\pi)

TR = 100 \times 20 = 2000

,

TC = 20(20) + (20)^2 + 200 = 1000

, Profit

= 2000 - 1000 = 1000

Naira

Economic profit is the difference between total revenue and total cost at equilibrium output.

Key Concept

Monopoly Profit Maximization Condition (MR = MC)

Estimated Time:2m 30s

Question 287Question →

In a given year, a country recorded a Nominal Gross Domestic Product (GDP) of $\$500\text{ billion}$ . The GDP deflator for the year was $125$ (with a base year index of $100$ ), and the total population was $50\text{ million}$ . What was the Real Per Capita Income of the country in dollars?

Show answer & explanation

Answer: 8000

Answer

The Real Per Capita Income of the country is $8,000.

To compute the Real Per Capita Income, Nominal GDP is first converted to Real GDP using the GDP deflator:

\frac{\$500\text{ billion}}{125} \times 100 = \$400\text{ billion}

. Dividing this real income figure by the population of

50\text{ million}

gives an average real per capita income of

\$8,000

.

Step-by-Step Solution

1

Deflate Nominal GDP to obtain Real GDP

Real GDP = $400 billion

Real GDP removes the inflationary effect measured by the GDP deflator.

2

Divide Real GDP by the population size

Real Per Capita Income = $8,000

Per capita real income measures average real economic output per person.

Key Concept

Real GDP and Per Capita Income Derivation

Estimated Time:1m 30s

Question 288Question →

A software engineer in Lagos currently earns a salary of $₦500,000$ per month. She is considering quitting her job for one year to pursue one of two mutually exclusive opportunities:

- Opportunity X: Establish an independent tech startup requiring an initial capital investment of $₦3,000,000$ , which she must withdraw from her fixed deposit account currently yielding $12\%$ per annum interest. The startup is expected to generate $₦15,000,000$ in total revenue over the year, with total operating expenses (office rent, server infrastructure, and wages) amounting to $₦6,500,000$ .
- Opportunity Y: Work as an overseas remote contractor earning a net salary of $₦900,000$ per month with zero capital investment required.

Calculate, in Naira ( $₦$ ), the economic opportunity cost of choosing Opportunity X for the year.

Show answer & explanation

Answer: 11160000

Answer

The economic opportunity cost of choosing Opportunity X for the year is ₦11,160,000.

The economic opportunity cost of an action is defined as the total benefit of the next best alternative foregone. By choosing Opportunity X, the software engineer gives up Opportunity Y (worth ₦900,000 × 12 = ₦10,800,000) as well as the interest her ₦3,000,000 capital would have earned in the fixed deposit account (12% of ₦3,000,000 = ₦360,000). Together, the total sacrificed value of this highest-ranked foregone package equals ₦10,800,000 + ₦360,000 = ₦11,160,000.

Step-by-Step Solution

1

Calculate the annual income of all alternative employment options available during the year.

Current Job annual income = ₦500,000 × 12 = ₦6,000,000. Opportunity Y annual income = ₦900,000 × 12 = ₦10,800,000.

Opportunity cost evaluates the sacrifice made regarding alternative choices foregone.

2

Calculate the foregone interest earned if funds remain in the fixed deposit account.

Foregone annual interest = 12% of ₦3,000,000 = ₦360,000.

Selecting Opportunity Y or retaining her current job would leave the ₦3,000,000 intact in savings, earning 12% interest.

3

Determine the net value of the single best foregone alternative option.

Value of Next Best Alternative (Opportunity Y + Savings Interest) = ₦10,800,000 + ₦360,000 = ₦11,160,000.

Economic opportunity cost is defined as the total value of the highest-valued alternative option foregone.

Key Concept

Opportunity Cost as the Next Best Alternative Foregone

Estimated Time:3m 0s

Question 289Question →

A consumer obtains an average utility ( $AU$ ) of $15$ utils from consuming $4$ units of bottled water. When the $5^{\text{th}}$ unit is consumed, the marginal utility ( $MU$ ) derived is $25$ utils. If consuming the $6^{\text{th}}$ unit brings the consumer's total utility ( $TU$ ) to $96$ utils, what is the average utility ( $AU$ ) per unit, in utils, when $6$ units are consumed?

Show answer & explanation

Answer: 16

Answer

The average utility per unit when 6 units are consumed is 16 utils.

To find the average utility after 6 units, determine total utility at 4 units (

15 \times 4 = 60

utils). Adding the marginal utility of the 5th unit (

25

utils) gives a total utility of

85

utils for 5 units. Since total utility for 6 units is given as

96

utils, dividing by quantity (

96 / 6

) yields an average utility of

16

utils.

Step-by-Step Solution

1

Calculate Total Utility for 4 units (

TU_4

)

TU_4 = 60

utils

Total utility is calculated by multiplying average utility by the quantity consumed:

TU_4 = AU_4 \times 4 = 15 \times 4 = 60

utils.

2

Calculate Total Utility for 5 units (

TU_5

)

TU_5 = 85

utils

Total utility of 5 units equals total utility of 4 units plus the marginal utility of the 5th unit:

TU_5 = TU_4 + MU_5 = 60 + 25 = 85

utils.

3

Identify Total Utility for 6 units (

TU_6

)

TU_6 = 96

utils

The total utility after consuming 6 units is given directly in the problem description as 96 utils.

4

Calculate Average Utility for 6 units (

AU_6

)

AU_6 = 16

utils

Average utility is calculated by dividing total utility by total units consumed:

AU_6 = \frac{TU_6}{6} = \frac{96}{6} = 16

utils.

Key Concept

Mathematical interrelationships between Total Utility (TU), Average Utility (AU), and Marginal Utility (MU)

Question 290Question →

A commercial bank receives an initial cash deposit of ₦50,000. If the central bank specifies a cash reserve ratio of 20%, what is the total amount of deposits created in the banking system?

Show answer & explanation

Answer: 250000

Answer

The total amount of deposits created in the banking system is ₦250,000.

The total amount of deposits created across the commercial banking system is calculated using the formula

\text{Total Deposit} = \frac{\text{Initial Deposit}}{\text{Cash Reserve Ratio}}

. Substituting the given values gives

\frac{₦50,000}{0.20} = ₦250,000

.

Step-by-Step Solution

1

Identify the given initial deposit and cash reserve ratio.

Initial Deposit = ₦50,000; Cash Reserve Ratio (CRR) =

20\% = 0.20

.

These parameters are required to compute credit expansion.

2

Apply the total credit creation formula.

\text{Total Deposit Expansion} = \frac{\text{Initial Deposit}}{\text{Cash Reserve Ratio}}

The commercial banking system creates secondary deposits equal to the initial primary deposit divided by the cash reserve ratio.

3

Perform the numerical calculation.

\text{Total Deposit Expansion} = \frac{50,000}{0.20} = 250,000

Dividing ₦50,000 by 0.20 yields ₦250,000.

Key Concept

Credit Creation and Total Deposit Expansion

Question 291Question →

An agricultural officer recorded the annual yield of palm oil (in metric tons) produced by a commercial farm over a 5-year period as follows: $12$ , $16$ , $18$ , $20$ , and $24$ . What is the standard deviation of the annual palm oil yield in metric tons?

Show answer & explanation

Answer: 4

Answer

The standard deviation of the palm oil yield over the 5-year period is

4\text{ metric tons}

.

The standard deviation is calculated by determining the mean yield (

18\text{ metric tons}

), taking the sum of the squared deviations from the mean (

80

), dividing by the total number of observations (

5

) to obtain a variance of

16

, and then taking the square root of

16

, which equals

4\text{ metric tons}

.

Step-by-Step Solution

1

Calculate the arithmetic mean (

\mu

) of the data set.

\mu = \frac{12 + 16 + 18 + 20 + 24}{5} = 18\text{ metric tons}

.

The mean is required to determine the deviations of individual data values.

2

Determine the squared deviations from the mean for each yield value.

(12-18)^2 = 36

,

(16-18)^2 = 4

,

(18-18)^2 = 0

,

(20-18)^2 = 4

, and

(24-18)^2 = 36

. The sum of these squared deviations is

80

.

Squaring deviations eliminates negative values so that dispersion around the mean can be aggregated accurately.

3

Compute the variance (

\sigma^2

) of the dataset.

\sigma^2 = \frac{80}{5} = 16

.

Variance represents the average of the squared deviations from the mean.

4

Compute the standard deviation (

\sigma

) by taking the square root of the variance.

\sigma = \sqrt{16} = 4\text{ metric tons}

.

Standard deviation measures dispersion in the original units of the data.

Key Concept

Standard Deviation of Ungrouped Data

Question 292Question →

The supply function for a commodity is given by $Q_s = -20 + 5P$ , where $Q_s$ is the quantity supplied in units and $P$ is the price in Naira ( $\text{N}$ ). What is the quantity supplied when the price per unit is $\text{N}10$ ?

Show answer & explanation

Answer: 30

Answer

The quantity supplied when the price is

\text{N}10

is 30 units.

Substituting

P = 10

into the linear supply equation

Q_s = -20 + 5P

gives

Q_s = -20 + 5(10) = 30

units, directly reflecting the positive relationship between price and quantity supplied stated by the law of supply.

Step-by-Step Solution

1

Substitute the given price into the supply function

Q_s = -20 + 5(10)

The price

P

is specified as 10 Naira in the problem statement.

2

Perform the multiplication and addition

Q_s = -20 + 50 = 30

Evaluating the linear expression gives the total quantity supplied at price level 10.

Key Concept

Linear Supply Function and Quantity Supplied Calculation

Estimated Time:45s

Question 293Question →

At a constant temperature,

1.0\text{ mole}

of carbon monoxide,

\text{CO}(g)

, and

1.0\text{ mole}

of steam,

\text{H}_2\text{O}(g)

, are introduced into a

2.0\text{ dm}^3

sealed vessel and allowed to reach equilibrium according to the equation:

\text{CO}(g) + \text{H}_2\text{O}(g) \rightleftharpoons \text{CO}_2(g) + \text{H}_2(g)

If

0.6\text{ mole}

of carbon dioxide, \text{CO}_2(g), is present at equilibrium, what is the numerical value of the equilibrium constant,

K_c

, at this temperature?

Show answer & explanation

Answer: 2.25

Answer

The numerical value of the equilibrium constant,

K_c

, is 2.25.

At equilibrium,

0.6\text{ mol}

of

\text{CO}_2

is present, which implies

0.6\text{ mol}

of

\text{H}_2

is also formed. The remaining amounts of reactants are

1.0 - 0.6 = 0.4\text{ mol}

for both

\text{CO}

and

\text{H}_2\text{O}

. Dividing by the vessel volume (

2.0\text{ dm}^3

) yields concentrations of

0.30\text{ mol dm}^{-3}

for products and

0.20\text{ mol dm}^{-3}

for reactants. Substituting these values into

K_c = \frac{[\text{CO}_2][\text{H}_2]}{[\text{CO}][\text{H}_2\text{O}]}

gives

K_c = \frac{0.30 \times 0.30}{0.20 \times 0.20} = 2.25

.

Step-by-Step Solution

1

Determine the equilibrium moles for all reactants and products

Equilibrium moles are:

\text{CO} = 0.4\text{ mol}

,

\text{H}_2\text{O} = 0.4\text{ mol}

,

\text{CO}_2 = 0.6\text{ mol}

, and

\text{H}_2 = 0.6\text{ mol}

.

Since 1 mole of

\text{CO}_2

is produced per mole of

\text{CO}

consumed, forming

0.6\text{ mol}

of

\text{CO}_2

consumes

0.6\text{ mol}

of

\text{CO}

and

0.6\text{ mol}

of

\text{H}_2\text{O}

, while producing

0.6\text{ mol}

of

\text{H}_2

.

2

Calculate equilibrium concentrations by dividing moles by volume (

2.0\text{ dm}^3

)

[\text{CO}] = 0.20\text{ mol dm}^{-3}

,

[\text{H}_2\text{O}] = 0.20\text{ mol dm}^{-3}

,

[\text{CO}_2] = 0.30\text{ mol dm}^{-3}

,

[\text{H}_2] = 0.30\text{ mol dm}^{-3}

.

Concentration is given by

C = \frac{n}{V}

where volume

V = 2.0\text{ dm}^3

.

3

Evaluate the equilibrium expression

K_c = \frac{[\text{CO}_2][\text{H}_2]}{[\text{CO}][\text{H}_2\text{O}]}

K_c = \frac{0.30 \times 0.30}{0.20 \times 0.20} = 2.25

.

Plugging in the calculated equilibrium concentrations into the law of mass action expression gives the value of

K_c

.

Key Concept

Calculating Equilibrium Constant (

K_c

) from Equilibrium Amounts

Question 294Question →

The table below presents the frequency distribution of monthly sales revenue (in $₦\text{'000}$ ) recorded by a sample of $50$ retail traders in a urban commercial center:

Sales Revenue ( $₦\text{'000}$ )	Number of Traders ( $f$ )
10 �� 19	8
20 – 29	12
30 – 39	16
40 – 49	10
50 – 59	4

Calculate the estimated median sales revenue (in $₦\text{'000}$ ) for the traders.

Show answer & explanation

Answer: 32.63

Answer

The estimated median sales revenue is 32.63 thousand Naira (

₦32,625

).

To calculate the median of grouped data, determine the total frequency (

N = 50

) and the median position (

\frac{N}{2} = 25

). The cumulative frequency distribution shows that the

25\text{th}

observation falls within the

30\text{ – }39

interval. Using the lower class boundary

L = 29.5

, cumulative frequency of prior classes

F = 20

, median class frequency

f = 16

, and class length

c = 10

, the formula

\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f}\right) \times c

yields

29.5 + \left(\frac{25 - 20}{16}\right) \times 10 = 32.625

, which equals

32.63

when rounded to two decimal places.

Step-by-Step Solution

1

Calculate the total frequency and median position

Total frequency

N = 50

, median rank =

\frac{N}{2} = 25

The median of a frequency distribution is located at the middle position

\frac{N}{2}

.

2

Determine cumulative frequencies and identify the median class

Cumulative frequencies are

8

,

20

,

36

,

46

, and

50

. The median class is 30 – 39

Since

20

items fall below

30

, the

25\text{th}

item lies within the

30\text{ – }39

class interval.

3

Extract interpolation parameters for the median class

Lower class boundary

L = 29.5

, preceding cumulative frequency

F = 20

, class frequency

f = 16

, class interval width

c = 10

The continuous class boundary between

29

and

30

is

29.5

, and interval width is

19.5 - 9.5 = 10

.

4

Compute the linear interpolation for grouped median

\text{Median} = 29.5 + \left(\frac{25 - 20}{16}\right) \times 10 = 29.5 + 3.125 = 32.625 \approx 32.63

Applying the standard grouped data median formula yields the precise interpolated value.

Key Concept

Grouped Data Median Calculation

Question 295Question →

In a given fiscal year, an economy records a National Income of ₦850 million. The accounting records reveal corporate profit taxes of ₦60 million, undistributed corporate profits of ₦40 million, social security contributions of ₦30 million, government transfer payments to households of ₦50 million, and personal direct taxes of ₦70 million. What is the Disposable Personal Income of this economy in millions of Naira?

Show answer & explanation

Answer: 700

Answer

700 million Naira

Starting from National Income of ₦850 million, we subtract corporate profit taxes (₦60 million), undistributed corporate profits (₦40 million), and social security contributions (₦30 million), while adding transfer payments (₦50 million) to arrive at Personal Income of ₦770 million. Subtracting personal direct taxes of ₦70 million yields a Disposable Personal Income of ₦700 million.

Step-by-Step Solution

1

Calculate Personal Income from National Income by deducting earnings not received by households (corporate taxes, retained earnings, social security payments) and adding income received but not earned in current production (transfer payments).

Personal Income = ₦850 million - ₦60 million - ₦40 million - ₦30 million + ₦50 million = ₦770 million

Personal Income measures the total earnings received by households from all sources prior to personal direct taxation.

2

Deduct personal direct taxes from Personal Income to find Disposable Personal Income.

Disposable Personal Income = ₦770 million - ₦70 million = ₦700 million

Disposable Personal Income represents the net amount remaining for spending and personal savings after paying direct taxes.

Key Concept

Derivation of Personal Income and Disposable Personal Income from National Income

Question 296Question →

A sample survey of 20 small-scale enterprises in an industrial cluster recorded their daily profit (in thousands of Naira, $\text{₦'000}$ ) with the following frequency distribution:

Daily Profit ( $x$ in $\text{₦'000}$ )	Number of Enterprises ( $f$ )
10	3
15	5
20	7
25	3
30	2

What is the mean daily profit of these enterprises in thousands of Naira?

Show answer & explanation

Answer: 19

Answer

The mean daily profit of the enterprises is 19 thousand Naira.

The arithmetic mean for a frequency distribution is calculated using

\bar{x} = \frac{\sum fx}{\sum f}

. Multiplying each daily profit by its frequency yields a total sum of

380

. Dividing by the total frequency of

20

enterprises gives a mean daily profit of

19

thousand Naira.

Step-by-Step Solution

1

Multiply each profit level by its corresponding frequency to get the total profit contribution per group

f \cdot x

values are 30, 75, 140, 75, and 60

Each profit value must be weighted by how many enterprises earned that amount

2

Sum all weighted profit values to find total combined profit

\sum fx = 380

The sum of

f \cdot x

gives the grand total daily profit for all surveyed enterprises

3

Sum all frequencies to obtain total count of enterprises

\sum f = 20

The mean requires dividing total profit by total sample size

4

Divide total weighted profit by total number of enterprises

\bar{x} = \frac{380}{20} = 19

Formula for discrete grouped mean is

\bar{x} = \frac{\sum fx}{\sum f}

Key Concept

Calculation of Mean from a Discrete Frequency Distribution

Question 297Question →

A firm operating in the short run incurs a total cost of $\text{₦}400$ when output is zero units. At an output level of $8$ units, its average total cost ( $ATC$ ) is $\text{₦}100$ . If the marginal cost ( $MC$ ) of producing the $9^{\text{th}}$ unit is $\text{₦}110$ and the marginal cost of producing the $10^{\text{th}}$ unit is $\text{₦}130$ , what is the average variable cost ( $AVC$ ) in Naira at an output level of $10$ units?

Show answer & explanation

Answer: 64

Answer

The average variable cost (AVC) of producing 10 units is 64 Naira.

By applying short-run cost identities, TFC is determined as 400 Naira from TC at Q = 0. TC at Q = 8 is 8 × 100 = 800 Naira. Adding the MC for the 9th (110) and 10th (130) units gives TC at Q = 10 as 1040 Naira. Subtracting TFC (400) gives TVC = 640 Naira. Dividing TVC by Q = 10 gives AVC = 64 Naira.

Step-by-Step Solution

1

Determine Total Fixed Cost (TFC)

TFC = 400 Naira

At an output level of zero (Q = 0), variable costs are zero, so Total Cost equals Total Fixed Cost.

2

Calculate Total Cost at 8 units (TC(8))

TC(8) = 800 Naira

Total Cost is equal to Quantity multiplied by Average Total Cost: TC = Q × ATC = 8 × 100 = 800.

3

Calculate Total Cost at 10 units (TC(10))

TC(10) = 1040 Naira

Add the Marginal Cost of the 9th unit (110) and 10th unit (130) to TC(8): TC(10) = 800 + 110 + 130 = 1040.

4

Calculate Total Variable Cost at 10 units (TVC(10))

TVC(10) = 640 Naira

Total Variable Cost is derived by subtracting Total Fixed Cost from Total Cost: TVC = TC - TFC = 1040 - 400 = 640.

5

Compute Average Variable Cost at 10 units (AVC(10))

AVC(10) = 64 Naira

Average Variable Cost is Total Variable Cost divided by quantity: AVC = TVC / Q = 640 / 10 = 64.

Key Concept

Short-Run Cost Relationships and Calculations

Question 298Question →

Given the demand function for a commodity as $Q_d = 100 - 5P$ and the supply function as $Q_s = 20 + 3P$ , where $P$ represents the price in Naira ( $₦$ ), calculate the equilibrium quantity.

Show answer & explanation

Answer: 50

Answer

The equilibrium quantity is 50 units.

At market equilibrium, quantity demanded equals quantity supplied (

Q_d = Q_s

). Setting

100 - 5P = 20 + 3P

gives

80 = 8P

, which solves to

P = 10

. Substituting

P = 10

into

Q_d = 100 - 5(10)

gives the equilibrium quantity of 50 units.

Step-by-Step Solution

1

Equate quantity demanded to quantity supplied to find the equilibrium condition

100 - 5P = 20 + 3P

Market equilibrium occurs where quantity demanded equals quantity supplied.

2

Solve for the equilibrium price (P)

P = 10 Naira

Group like terms: 100 - 20 = 5P + 3P => 80 = 8P, yielding P = 10.

3

Substitute the equilibrium price into the demand function to find equilibrium quantity

Q = 50 units

Qd = 100 - 5(10) = 50 units.

Key Concept

Market Equilibrium Price and Quantity

Question 299Question →

In a local agricultural market, there are $25$ identical cassava producers operating under competitive conditions. The individual supply function for each producer is expressed as $Q_i = -15 + 4P$ , where $Q_i$ represents the quantity supplied by a single producer in bags and $P$ is the market price per bag in Naira ( $\text{₦}$ ). If the total aggregate market quantity supplied is $1,325$ bags, what is the prevailing market price per bag in Naira?

Show answer & explanation

Answer: 17

Answer

The prevailing market price per bag is ₦17.

To calculate the prevailing market price, aggregate individual producer supply into the market supply equation by multiplying by the total number of producers:

Q_M = 25 \times (-15 + 4P) = -375 + 100P

. Substituting the market output of

1,325

bags gives

1325 = -375 + 100P

. Rearranging terms results in

100P = 1700

, which gives

P = 17

Naira.

Step-by-Step Solution

1

Aggregate individual producer supply functions to find the total market supply equation.

Q_M = 25 \times (-15 + 4P) = -375 + 100P

Market supply is the horizontal summation of all individual producers' supply functions in a competitive market.

2

Substitute the total market quantity supplied into the market supply equation.

1325 = -375 + 100P

The aggregate market quantity supplied is given as 1,325 bags.

3

Isolate the price variable P to compute the market price.

P = 17

Adding 375 to both sides yields

100P = 1700

, and dividing by 100 gives

P = 17

Naira.

Key Concept

Aggregation of Individual Supply Functions to Derive Market Supply

Estimated Time:2m 0s

Question 300Question →

Country Y recorded visible exports of $\$600\text{ million}$ and visible imports of $\$850\text{ million}$ during a financial year. Its invisible exports (services rendered abroad) totaled $\$250\text{ million}$ , while its invisible imports (services received) stood at $\$100\text{ million}$ . What is Country Y's current account balance in millions of dollars? (Note: Express a deficit as a negative value, e.g., -100).

Show answer & explanation

Answer: -100

Answer

The current account balance is

-\$100\text{ million}

(a current account deficit of

\$100\text{ million}

).

The Current Account balance combines net visible trade (goods) and net invisible trade (services). Net visible trade is

\$600\text{ million} - \$850\text{ million} = -\$250\text{ million}

(a trade deficit). Net invisible trade is

\$250\text{ million} - \$100\text{ million} = +\$150\text{ million}

(a services surplus). Adding these yields

-\$250\text{ million} + \$150\text{ million} = -\$100\text{ million}

, representing a overall Current Account deficit of

\$100\text{ million}

.

Step-by-Step Solution

1

Calculate the visible trade balance (Balance of Trade).

Visible balance =

\$600\text{ million} - \$850\text{ million} = -\$250\text{ million}

.

Visible trade balance equals earnings from visible exports minus payments for visible imports.

2

Calculate the invisible trade balance.

Invisible balance =

\$250\text{ million} - \$100\text{ million} = +\$150\text{ million}

.

Invisible balance equals receipts from services rendered abroad minus payments for services received.

3

Combine visible balance and invisible balance to get the Current Account balance.

Current Account Balance =

-\$250\text{ million} + \$150\text{ million} = -\$100\text{ million}

.

The Current Account comprises both the trade balance in visible goods and the balance on invisible services and transfers.

Key Concept

Current Account Balance Calculation in Balance of Payments

Estimated Time:1m 0s

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept