Share this:
Linear programming Questions
1. A tailoring business makes two types of garments A and B. Garment A requires 3 metres of material while garment B requires 2 ½ metres of material. The business uses not more than 600 metres of material daily in making both garments. It must make not more than 100 garments of type A and not less than 80 of type B each day.
(a) Write down four inequalities from this information. (3mks)
(b) Graph these inequalities. (3mks)
(c) If the business makes a profit of shs 80 on garment A and a profit of shs 60 on garment B, how many garments of each type must it make in order to maximize the total profit? (4mks)
2. A man bakes two types of cakes, queen cakes and marble cakes. Each week he bakes x
queen cakes and y marble cakes. The number of cakes baked are subject to the following
conditions; 30x + 20y ≤ 4800, 30x + 40y ≥ 3600 and 10x >30y
He makes a profit of shs.10 on each queen cake and shs.12 on each marble cake.
(i) Draw a graph to represent the above information on the grid provided
(ii) From the graph, determine how many cakes of each type he should make to
maximize his weekly profit
(iii) Calculate the maximum profit
(iv) If he is to make a weekly profit of at least shs.600, find the least number of marble
cakes he should bake
3. A company produces shirts and jerseys using two types of machines. Every shirt made
requires 2 hours on machine A and 2 hours on machine B. Every Jersey made requires
3hours on machine A and I hour on machine B. In one day the time limit on machine A
is 24hours but that on machine B is 12hrs. The number of Jerseys produced must not be
more than the shirts produced in one day. The company makes a profit of shs.200 on each
shirt and shs.200 on each Jersey. The company produces x shirts and y jerseys per day
(a) Write down four inequalities which must be satisfied by x and y and represent these
inequalities on a grid
(b) Find the values of x and y which will give the company maximum daily profit and
also state the maximum profit
4. A trader makes two types of chair, ordinary and special chairs. The cost of each ordinary
chair is shs.300 while each special chair costs shs.700. He is prepared to spend not more than shs.21,000. It is not viable for hi m to make less than 20 chairs. Ordinary chairs must be less
than twice the special chairs but more than 15. By taking the number of ordinary chairs as x
and special chairs as y:
(a) Write down all the inequalities in x and y
(b) Draw the inequalities on the grid provided
(c) He sells a special chair at a profit of shs.140 while ordinary chairs at a profit of shs.120;
Determine the maximum possible profit
5. A school has to take 384 people for a tour. There are two types of buses available.
Type X and type Y. Type X can carry 64 passengers and type Y can carry 48 passengers.
They have to use at least 7 buses.
a) Form all linear inequalities which will represent the above information .
b) On the grid provided, draw the inequalities and shade the unwanted region.
b) The charges for hiring the buses are
Type X: shs.25,000
Type Y: shs.20,000
Use your graph to determine the number of buses of each type that should be hired to
minimize the cost.
6. A shoe maker makes two types of shoes A and B. He takes 3 hours to make one pair of type
A and 4 hours to make one pair of type B. He works for a maximum of 120 hours to make x
pairs of type A and y pairs of type B. It costs him Kshs. 400 to make a pair of type A and
Kshs.150 to make a pair of type B. His total cost does not exceed kshs.9000. He must make
at least 8 pairs of type A and 12 pairs of type B.
(a) Write down four inequalities representing the information above
(b) On the grid provided represent the inequalities and shade the unwanted regions
(c) The shoe maker makes a profit of kshs.40 on each pair of type A and kshs.70 on each pair
7. A theatre has a seating capacity of 250 people. The charges are shs.100 for an ordinary
seat and shs.160 for a special seat. It costs shs.16,000 to stage a show and the threatre
must make a profit. There is never more than 200 ordinary seats and for a show to take
place at least 50 ordinary chairs must be occupied. The number of special seats is always
less than twice the number of ordinary seats.
a) taking x to be the number of ordinary seats and y the number of special seats ,
write down all the inequalities representing the information above.
b) On the grid provided, draw the graph to show the inequalities in (a) above
c) Determine the number of seats of each type that should be booked in order to maximize
the profit.
8. A man sells two types of ice creams in cups and sticks. He can store less than ten packets
in his cooling box. He sells more cups than sticks but less than 3 items as many cups as sticks.
He also knows that he will sell more than 3 packets of sticks. His profit is shs.3.00 on a packet
of cups and shs.2.00 on a packet of sticks.
(a) Form inequalities to represent the above information:
(Let x – packets of cups and
y – packets of sticks)
(b) On the grid provided graph the inequalities to satisfy the required condition
(c) How many packets of cups and sticks should the man put in his box to give him the highest
profit?
9. A shopkeeper bought 50 pangas and 30 jembes :
(a) From a wholesalers for shs.4,260. He had bought half as many jembes and 5 pangas less,
he would have paid shs.1290 less. Had the shopkeeper bought form wholesaler B, he would
have paid 10% more a panga and 15% less for a jembe. How much would he have Ho saved if he
had bought the 50pangas and 30 jembes from wholesalers B
(b) The price of a suit if marked at shs.5000. A discount
10. The games master whishes to hire two matatus for a trip. The operators have a Toyota
which carries 10passengers and a Kombi which carries 20 passengers. Altogether 120
people have to travel. The operators have only 20litres of fuel and the Toyota consumes
4 litres on each round trip and the Kombi 1 litre on each round trip. If the Toyota makes x
round trips and the kombi y round trips;
(a) write down four inequalities in x and y which must be satisfied
b) represent the inequalities graphically on the grid provided
(c) The operators charge shs.100 for each round trip in the Toyota and shs.300 for each
round trip in the kombi;
(i) determine the number of trips made by each vehicle so as to make the total cost a
Minimum
(ii) find the minimum cost
11. The velocity of a particle Vm/s moving in a straight line after t seconds is given by
V = 3t^{2} – 3t – 6. Find the distance covered by the particle between t = 1 and t = 4seconds
Linear programming Answers
1.  (a) let the No. of garments of type A be x and those of type B be y (i) 3x + 2 ½ y 600 (material) (ii) x 1000 y 80 x 0 (b) Lines drawn 3x + 2 ½ y = 600 x = 100 x = 80 x = 0 (c) The object function is P = 80x + 60y where P = total profit Either drawn a search line by choosing an appropriate value of P e.g 12000 = 80x + 60y or inspect for maximum profit using points further from origin maximum profit 100 garment og type A 120 garments of type B 
B1 B2 B1
B1
B1
B1
B1
B1
B1 B1 
all any two
lines and shading
lines and shading
forregion indicated
objective function
Use of search line or inspection

1. 30x + 20y 4800……..(i)
30x + 40y 3600……..(ii)
10x 30 y…………(iii)
x 0 y 0
objective function 10 x+ 12 y = K
3x + 2y = 480 3x + 4y = 360 x = 3y
X  40  60  80  X  20  40  60  X  30  45  60  
y  180  150  120  Y  75  60  45  Y  10  15  20 
(ii) consider (60,40)
10 (60) + 12(40) = 600 +480
=1080
10x + 12y = 108 0
5x + 6y = 540 – search line
X  20  40  60 
y  73  57  40 
Maximum profit at ( , 240)
No queen cake , 240 marble cakes
(iii) 240 X 12 = sh. 2880
(iv) 10x + 12y 600 10x + 12y = 600
5x + 6y = 300
X   12  60 
y  50  40  0 
2. Machine A Machine B
Shirts Jerseys Shirts Jerseys
No. x y x y
Hrs. @2hrs @3hrs @2hrs @1hr
(i) 2x + 3y 24
(ii) 2x + y 12
(iii) y > x
(iv) x > 0
y > 0
Max pt(3,6)
Max profit = 22 x 3 + 200 x 6
= 600 + 1200
= Shs. 1800
3. (a) 3x + 7y 210
x + y 20
x < 2y
x > 15
(b) refer
(c) 120x + 140y = 120 x 130 + 140 x 10
Profit = shs.5960
x = 31
y = 16
4. Passengers
64x + 48y ≥ 384 i.e. 8x + 6y ≥ 48
x > 0
y > 0
x + y ≥ 7
Cost equation
Total cost = 2500x + 20000y
(3,4)
3 type x
4 type y
5. 3x + 4y ≤120
400x + 150y ≥ 9000
x ≥ 8 y >12
(b)(i) 3x + 4y 20
(ii) 40x + 15y 900
(iii) x 8
(iv) y 12
Points  Objective function 40x + 70y  Profit 
(i) (8,24)  320 + 1680  2000 
(ii) (24, 12)  960 + 840  1800 
(iii) (8, 12)  320 + 840  1160 
(c) 12 type x
12 type y
Profit = 40(24) + 70(12)
= 1800
6. 100 x = 160y = 16000 5 x 200 + 8 x 50
=100 x 200 + 160 x 50 1000 + 4000
20000 + 8000 10 x 200 + 16 x 50=
28000/= 10x + 16y = 1600
5x + 8y = 800
5 x 20 + 100
8y = 800 – 100
y = 700
8
^{800}/_{5} = 160
a) y<2x, 50≤x≤200 x >100
y >0, x+y ≤250, 100x + 160y ≥ 16000
b) See graph
profit?
7. x + y 10
y 3x
y > 3
 Obejctive function 3x = 2y = I or use of serach line
5 packets of cups and 4packets of stucks
x  y  Profit 
2 2 3 3 3 4 4 5  4 5 4 5 6 4 5 4  14 16 17 19 21 20 22 23 
8. Panga – P, Jembe J
(a) 50P + 30J = 4260
50P + 15J = 1290
50P + 30J = 4260
10P + 30J = 1290
40P = 1680
P = 168 = 42
4
50(42) + 30J = 4260
 + 30J = 4260
30J = 2160
J = (2160)
30
J = 72
Wholesaler
110 x 42 = shs.46.50 = pangas
100
85 x 72 = shs 60 = jembes
100
For B
50 x 46.50 + 30 x 61.2
2310 + 1836 = 4146
Saving = 4260
4116
144
(b) Discount 5000 – 3500 = 1500
% discount = 1500 x 100
5000
= 30%
9. a) X ≥ 0, y = ≥ 0
10x + 20y ≥ 120
4x + y ≥ 20
b) On the graph.
c) i) (4,4)
4 x 100 + 4 x 300
400 + 1200 = 1600
10. Distance Covered = (3t^{2} – 3t 6)dt
= t^{3} – ^{3}/_{2} t^{2} – 6t 4 ^{4 }
^{1}
4^{3} – 3 (4)^{2} – 6(4) – 1^{3 }– 3(1)^{2} – 6(1)
2
16 – ^{13}/_{2}=_{ }