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Questions and Answers of Mathematics Economics Business

Write down the percentage changes corresponding to the following scale factors:(a) 1.04 (b) 1.42 (c) 0.86(d) 3.45 (e) 1.0025 (f) 0.04
Write down the scale factors corresponding to(a) an increase of 19%;(b) an increase of 250%;(c) a decrease of 2%;(d) a decrease of 43%.
A firm has 132 female and 88 male employees.(a) What percentage of staff are female?(b) During the next year 8 additional female staff are employed. If the percentage of female staff is now 56%, how
Calculate each of the following:(a) 5% of 24 (b) 8% of 88 (c) 48% of 4563 (d) 112% of 56
Express the following percentages as fractions in their simplest form:(a) 35% (b) 88% (c) 250% (d) 171/2% (e) 0.2%
Show that the following production function is homogeneous and state whether it displays decreasing, increasing or constant returns to scale.f (K,L) = (K2 + L2)eK/L
Future sales of two products A and B are given by SA = 5e0.01t and SB = 2e0.02t. Find the time, t, when sales of the two products are the same.
Solve the following equations:(a) ln x = 5 (b) ln x = 0
The value of a second-hand car reduces exponentially with age, so that its value $y after t years can be modelled by the formula y = Ae−axIf the car was worth $50 000 when new and $38 000 after two
Solve each of the following equations. (Round your answer to two decimal places.)(a) ex = 5.9 (b) ex = 0.45 (c) ex = −2(d) e3x = 13.68 (e) e−5x = 0.34 (f) 4e2x = 7.98
Use the rules of logs to express each of the following as a single logarithm:(a) ln x + 2 ln x (b) 4 ln x − 3 ln y + 5 ln z
Use the rules of logs to expand each of the following:(a) ln xy (b) ln xy4 (c) ln (xy)2(d) In x5/y7(e) In√x/y(f) In √xy3/z
The number of items, N, produced each day by an assembly-line worker, t days after an initial training period, is modelled byN = 100 − 100e−0.4t(1) Calculate the number of items produced daily(a)
If two firms A and B use the same labour input, L, their output in the short term is given by QA = 108√L and QB = 4L2, respectively. Find the non-zero value of L which produces the same level of
The number of complaints, N, received by a small company each month can be modelled byN = 80log10(7 + 10t)where t denotes the number of months since the company’s launch.(a) Estimate the number of
(1) State the values of x that satisfy the following equations:(a) 81 = 3x (b) 1/25 = 5x (c) 161/2 = 2x(2) Use the rules of indices to simplify: хву? |(a) х x'уб .99,4 x°y+ (b)
(1) State the values of(a) log232 (b) log9 (1/3)(2) Use the rules of logs to express 2logbx = 4logby as a single logarithm.(3) Use logs to solve the equation 10(1.05)x = 300 Give your answer
Solve the following equations for x. Give your answers to two decimal places.(a) 5x = 8 (b) 10x = 50 (c) 1.2x = 3 (d) 1000 × 1.05x = 1500
Express the following in terms of logbx and logby:(a) logbx2y(b) logb(x/y2)(c) logbx2y7
Use the rules of logs to express each of the following as a single log:(a) logbx + logbz(b) 3logbx − 2logby(c) logby − 3logbz
Write down the value of(a) logbb2 (b) logbb (c) logb1 (d) logb √b (e) logb(1/b)
Write down the values of x which satisfy each of the following equations:(a) 5x = 25 (b) 3x = 1/3(c) 2x = 1/8(d) 2x = 64 (e) 100x = 10 (f) 8x = 1
Which of the following production functions are homogeneous? For those functions which are homogeneous, write down their degrees of homogeneity and comment on their returns to scale.(a) Q =
For the production function, Q = 200K1/4L2/3, find the output when(a) K = 16, L = 27 (b) K = 10 000, L = 1000
Write the following expressions using index notation: (a) VE (b)= (0) JE (f) xVF (e) х
Use the rules of indices to simplify
(1) Without using your calculator, evaluate(a) 82 (b) 21 (c) 3−1(d) 170 (e) 11/5 (f) 361/2 (g) 82/3 (h) 49−3/2
The demand function for a firm’s product is given by P = 60 − Q. Fixed costs are 100, and the variable costs per good are Q + 6.(a) Write down an expression for total revenue, TR, in terms of Q
Sketch, on the same diagram, graphs of the total revenue and total cost functions,TR = −2Q2 + 14QTC = 2Q + 10(1) Use your graphs to estimate the values of Q for which the firm(a) Breaks even;(b)
A taxi firm charges a fixed cost of $10 together with a variable cost of $3 per mile.(a) Work out the average cost per mile for a journey of 4 miles.(b) Work out the minimum distance travelled if the
Find an expression for the profit function given the demand function2Q + P = 25and the average cost functionAC = 32/Q + 5Find the values of Q for which the firm(a) breaks even;(b) makes a loss of 432
The total cost of producing 500 items a day in a factory is $40 000, which includes a fixed cost of $2000.(a) Work out the variable cost per item.(b) Work out the total cost of producing 600 items a
The total cost, TC, of producing 100 units of a good is 600 and the total cost of producing 150 units is 850. Assuming that the total cost function is linear, find an expression for TC in terms of Q,
Given that fixed costs are 1 and that variable costs are Q + 1 per unit, express TC and AC as functions of Q. Hence sketch their graphs.
Given that fixed costs are 500 and that variable costs are 10 per unit, express TC and AC as functions of Q. Hence sketch their graphs.
Given the following total revenue functions, find the corresponding demand functions:(a) TR = 50Q − 4Q2 (b) TR = 10
Given the following demand functions, express TR as a function of Q and hence sketch the graphs of TR against Q:(a) P = 4 (b) P = 7/Q (c) P = 10 − 4Q
(a) If the demand function of a good is given byP = 80 − 3Qfind the price when Q = 10, and deduce the total revenue.(b) If fixed costs are 100 and variable costs are 5 per unit, find the total cost
A clothing supplier sells T-shirts to retailers for $7 each. If a store agrees to buy more than 30, the supplier is willing to reduce the unit price by 3 cents for each shirt bought above 30, with a
Given the supply and demand functionsdetermine the equilibrium price and quantity. P = Q + 20s + 7 P = -Q, + 25
Given the quadratic supply and demand functionsdetermine the equilibrium price and quantity. P = Q + 2Qs + 12 P = -Q% – 4Qp + 68 |
Use a sign diagram to solve the following inequalities: x + 4 (b) (x – 1)(x + 1) = 0 (a) x(x – 3) > 0 (c) c)
The production levels of coffee in Mexico, Q (in suitable units) depends on the average summer temperature, T (in °C). A statistical model of recent data shows that Q = −0.046T2 + 2.3T + 27.6.(a)
Use the results of Question 5 to solve each of the following inequalities:(a) x2 − 16 ≥ 0(b) x (100 − x) > 0 (c) −x2 + 22x − 85 ≥ 0(d) x2 − 18x + 81 ≤ 0(e) 2x2 + 4x + 3 > 0
Sketch the graphs of the quadratic functions given in Question 4.Data from Question 4 |(a) f(x) = x² – 16 (d) f(x) = x² – 18x + 81 (c) f(x) = –x² + 22x – 85 (b) f(x) = x(100 – x) (e)
Solve the equation f (x) = 0 for each of the following quadratic functions:(a) f (x) = x2 − 16(b) f (x) = x(100 − x) (c) f (x) = −x2 + 22x − 85(d) f (x) = x2 − 18x + 81 (e) f (x)
Use ‘the formula’ to solve the following quadratic equations. (Round your answers totwo decimal places.)(a) x2 − 5x + 2 = 0(b) 2x2 + 5x + 1 = 0(c) −3x2 + 7x + 2 = 0(d) x2 − 3x − 1 = 0(e)
Write down the solutions of the following equations:(a) (x − 1)(x + 3) = 0 (b) (2x − 1)(x + 10) = 0 (c) x(x + 5) = 0(d) (3x + 5)(4x − 9) = 0(e) (5 − 4x)(x − 5) = 0
Solve the following quadratic equations:(a) x2 = 81 (b) x2 = 36 (c) 2x2 = 8(d) (x − 1)2 = 9 (e) (x + 5)2 = 16
Given thatG = 50I = 40C = 0.75Yd + 45T = 0.2Y + 80calculate the equilibrium level of national income.
A consumption function is given by C = aY + b.It is known that when Y = 10, the value of C is 28, and that when Y = 30, the value of C is 44.By solving a pair of simultaneous equations, find the
For a closed economy with no government intervention, the consumption function is C = 0.6Y + 30 and planned investment is I = 100 Calculate the equilibrium level of(a) national income;(b)
Write down expressions for the savings function given that the consumption function is(a) C = 0.9Y + 72(b) C = 0.8Y + 100
If the consumption function is given by C = 0.7 Y + 40 state the values of (a) autonomous consumption;(b) marginal propensity to consume.Transpose this formula to express Y in terms of C and
If the national income, Y, is 1000 units, then consumption, C, is 800 units. Also whenever income rises by 100, consumption increases by 70. Assuming that the consumption function is linear:(a) state
If the consumption function is given by C = 4200 + 0.75Y, state the marginal propensity to consume and deduce the marginal propensity to save.
Make x the subject of the formulay = 3/x - 2
Make x the subject of each of the following formulae: х (b) y = (x + 4)/3 (а) у — 9х - 6 (c) y = 1 (e) y * 4 (Г) у — х (d) y + 8 х+2 Зх — 7 ||
Transpose the formulae:(a) Q = aP + b to express P in terms of Q(b) Y = aY + b + I to express Y in terms of I(c) Q = 1/aP + b to express P in terms of Q
Draw flow charts for each of the following formulae: |(a) y = 5x + 3 |(e) y =+7 (b) y = 5(x + 3) (c) y = ár – 9 (9) y = (d) y = 4x² – 6
Write down the formula representing each of the following flow charts (a) double add 5 х (b) add 5 double х (c) multiply by 5 reciprocate square х (d) multiply by 2 add 4 subtract 3 square
Make Q the subject ofP = 2Q + 8Hence find the value of Q when P = 52.
The demand and supply functions for two interdependent commodities are given bywhere QDi, QSi and Pi denote the quantity demanded, quantity supplied and price of good i, respectively. Determine the
The demand and supply functions of a good are given byP = −3QD + 48P = 1/2QS + 23Find the equilibrium quantity if the government imposes a fixed tax of $4 on each good.
(a) Copy and complete the following table of values for the supply functionP = 1/2Q + 20Hence, or otherwise, draw an accurate sketch of this function using axes with values of Q and P between 0 and
The demand for a good priced at $50 is 420 units, and when the price is $80, demand is 240 units. Assuming that the demand function takes the form Q = aP + b, find the values of a and b.
The demand, Q, for a certain good depends on its own price, P, and the price of an alternative good, PA, according to Q = 30 − 3P + PA(a) Find Q if P = 4 and PA = 5.(b) Is the alternative good
The demand function of a good isQ = 100 − P + 2Y + 1/2Awhere Q, P, Y and A denote quantity demanded, price, income and advertising expenditure, respectively.(a) Calculate the demand when P = 10, Y
Sketch a graph of the supply function P = 1/3Q + 7 Hence, or otherwise, determine the value of(a) P when Q = 12(b) Q when P = 10(c) Q when P = 4
If f (x) = 3x + 15 and g(x) = 1/3x − 5, evaluate(a) f(2) (b) f(10) (c) f(0)(d) g(21) (e) g(45)(f) g(15)What word describes the relationship between f and g?
Use the elimination method to attempt to solve the following systems of equations.Comment on the nature of the solution in each case.(a) −3x + 5y = 49x − 15y = −12 (b) 6x − 2y = 315x −
Sketch the following lines on the same diagram:2x − 3y = 6, 4x − 6y = 18, x -3/2y = 3Hence comment on the nature of the solutions of the following systems of equations: (а) 2r - Зу— 6 3 Зу
The total annual sales of a book in either paper or electronic form are 3500. Each paper copy of the book costs $30 and each e-book costs $25. The total cost is $97 500.(a) If x and y denote the
Use the elimination method to solve the following pairs of simultaneous linear equations: (a) – 2r + y = 2 2r + y = -6 (b) 3x + 4y = 12 x + 4y = 8 (c) 2x + y = 4 4x 3y = 3 (d) x + y = 1 6x + 5y = 15
Monthly sales revenue, S (in $), and monthly advertising expenditure, A (in $), are modelled by the linear relation, S = 9000 + 12A.(a) If the firm does not spend any money on advertising, what is
The number of people, N, employed in a chain of cafes is related to the number of cafes, n, by the equation:N = 10n + 120(a) Illustrate this relation by plotting a graph of N against n for 0 ≤ n
A taxi firm charges a fixed cost of $4 plus a charge of $2.50 a mile.(a) Write down a formula for the cost, C, of a journey of x miles.(b) Plot a graph of C against x for 0 ≤ x ≤ 20.(c) Hence, or
Use the slope–intercept approach to produce a rough sketch of the following lines:(a) y = −x (b) x − 2y = 6
State the value of the slope and y intercept for each of the following lines:(a) y = 5x + 9 (b) y = 3x − 1 (c) y = 13 − x(d) −x + y = 4 (e) 4x + 2y = 5 (f) 5x − y = 6
Solve the following pairs of simultaneous linear equations graphically:(a) −2x + y = 22x + y = −6 (b) 3x + 4y = 12 x + 4y = 8 (c) 2x + y = 4 4x − 3y = 3 (d) x + y = 16x
If 4x + 3y = 24, complete the following table and hence sketch this line. х y 3
For the line 3x − 5y = 8, (a) find the value of x when y = 2;(b) find the value of y when x = 1.Hence write down the coordinates of two points which lie on this line.
By substituting values into the equation, decide which of the following points lie on the line, x + 4y = 12:A(12, 0), B(2, 2), C(4, 2), D(−8, 5), E(0, 3)
An airline charges $300 for a flight of 2000 km and $700 for a flight of 4000 km.(a) Plot these points on graph paper with distance on the horizontal axis and cost on the vertical axis.(b) Assuming a
On graph paper draw axes with values of x and y between −3 and 10, and plot the following points:P(4, 0), Q(−2, 9), R(5, 8), S(−1, −2) Hence find the coordinates of the point of intersection
(a) Solve the equation 6(2 + x) = 5(1 − 4x)(b) Solve the inequality3x + 6 ≥ 5x − 14
Simplify the following algebraic expression:4/x2y / 2x/y
Simplify the following inequalities:(a) 2x > x + 1 (b) 7x + 3 ≤ 9 + 5x (c) x − 5 > 4x + 4 (d) x − 1 < 2x − 3
Which of the following inequalities are true?(a) −2 < 1 (b) −6 > −4 (c) 3 < 3(d) 3 ≤ 3 (e) −21 ≥ −22 (f) 4 < 25
Solve each of the following equations. If necessary give your answer as a mixed fraction reduced to its lowest terms. (a) x + 2 = 7 (b) 3x = 18 (d) x – 4 = -2 (9;-2 (9)-7= 3 (g) (h) 3(x – 1) = 2
Work out each of the following, simplifying your answer as far as possible: (e) (c) (b) (d) (a) 4gh 2g (f) (G) (g) 9h
It takes 11/4 hours to complete an annual service of a car. If a garage has 471/2 hours available, how many cars can it service?
(1) Without using a calculator, work out the following, giving your answer in its lowest terms:(2) Use your calculator to check your answers to part (1). (d) (e) 6. 6. (1) 3 ÷ (g) 3. 2/3
Which one of the following algebraic fractions can be simplified? Explain why the other two fractions cannot be simplified.x - 1/2x - 2, x - 2/x + 2, 5t/10t - s
By factorising the numerators and/or denominators of each of the following fractions, reduce each to its lowest terms:(a) 2p/4q + 6r(b) x/x2 - 4x(c) 3ab/6a2 + 3a(d) 14d/21d - 7de(e) x + 2/x2 - 4
Reduce each of the following algebraic fractions to their lowest terms:(a) 6x/9(b) x/2x2(c) b/abc(d) 4x/6x2y(e) 15a2b/20ab2
In 2011 in the United States, 35 out of every 100 adults owned a smartphone. By 2013 this figure increased to 56 out of every 100.(a) Express both of these figures as fractions reduced to their
Reduce each of the following numerical fractions to their lowest terms:(a) 13/26(b) 9/12(c) 18/30(d) 24/72(e) 36/27
Write down a formula for each situation:(a) A plumber has a fixed call-out charge of $80 and has an hourly rate of $60. Work out the total charge, C, for a job that takes L hours in which the cost of
A law firm seeks to recruit top-quality experienced lawyers. The total package offered is the sum of three separate components: a basic salary which is 1.2 times the candidate’s current salary

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