Question 1: Example
Consider the following sequence:
𝒙𝟏 𝒚𝟏 + 𝒙𝟐 𝒚𝟐 + 𝒙𝟑 𝒚𝟑 + 𝒙𝟒 𝒚𝟒 + 𝒙𝟓 𝒚𝟓
Write it in summation notation.
Answer
Tips:
• First notice that the x and y subscripts change from 1 to 5 in the sequence.
• Also notice that the subscript is the only thing that changes in the sequence while all
else remains the same
𝒏
• Writing something in summation notation means using the summation symbol 𝒊+𝟏 𝒙𝒊 or
𝒏 𝒏
𝒊+𝟎 𝒙𝒊 or 𝒊+-𝟏 𝒙𝒊 depending on what changing number/subscript your given sequence
starts from.
ü So we already know that our “i” starts at 1 and ends at 5. Meaning our symbol will look
something like this:
𝟓
𝒊+𝟏 𝒙𝒊
ü From the summation symbol we know that the “i” represents the subscripts that are
changing meaning that the general term of our sequence is 𝒙𝒊 𝒚𝒊
ü Putting these two together, we finally get:
𝟓
𝒙𝒊 𝒚𝒊 𝒊
𝒊+𝟏
Question 2: Example
The number of girls studying at UNISA in second semester has increased in the ratio 3:2. If
the number of girls at the start of first semester were only 20, how many of girls do we have
now?
Answer
Tips:
• Another way of writing a ratio is as a fraction.
𝟓
e.g. 𝟓 ∶ 𝟐 → = 𝟐. 𝟓
𝟐
• Sometimes they can give the ratio as words:
, 𝟓
e.g. the ratio of 5 buttons to 2 shirts → 𝟓 ∶ 𝟐 → = 𝟐. 𝟓
𝟐
ü Convert your ratio to a fraction:
𝟑
𝟑∶𝟐→ = 𝟏. 𝟓
𝟐
ü The question says the number has increased by this ratio. When something increases by
something, we must multiply the two:
So we get: 𝟐𝟎×𝟏. 𝟓 = 𝟑𝟎
The number of girls in second semester has now increased to 30.
Question 3: Example
How many ways are there to select a committee to develop a discrete mathematics course at a
school if the committee is to consist of 3 faculty members from the mathematics department
and 4 from the computer science(CS) department, if there are 9 faculty members of the math
department and 11 of the CS department?
Answer
Tips:
• First consider if the order is important, if it is then we know to use a permutation and if
it is not then we know to use a combination.
• Consider the number of ways to choose the faculty members from the mathematics
department, and then consider the number of ways to choose the faculty members from
the CS department
ü We can pick the 3 faculty members out of the total 9 from the maths department as
follows:
𝟗!
𝒙𝟗 𝑪𝟑 = = 𝟖𝟒
𝟗 − 𝟑 !×𝟑!
ü We can pick the 4 faculty members out of the total 11 from the CS department as
follows:
𝟏𝟏!
𝒙𝟏𝟏 𝑪𝟒 = = 𝟑𝟑𝟎
𝟏𝟏 − 𝟒 !×𝟒!
Therefore, there are:
𝟗! 𝟏𝟏!
𝒙𝟗 𝑪𝟑 ×𝒙𝟏𝟏 𝑪𝟒 = × = 𝟖𝟒×𝟑𝟑𝟎 = 𝟐𝟕𝟕𝟐𝟎 𝒘𝒂𝒚𝒔
𝟗 − 𝟑 !×𝟑! 𝟏𝟏 − 𝟒 !×𝟒!
,Question 4: Example
On each delivery, a postman delivers letters to 3 of the 12 suburbs in his territory. In how
many different ways can he schedule his route?
Answer
Tips:
• The order in which he delivers to the suburbs is not mentioned, meaning that this is a
combination
ü The number of ways in which he can schedule his route is:
𝟏𝟐! 𝟏𝟐!
𝒙𝟏𝟐 𝑪𝟑 = × = 𝟐𝟐𝟎 𝒘𝒂𝒚𝒔
𝟏𝟐 − 𝟑 !×𝟑! 𝟗!×𝟑!
Question 5 & 6: Examples
Please see the hints posted on the main QMI1500 site posted by the lecturer before
proceeding to the examples below.
Example 1
Find the volume of the triangular prism shown in the diagram:
Solution:
ü 𝑽𝒐𝒍𝒖𝒎𝒆 𝒐𝒇 𝒂 𝒑𝒓𝒊𝒔𝒎 = 𝑨𝒓𝒆𝒂 𝒐𝒇 𝒃𝒂𝒔𝒆 ×𝒉𝒆𝒊𝒈𝒉𝒕
ü The base is a triangle, so we must find the area of the triangle which is:
𝟏 𝟏
𝑨= ×𝒃𝒂𝒔𝒆×𝒉𝒆𝒊𝒈𝒉𝒕 = ×𝟗×𝟏𝟐 = 𝟓𝟒𝒄𝒎𝟐
𝟐 𝟐
Hence,
𝑽 = 𝑨×𝒉 = 𝟓𝟒×𝟏𝟖 = 𝟗𝟕𝟐𝒄𝒎𝟑
, Example 2
Consider the following two similar triangles:
If 𝐴𝐵 = 6, 𝐷𝐸 = 3 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝐷𝐸𝐹 = 6𝑐𝑚_ . What is the area of ABC
Solution:
ü The ratio of the given sides is:
𝑨𝑩 𝟔
= =𝟐
𝑫𝑬 𝟑
ü If you follow the hints posted by the lecturer you will find out that if ratio of the sides of
two similar triangles is 𝒙 then the ratio of the areas of the triangles is 𝒙𝟐
Therefore,
𝑨𝒓𝒆𝒂 𝒐𝒇 𝑨𝑩𝑪 = 𝟐𝟐 ×𝑨𝒓𝒆𝒂 𝒐𝒇 𝑫𝑬𝑭
= 𝟒×𝟔 = 𝟐𝟒𝒄𝒎𝟐
Question 7: Example
There are 3 grade 12 classes at John’s high school, class one has 50 students, class two has 60
𝟏
students and class 3 has 80 students. An examiner decides to take a sample of of the entire
𝟐
grade 12 to test the new syllabus. All classes must be represented in the sample. Show how
many students will be sampled in each class
Answer
ü 𝑻𝒐𝒕𝒂𝒍 𝒈𝒓𝒂𝒅𝒆 𝟏𝟐 𝒔𝒕𝒖𝒅𝒆𝒏𝒕𝒔 = 𝟓𝟎 + 𝟔𝟎 + 𝟖𝟎 = 𝟏𝟗𝟎
ü 𝑷𝒓𝒐𝒑𝒐𝒓𝒕𝒊𝒐𝒏𝒂𝒍 𝒔𝒂𝒎𝒑𝒍𝒆 𝒔𝒊𝒛𝒆 = 𝟏𝟗𝟎×𝟎. 𝟓 = 𝟗𝟓
ü Representation of each class will be:
𝟓𝟎
𝑪𝒍𝒂𝒔𝒔 𝟏 = ×𝟗𝟓 = 𝟐𝟓
𝟏𝟗𝟎
Consider the following sequence:
𝒙𝟏 𝒚𝟏 + 𝒙𝟐 𝒚𝟐 + 𝒙𝟑 𝒚𝟑 + 𝒙𝟒 𝒚𝟒 + 𝒙𝟓 𝒚𝟓
Write it in summation notation.
Answer
Tips:
• First notice that the x and y subscripts change from 1 to 5 in the sequence.
• Also notice that the subscript is the only thing that changes in the sequence while all
else remains the same
𝒏
• Writing something in summation notation means using the summation symbol 𝒊+𝟏 𝒙𝒊 or
𝒏 𝒏
𝒊+𝟎 𝒙𝒊 or 𝒊+-𝟏 𝒙𝒊 depending on what changing number/subscript your given sequence
starts from.
ü So we already know that our “i” starts at 1 and ends at 5. Meaning our symbol will look
something like this:
𝟓
𝒊+𝟏 𝒙𝒊
ü From the summation symbol we know that the “i” represents the subscripts that are
changing meaning that the general term of our sequence is 𝒙𝒊 𝒚𝒊
ü Putting these two together, we finally get:
𝟓
𝒙𝒊 𝒚𝒊 𝒊
𝒊+𝟏
Question 2: Example
The number of girls studying at UNISA in second semester has increased in the ratio 3:2. If
the number of girls at the start of first semester were only 20, how many of girls do we have
now?
Answer
Tips:
• Another way of writing a ratio is as a fraction.
𝟓
e.g. 𝟓 ∶ 𝟐 → = 𝟐. 𝟓
𝟐
• Sometimes they can give the ratio as words:
, 𝟓
e.g. the ratio of 5 buttons to 2 shirts → 𝟓 ∶ 𝟐 → = 𝟐. 𝟓
𝟐
ü Convert your ratio to a fraction:
𝟑
𝟑∶𝟐→ = 𝟏. 𝟓
𝟐
ü The question says the number has increased by this ratio. When something increases by
something, we must multiply the two:
So we get: 𝟐𝟎×𝟏. 𝟓 = 𝟑𝟎
The number of girls in second semester has now increased to 30.
Question 3: Example
How many ways are there to select a committee to develop a discrete mathematics course at a
school if the committee is to consist of 3 faculty members from the mathematics department
and 4 from the computer science(CS) department, if there are 9 faculty members of the math
department and 11 of the CS department?
Answer
Tips:
• First consider if the order is important, if it is then we know to use a permutation and if
it is not then we know to use a combination.
• Consider the number of ways to choose the faculty members from the mathematics
department, and then consider the number of ways to choose the faculty members from
the CS department
ü We can pick the 3 faculty members out of the total 9 from the maths department as
follows:
𝟗!
𝒙𝟗 𝑪𝟑 = = 𝟖𝟒
𝟗 − 𝟑 !×𝟑!
ü We can pick the 4 faculty members out of the total 11 from the CS department as
follows:
𝟏𝟏!
𝒙𝟏𝟏 𝑪𝟒 = = 𝟑𝟑𝟎
𝟏𝟏 − 𝟒 !×𝟒!
Therefore, there are:
𝟗! 𝟏𝟏!
𝒙𝟗 𝑪𝟑 ×𝒙𝟏𝟏 𝑪𝟒 = × = 𝟖𝟒×𝟑𝟑𝟎 = 𝟐𝟕𝟕𝟐𝟎 𝒘𝒂𝒚𝒔
𝟗 − 𝟑 !×𝟑! 𝟏𝟏 − 𝟒 !×𝟒!
,Question 4: Example
On each delivery, a postman delivers letters to 3 of the 12 suburbs in his territory. In how
many different ways can he schedule his route?
Answer
Tips:
• The order in which he delivers to the suburbs is not mentioned, meaning that this is a
combination
ü The number of ways in which he can schedule his route is:
𝟏𝟐! 𝟏𝟐!
𝒙𝟏𝟐 𝑪𝟑 = × = 𝟐𝟐𝟎 𝒘𝒂𝒚𝒔
𝟏𝟐 − 𝟑 !×𝟑! 𝟗!×𝟑!
Question 5 & 6: Examples
Please see the hints posted on the main QMI1500 site posted by the lecturer before
proceeding to the examples below.
Example 1
Find the volume of the triangular prism shown in the diagram:
Solution:
ü 𝑽𝒐𝒍𝒖𝒎𝒆 𝒐𝒇 𝒂 𝒑𝒓𝒊𝒔𝒎 = 𝑨𝒓𝒆𝒂 𝒐𝒇 𝒃𝒂𝒔𝒆 ×𝒉𝒆𝒊𝒈𝒉𝒕
ü The base is a triangle, so we must find the area of the triangle which is:
𝟏 𝟏
𝑨= ×𝒃𝒂𝒔𝒆×𝒉𝒆𝒊𝒈𝒉𝒕 = ×𝟗×𝟏𝟐 = 𝟓𝟒𝒄𝒎𝟐
𝟐 𝟐
Hence,
𝑽 = 𝑨×𝒉 = 𝟓𝟒×𝟏𝟖 = 𝟗𝟕𝟐𝒄𝒎𝟑
, Example 2
Consider the following two similar triangles:
If 𝐴𝐵 = 6, 𝐷𝐸 = 3 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝐷𝐸𝐹 = 6𝑐𝑚_ . What is the area of ABC
Solution:
ü The ratio of the given sides is:
𝑨𝑩 𝟔
= =𝟐
𝑫𝑬 𝟑
ü If you follow the hints posted by the lecturer you will find out that if ratio of the sides of
two similar triangles is 𝒙 then the ratio of the areas of the triangles is 𝒙𝟐
Therefore,
𝑨𝒓𝒆𝒂 𝒐𝒇 𝑨𝑩𝑪 = 𝟐𝟐 ×𝑨𝒓𝒆𝒂 𝒐𝒇 𝑫𝑬𝑭
= 𝟒×𝟔 = 𝟐𝟒𝒄𝒎𝟐
Question 7: Example
There are 3 grade 12 classes at John’s high school, class one has 50 students, class two has 60
𝟏
students and class 3 has 80 students. An examiner decides to take a sample of of the entire
𝟐
grade 12 to test the new syllabus. All classes must be represented in the sample. Show how
many students will be sampled in each class
Answer
ü 𝑻𝒐𝒕𝒂𝒍 𝒈𝒓𝒂𝒅𝒆 𝟏𝟐 𝒔𝒕𝒖𝒅𝒆𝒏𝒕𝒔 = 𝟓𝟎 + 𝟔𝟎 + 𝟖𝟎 = 𝟏𝟗𝟎
ü 𝑷𝒓𝒐𝒑𝒐𝒓𝒕𝒊𝒐𝒏𝒂𝒍 𝒔𝒂𝒎𝒑𝒍𝒆 𝒔𝒊𝒛𝒆 = 𝟏𝟗𝟎×𝟎. 𝟓 = 𝟗𝟓
ü Representation of each class will be:
𝟓𝟎
𝑪𝒍𝒂𝒔𝒔 𝟏 = ×𝟗𝟓 = 𝟐𝟓
𝟏𝟗𝟎
