MATHEMATICS (EXTENDED) 0580
IGCSE MAY/JUNE 2021
LINEAR
PROGRAMMING
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NOTES: CHAPTER 5 LINEAR PROGRAMMING
Linear Programming is a branch of Mathematics that deals with systems of linear inequalities (called
constraints) used to findi the maximum or minimum values of the object function.
Applications of the Linear Programming are evident in the field of:
3 common stages involved in solving Linear Programming problems:
Ø Interpret the information given as a system of inequalities and display them graphically
at least implies >
less than implies <
at most implies <
more than implies >
Ø Investigate some characteristics of the points in the unshaded solution region (region R)
o Utilize “corner points” of R
Ø Find the maximum/minimum value according to the object function needed.
EXAMPLE:
Miguel has $100 to spend on blue and red pens.
The cost of each blue pen is $6 while a red pen costs $8.
He must buy at least 6 pens of each color, at least 13 pens in total.
(a) Represent the information above in inequalities.
(b) Show the graph of how he can possibly buy the blue and red pens.
(c) If he can sell each blue pen at $8 and each red pen $12, find number of blue and red pens that
he must buy to have the most profit.
Working:
Let x be the number of blue pens and y be the number of red pens that Miguel buys.
If each blue pen is $6, the total cost of x number of blue pens is 6x.
If each red pen is $8, the total cost of y number red pens is 8y.
If Miguel has a maximum of $100 to spend, the total cost of x blue pens and y red pens is
6x + 8y < 100
3x + 4y < 50 (simplest form by dividing 6x + 8y < 100 by 2) à Inequality #1
If Miguel must buy at least 6 pens of each color:
x>6 à
Inequality #2
y>6 à
Inequality #3
Profit: 2x + 4y
If he sells each blue pen at $8, his profit for every blue pen is $2.
His profit for all blue pens à $(2x)
If he sells each red pen at $12, his profit for every red pen is $4.
His profit for all red pens à $(4y)
Test Pointà
(6,6) à
(8,6) à
(6,8) à
(7,7) à
Profit
$36
$40
$44
$42
His total profit is then $(2x + 4y).
Hence, he must buy 6 blue pens and 8 red pens.
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SEKOLAH BUKIT SION - IGCSE MATH REVISION
1. Given the inequalities:
x + y < 11
y > 3 and y < x.
Find the point having whole number coordinates and satisfying these inequalities which
gives:
(a) the maximum value of x + 4y
Answer: ………………………………………… [2]
(b) the minimum value of 3x + y
Answer: ………………………………………… [2]
2. Given:
3x + 2y > 24;
x + y < 12;
y < ½ x;
y>1
Find the point having whole number coordinates and satisfying these inequalities which
gives:
(a) the maximum value of 2x + 3y
Answer: ………………………………………… [2]
(b) the minimum value of x + y
Answer: ………………………………………… [2]
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SEKOLAH BUKIT SION - IGCSE MATH REVISION
3.
The region R contain points which satisfy the inequalities
y≤
!
"
#+4
y≥3
and
x + y ≥ 6.
On the grid, label with the letter R the region which satisfy these inequalities.
You must shade the unwanted regions.
[3]
4. Pablo plants x lemon trees and y orange trees.
(a) (i) He plants at least 4 lemon trees.
Write down an inequality to show this information.
Answer: ………………………………………… [1]
(ii) Pablo plants at least 9 orange trees.
Write down an inequality to show this information.
Answer: ………………………………………… [1]
(iii) The greatest possible number of trees he can plant is 20.
Write down an inequality in x and y to show this information.
Answer: ………………………………………… [1]
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SEKOLAH BUKIT SION - IGCSE MATH REVISION
(b) Lemon trees cost $5 each and orange trees cost $10 each.
The maximum Pablo can spend is $170.
Write down an inequality in x and y and show that it simplifies to x + 2y ≤ 34.
[2]
(c) (i) On the grid below, draw four lines to show the four inequalities and
shade the unwanted region.
[4]
(ii) Calculate the smallest cost when Pablo buys a total of 20 trees.
Answer: ………………………………………… [2]
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SEKOLAH BUKIT SION - IGCSE MATH REVISION
5. (a) Luke wants to buy x goats and y sheep.
(i) He wants to buy at least 5 goats.
Write down an inequality in x to represent this condition.
Answer: ………………………………………… [1]
(ii) He wants to buy at least 11 sheep.
Write down an inequality in y to represent this condition.
Answer: ………………………………………… [1]
(iii) He wants to buy at least 20 animals.
Write down an inequality x and y to represent this condition.
Answer: ………………………………………… [1]
(b) Goat costs $4 and sheep costs $8.
The maximum Luke can spend is $160.
Write down an inequality in x and show that it simplifies to x + 2y < 40
[2]
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SEKOLAH BUKIT SION - IGCSE MATH REVISION
(c) (i) On the grid below, draw four lines to show the four inequalities and
shade the unwanted regions.
(ii) Work out the maximum number of animals that Luke can buy.
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[5]
[2]
SEKOLAH BUKIT SION - IGCSE MATH REVISION
6.
Write down the 3 inequalities which define the unshaded region.
Answer: ………………………………………… [1]
Answer: ………………………………………… [1]
Answer: ………………………………………… [4]
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7. Sima sells x biscuits and y cakes.
(a) (i) She sells at least 100 biscuits. Write down an inequality in x.
Answer: ………………………………………… [1]
(ii) She sells at least 120 cakes. Write down an inequality in y.
Answer: ………………………………………… [1]
(iii) She sells a maximum of 300 biscuits and cakes altogether.
Write down an inequality in x and y.
Answer: ………………………………………… [1]
(iv) Sima makes a profit of 40 cents on each biscuit and 80 cents on each cake.
Her total profit is at least $160.
Show that x + 2y ≥ 400.
Answer: ………………………………………… [2]
(b) On the grid below, draw four lines to show the four inequalities and shade
the unwanted regions.
[4]
(c) Calculate Sima’s maximum profit.
Give your answer in dollars.
Answer: ………………………………………… [2]
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