ST.

ANDREW EXAMINATION BOARD

END OF YEAR ASSESSMENT 2025

SUBJECT: MATHEMATICS (456/1)

CLASS: S3

TIME: 2 HOURS 15 MINUTES

Instructions

This paper has two sections; A and B. It consist of six examination items Section A has two compulsory
items

Section B has two Parts; I and II. Answer one item from each part.

Answer four examination items in all

Any additional item(s) answered will not be scored

All answers must be written in the Answer sheet(s) provided Graph paper is provided

SECTION A

Answer both items in this section

Item 1

Wisdom has four children namely, Beatrice, Bonny, Peter and Martha. He was earning UGX, 1,950,000
and his salary has been increased by 15%. He is to spend all his new salary to pay fees for his four
children. Beatrice is to get 20% of the money; Bonny will get half of the remaining. The rest is to be
shared equally between Peter and Martha The father has been invited to attend a workshop and left the
money in the Bank. At the time for going back to school he had not yet returned. He directed his wife to
get the ATM card from the drawer wrapped in a paper with the secret number 43 in base ten. The actual
PIN is to be obtained by converting this number to base three.

Task

Help the family to determine; 2

a) How much money each child will get

b) The actual PIN that the family will use to withdraw the money

Item 2

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A carpenter has been contacted to make tables and chairs. At least 10 tables and over 35 chairs are
needed. In total less than 50 items are to be made. A chair costs UGX 60,000 and a table costs UGX
180,000 to produce. The contractor has atleast UGX 9,000,000 available for the purchases.

Task

a) Express the conditions given to the carpenter as inequalities and equations

b) Help the contractor to identify the feasible region with the possible number of chairs and tables on a
Cartesian plane.

SECTION B

This section has two parts 1 and II Part I Answer one item from this part Item 3 A researcher has carried
out survey in the department of languages at a university. A student of languages has a 65% of getting
an international job. The research found out that 20 students spoke German, 10 spoke French and
German, 9 spoke French and Kiswahili, 7 spoke Kiswahili and German only. Two could not speak any of
the three languages, 22 could speak at least two of the languages, 11 could either speak German or
Kiswahili but not French, 12 could speak only one language. The research report has been given to the
Head of languages but he has failed to establish how many students were interviewed

Task

a) Help the head of languages to determine the number of students who were interviewed by the
researcher.

b) Determine if a student who speaks at least two languages will get an International job.

Give a reason for your answer.

Item 4

A teacher has given a test to his class and the scores are shown below

48 44 33 52 54 44 53 59 37 32

49 37 35 51 38 40 53 35 46 48

62 48 32 54 54 46 59 45 35 42

The teacher would like to; Presents the data in a frequency distribution table using class interval of
Determine the score that is half- way the distribution Determine the number of students who will pass
the test with a score above 49.5

Task Help the teacher to;

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a) Represent the data in a frequency distribution table

b) Determine the score that is half –way the distribution

c) Determine the number of students who passed the test

PART II

Answer one item from this part

Item 5

Rati is a village member who keeps cattle and rears chicken. He lacks feeding troughs for both cattle and
chicken. He is planning to make them. He bought a metallic sheet measuring 4 metres by 2 metres which
he intends to use. A teacher tells Rati that; The trough for feeding cattle is an open cuboid which is 1.2
metres long, 50 cm wide and 80 cm high. The trough for feeding chicken is conical in shape whose base
radius is 50 cm and slant height is 80 cm Rati is now wondering if the metal sheet he bought will be
enough and how much feed can fill each of the two troughs. Given that 1 kg of feeds occupies 12000
cm3

Task

Help Rati to determine

a) If the sheet is going to be enough for both feeding troughs

b) The amount of feed that can fill each feeding trough for cattle

Item 6

A compound designer is to beautify a compound with flower gardens. The flower gardens are
rectangular in shape. The design of the first garden is at A(1,1) B (4,1), C(4,2) and D(1,2). The second
garden A 1B 1C 1D 1 is reflection of garden ABCD in the x-axis. Another garden A11B 11C 11D 11 is
obtained by transforming ABCD using the matrix ( −2 0 0 − 2 ) The designer wishes to draw the
geometrical representation of the gardens before he begins the beautification. The actual dimensions of
the first garden are 4 metres long by 3 metres wide.

Task

Help the designer to;

a) Draw the complete design on a graph

b) Determine the actual area of the largest garden in square centimeters.

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 SECTION A

(Attempt all in this section)

Item one

Janet is a rich woman intending to invest $4500, $5000, and $9000 into a poultry farm, bakery and
supermarket respectively. In this, he wants to ensure that invested income returns a profit of 5%, 7%
and 6% from every respective business per month. However, he is still challenged in computing the
monthly profits and annual expenditure she will invest in these three projects, so as to make better
decisions. Meanwhile Janet intends to be visiting the farm, bakery and supermarket every after four, six
and eight days respectively. He still wonders on how many times he will have to visit the three projects
on the same day in every month.

Task:

a. Help James to compute his monthly profits and annual expenditure he will make in the three
projects b. How many times will Janet have to visit the three projects on the same day in every
month?
Item two
Kelly has been employed on a certain firm whose policy increases workers’ salary at a constant
rate per week. If he received $230 and $410 at the end of his first and fourth week respectively.
He wishes to know how long it will take for him to earn at least $1600. Besides, Kelly will have to
fly his jet north east 28km from home to the gas station and thereby fly south east 45km to his
work place. To minimize costs, Kelly wishes to find the shortest route from the work place to his
home and the direction in which he must go through.
Task
a. How long it will take for him to earn at least $1600?
b. Describe the shortest route from Kelly’s work place back to his home.

SECTION B
PART I
(Attempt only one question from this section)
Item three
James has been sent to a certain region to find out whether it is necessary to set up a new office
for processing National Identity Cards, Driving Permits and Passports, if the probability of a
person having only one document is less than 50% James randomly interviewed 75 people in
that region. He found that 18 people did not have any of the three documents, 30 people had
Driving Permits and 20 people had Passports. The number of people who had both National
Identity Cards and Driving Permits only, was twice those who had Passports and Driving Permits
but no National Identity Cards. 4 people had Passports only. 3 people had all the three
documents and 12 people had Driving Permits only. James has to report about the number of
people who have National Identity Cards but he has some missing information.

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Task:
a. Help James to find the total number of people with National Identity Cards.
b. Determine if there is need for a new office in the region. Give a reason for your answer.

Item four
A fast-food company plans to introduce fried chicken on the menu. It employs Luttu to
investigate people’s opinion on the weight of the package which will cost Ugx.12,000. Luttu
decides to conduct a survey on a random sample of customers to gather their opinions and
there after use the average as the suitable weight. The collected data was summarized as below.

Weight 1-10 11-20 21-30 31-40 41-50

Task.
a. Giving a reason based on computations, suggested the suitable weight of fried chicken to be
put on the menu.
b. The supervisor wishes to know the number of customers who prefer a weight greater than
25grams. Help him to estimate the number of customers basing on the above information.

PART II
(Attempt only one question from this section)

Item five
A welder wishes to fit an ironic equilateral triangle onto a semicircular aluminum piece to form a
strong structure that occupies a space of a hundred square meters. For beauty purposes, he has
prepared six and ten liters of blue and brown paint respectively to paint the iron and aluminum
metal space. When he was informed that one liter for each of the prepared paint can only color
ten square meters, he wishes to know whether the available paint will be enough. For better
space planning, the welder also wishes to know the total length derived by measuring the
boundaries of the formed formed.

Task
a. Analyze the information and tell if the prepared paint will be enough.
b. Determine the total length derived by measuring the boundaries of the formed structure.

Item six
Two walls are to be constructed facing each other at angles of 45° and 60° respectively via a flat
ground, such that they six meters horizontally far from each other. To increase walls’ strength,
an architect is to fix a vertical pillar from the horizontal ground to hold the walls at their meeting

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point. Since every square meter will cost about $100, the project manager wishes to estimate
the least amount of money required to construct the vertical pillar.

Task. Help the project manager to estimate the least amount of money required to construct
the vertical pillar
ministry's farmer support initiative.

ITEM 10
To address concerns about battery durability, Uganda Batteries Limited (UBL), a trusted
manufacturer since 1967, conducted a thorough test on a random sample of 50 batteries. Their
experts carefully selected and examined these batteries, yielding the following results (rounded
to the nearest minute):
423 369 387 411 393 394 405 369
372 410 371 377 389 409 392 408
409 396 431 391 431 401 363 391
405 382 396 381 438 422 400 381
399 415 428 422 397 399 401 398
396 372 410 419 386 390 362 373 391 402
The director has decided to withdraw batteries with a life equal to or less than the average
lifespan of the tested samples and has directed the experts to manufacture only batteries that
achieve at least 99 % of the median life of the 50 tested batteries.

TASK
a) (i) Organize the data into intervals of 10 using a statistical table and analyze the trends to
recommend the most effective battery replacement strategy to the director
(ii) Elaborate on the reasoning that led to your conclusion in a) i
b) (i) Develop a graphical display to illustrate the data, allowing the director and their team to
estimate the median, visualize and analyze the information
(ii) Identify the target battery lifespan for manufacturing, as recommended by the director.
(iii) Analyze the graph and explain the situation, backing your argument with data and logical
reasoning. Aid the manager in recognizing the chance of selecting a battery with a lifespan
greater than or equal to the median value.

ITEM 11
The recently concluded Uganda Secondary Schools Sports Association (USSSA) tournament was
largely dominated by schools from the western region, prompting head teachers of participating
schools to request detailed reports from their games’ teachers on transportation and prize
monies received during the football competition The four schools that dominated include; Fort
Porto SS, Tororo SS, Nyakasura HS, and Kyogera HS. Due to limited funds, the four schools
decided to use two buses; Fort Portal ss and Tororo ss used the Tausi bus which charges UGX.
24,000 per Km while Kyogera HS and Nyakasura HS used Global coaches that charge UGX.
28,000 per Km. On the tournament day, Tausi Bus embarked on its journey from Mbarara to

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Kampala at 4:30 a.m., cruising at a steady 80 Km/hr and arriving in Kampala at 9:00 a.m.
Simultaneously, Global Bus set off from Sanga town, 50 Km from Mbarara, at 4:30 a.m. and
traveled at a constant 50 Km/hr for 3 hours and 30 minutes before pausing for 30 minutes. It
then resumed its journey at a steady 67.5 Km/hr until it reached Kampala, with the bus fare
being equally distributed among the participating schools that used the bus. Upon arrival in
Kampala, the four schools competed in a two-round football tournament 1 st round 2 nd round
The tournament followed a standard points system: three points for a victory, one point for a
draw, and zero points for a defeat. Additionally, the four teams shared a prize pot of UGX
24,000,000, allocated proportionally to their points tally

TASK
a) (i) Assist the games teachers in plotting the buses' routes on a graph, enabling a more helpful
evaluation of their journey.
(ii) Provide the games teachers with the information needed to determine each school's
transportation expenditure.
(iii) Ascertain the first bus to arrive in Kampala and the time gap between its arrival and the
subsequent bus. b) Ascertain the winning and last teams and amount given to each team that
participated in the tournament.

ITEM 12
The school administration is contemplating either adding another teacher, buying more books,
or both. The school administration will do both if the average for this year’s performance at the
end of term one is lower than that of the previous year. According to the library survey report,
last year's students primarily utilized three mathematics textbooks: Longhorn (L), Baroque (B),
and Active Mathematics (A). Notably, students who did not use any of these three textbooks
struggled significantly in the subject, resulting in poor grades. This year, out of 35 students, the
textbook usage breaks down as follows: 13 students used Longhorn, 20 used Baroque, and 17
used Active Mathematics. Additionally, 9 students used both Longhorn and Active Mathematics,
7 used both Longhorn and Baroque, and 8 used both Baroque and Active Mathematics. Notably,
5 students did not use any of the three textbooks; The school administration has a budget to
double the number of students who utilize all three textbooks.

TASK:
a) Provide a data-driven analysis to inform the school administration's decision on whether to
hire an additional teacher, purchase more books, or implement both solutions.
b) (i) Guide the school administration on the number of books they can purchase, consistent
with their initial budgetary framework.
(ii) Do you have any other recommendation to the school administration regarding the book
purchase? Explain your reasoning.

ITEM 13.

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A non-governmental organization aims to teach French, German, and English to lower primary
school children in its community schools. The organization intends to offer language instruction
in French, German, and English to students in their schools. They plan to offer permanent
positions to candidates who can teach all three languages, while those who can teach one or
two languages will be hired on a contract basis. To fill these positions, they are soliciting
applications from qualified teachers. Out of the applications received, 29 candidates can teach
French, with 7 able to teach French only and 22 able to teach French plus one or both of the
other languages. 27 candidates can teach German, with 9 able to teach German only and 18 able
to teach German plus one or both of the other languages. 30 candidates can teach English, with
11 able to teach English only and 19 able to teach English plus one or both of the other
languages. The organization will only consider candidates who are proficient in at least two
languages for oral interviews. Your friend has been tasked with identifying the eligible
candidates and needs your assistance in analyzing the data to determine the number of
candidates who meet the criteria.

TASK
(a) Apply your mathematical expertise to help your friend know the number of candidates
eligible for an oral interview.
(b) The organization has a program for teachers who can teach all three languages (French,
German, and English). They'll be rotated among schools and paid more. What's the likelihood of
a random applicant being in this program?

ITEM 14
In a certain town, there is a section of the road where many accidents occur and the residents
believe it is due to over speeding so they have requested the authorities to build humps along
that section, the chairperson of the roads committee has decided to do some research so a
checkpoint has been put at that section to measure the speed of 50 vehicles passing that point.
They will put humps if the research shows that the percentage of vehicles passing that point at a
speed greater than the speed limit is greater than those who abide by the speed limit. The road
sign shows a speed limit of 55km/hr for that section. The results for the 50 vehicles sampled are
shown in the table below. Speed(km/hr.) 20-30 30- 40 40- 50 50- 60 60- 70 70- 80 80- 90 90- 100
Number of vehicles 5 8 7 9 6 5 4 6

TASK;
(a) Assist the chairperson in determining the average speed at which vehicles pass that point.
(b) Present a graphical analysis to guide the committee's choice of implementing traffic calming
measures.

ITEM 15
The headteacher of a certain school wants to hold a meeting on one of the following days:
Monday, Tuesday, or Wednesday. The purpose of the meeting is to communicate something
important. The headteacher wants to schedule the meeting in a way that the probability of

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some teachers (out of 70) not attending the meeting is less than 0.5 (or 50%). The following
information shows which teachers attend school on Monday, Tuesday, and Wednesday and
which teachers are absent:10 teachers come on Monday only. Ten teachers attend school only
on Tuesday, twelve teachers attend school only on Wednesday, eight teachers attend school on
both Monday and Wednesday, and Seven teachers attend school on both Monday and Tuesday.
Nine teachers attend school on both Tuesday and Wednesday. Three teachers attend school on
all three days. Some teachers do not attend school on any of the three days 10 | P a g e

TASK:
a) Is it advisable for the head teacher to hold the meeting on any of the days he has chosen?
Support your answer with a reason.
b) Which day would you recommend out of the three options, and what is the basis for your
recommendation?

ITEM 16.
A sports organization is selecting team members to participate in marathon competitions from a
group of 60 individuals. The selection process will occur in two phases. In Phase One,
participants who complete the race within 137 minutes or less will qualify for Phase Two. Then,
in Phase Two, those who finish within 122 minutes or less will be selected to participate in the
actual competitions, which is the ultimate goal. The following is a breakdown of the times
achieved by the participants in Phase One; Finish time (mins) 120 - 124 125 -129 130 -134 135 -
139 140 -144 Number of people 15 14 13 11 7

TASK:
a) (i) How many participants advanced to Phase Two?
(ii) What is the likelihood that some of the qualifiers from Phase Two will go on to participate in
the final competitions?
(iii) Based on the probability value, what is the likelihood of the organization finding suitable
participants for the competitions from the group?

ITEM 17.
Maria, a surveyor, embarks on a journey from Mukono to a Kampala construction site to
perform crucial soil testing, as the soil's sand content plays a critical role in ensuring foundation
stability and preventing potential settlement or foundation failure due to excessive sand.
Maria's journey begins with a 30 Km stretch on a bearing of 080° to Kalagi, followed by a 330°
turn and a 40 Km drive to Gayaza. Finally, she heads on a bearing of 30° to reach the
construction site in Kampala which is on a bearing of 020° from her starting point in Mukono.
Upon arrival, she collects soil samples at various depths and records the sand content
percentage in the table below: Soil depth (x) 35 65 55 25 45 75 20 90 51 60 Percentage of sand
(y) 86 70 84 92 79 68 96 58 86 77
Maria needs to create an appropriate graph to visualize the relationship between depth and
sand content and calculate the total cost of surveying materials; including 50 meters of

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measuring tape at UGX. 10,000, 20 soil sampling bags at UGX. 5,000 each, and fuel at UGX. 6,000
per Km that she traveled. She will submit the calculations to apply for funding from her
company. 11 | P a g e

TASK
a) Help Maria draw a precise diagram illustrating her journey, including bearings and distances.
b) (i) Develop a scatter plot to illustrate the relationship between depth and sand content,
aiding Maria in her data analysis.
(ii) Describe the relationship between soil depth and sand percentage, including any trends or
patterns you observe.
(iii) Plot a line of best fit through the scatter diagram data, and uses it to: - Predict the sand
percentage at a depth of 31 cm - Estimate the depth at which the sand percentage is 54%.
c) Assist Maria in preparing a budget proposal to fund her project activities, including her return
trip to Mukono via the same route. (20 marks)

SECTION B:
ITEM 18.
The school club is holding a fundraiser by selling chappattis, which they've made and packaged
in special containers. Each chapatti is 14 cm in diameter, has a uniform thickness of 6 cm, and is
divided into six slices. The containers, shaped like a chapatti slice, each have a marked price of
UGX 1000 and can hold 14,784 cubic centimeters. The cost of making each slice is UGX 140. By

containers gives a 10% discount to whoever buys by cash terms. ( take 𝜋 = 22 7 )

TASK
a) (i) Help the members of a school club to know the number of slices that can be loaded into
each container.
(ii) Ascertain the number of containers needed to pack the entire batch of slices.
b) (i) How much money was generated from the fundraising event?
(ii) Provide a calculation-based recommendation to the club on whether to continue this
business venture in the future, given that the containers were bought on cash terms.

ITEM 19
Stanbic Bank, a prominent African financial institution, seeks to revamp its logo to align with its
values and appeal to a newer, younger demographic generation. The current logo, a triangle
with 12 | P a g e coordinates A(2, 3), B(4, 1), and C(1, 2) on a white rectangular background, is
due for a refresh. The bank's graphic designer has suggested the following design modifications
to enhance the logo; Keep the original triangle in place, but turn it 90 degrees counterclockwise
around the origin. Then, mirror the resulting triangle across the horizontal axis. Next, scale up
the new triangle by a factor of 3 about the center (-5, -2), creating a logo with four triangles.
Paint only the enlarged triangle with a red-to-white ratio of 3:5, using red paint that costs UGX

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20,000 per square centimeter and white paint that costs UGX 15,000 per square unit. The bank
has set a budget limit of UGX 205,000 per logo for painting.
TASK
a) (i) Assist the designer in creating a precise layout of the logo, showcasing the exact
placement of the four triangles on the same material.
(ii) Specify the exact vertices of the new triangles.
b) Using data-driven insights, recommend to the bank owners whether to adjust their allocation
for logo painting expenses.

ITEM 20.
Uganda Crop Care Limited (UCCL) has secured a contract to supply liquid fertilizer in Kenya, with
a requirement to package it in cylindrical tanks measuring 15 meters in height and 4 meters in
radius. Currently, the company stores its liquid fertilizer in metallic buckets with dimensions of
10 meters in height, 1 meter in lower radius, and 3 meters in upper radius. To fulfill the order,
UCCL needs to determine the number of buckets required to fill 100 tanks. Each metallic bucket
costs UGX 8000 to manufacture, and the company sells the fertilizer at UGX 3600 per liter. The
manager at UCCL needs to calculate the number of buckets needed and evaluate the cost
implications.

TASK
a) Ascertain the number of buckets needed to fill all the required tanks.
b) Establish the total cost that will be required to manufacture the required metallic buckets.
c) Based on calculations, evaluate UCCL’s potential for success.

ITEM 21.
In Mukono town, a mobile vendor, previously operating outdoors, has been hindered by
inclement weather conditions affecting their seating area. To combat this, they have decided to
acquire four ‘three-sided table surfaces’, identical in size, to provide a shielded space. Each table
will feature a central hole to accommodate a circular umbrella, covering the tabletop's vertices.
With a side that is distinct from the two equal sides, measuring 4.5 meters, the tables will form
isosceles triangles with two equal angles of 65 degrees each. The furniture maker quotes a price
of Shs. 300,000 per table, offering a 5% discount for cash payments, providing a more conducive
and protected environment for the vendor's operations. 13 | P a g e The mobile vendor requires
the tables to be fabricated within a tight 48-hour deadline. The furniture maker began working
on the tables on Monday at 7: 30 am, with a schedule of 7:30 am to 4: 00 pm, including a 30-
minute break from 1:30 pm to 2:00 pm. Each table surface requires 5 hours of work to be
completed.

TASK
a) Assist the furniture maker in drafting designs for the tabletop surfaces that meet the mobile
vendor’s requirements. b) How much will the mobile vendor pay in cash for the four tables? c)

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When will the tables be ready for delivery? d) Recommend the minimum umbrella radius to the
mobile vendor for optimal table shading.

ITEM 24
Your father is one of the organizers of a marathon they want to draw the map of the route that
the participants will during the race. At their chosen starting point they chose to take a road that
turned E30oS and they moved for 5km where they set up point B which will be used as a
checkpoint, they then turned through 235o moving a distance of 9km to point C which will be
the finishing point, however on returning to the office they decided that the finishing point
should be put at point A to cut costs of organizing the two places but they were not sure of the
details of that route from C to A that had to be included on the map. They want to hire a vehicle
that will be used to film the racers, the vehicle available consumes 2 litres per km and a litre
costs UGX.5500, the owner of the vehicle has asked for UGX.500,000 plus fueling the vehicle for
the total distance to be covered but the vehicle owner plans to buy fuel from the fuel station
where he is given a discount of 5% for every 100,000 worth of fuel he buys since he is a regular
customer.

TASK;
(a) Help your father determine the direction from point C to the new finishing point A that will
be shown on the map to be drawn.
(b) (i) You are required to determine the total cost of hiring the vehicle.
(ii) Do you think the vehicle owner will save some money on fuel if so how much?

ITEM 25
A man took out a loan of 5,000,000 Ugandan shillings from the bank at a compound interest
rate of 20% and was supposed to repay it within 3 years. However, with the deadline
approaching, he has yet to raise the necessary amount. He decides to sell one of his plots of land
for 14,000,000 Ugandan shillings to raise the funds to pay off the loan. However, the buyer
negotiates a 10% discount on the price. If he accepts the offer and sells the land, the broker will
charge a 5% commission on the sale price. Additionally, the LC1 is requesting 350,000 Ugandan
shillings to sign the agreements. He is uncertain whether the remaining amount will be sufficient
to cover the loan he needs to repay. He needs to pick up the child from school by 5:00 pm.
However, it's currently 4:45 pm, and he's just leaving the broker's house, which is located a
certain distance away from the school (as shown below). Considering he drives at an average
speed of 50 km/h, he's unsure if he'll arrive on time.
TASK:
a) (i) What is the outstanding loan amount he needs to settle with the bank?
(ii) Will he be able to pay off the loan if he sells the land?
b) Will he arrive at the child's school on time?

ITEM 26

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Moses, an employee at a gift box manufacturing company, has been tasked with designing a
foldable gift box with square faces and a capacity of 2744 cubic centimeters. He's struggling to
create a sketch that will guide him in arranging the faces of the box to fold and close, as well as
determining the dimensions of each cardboard piece to cut and join to achieve the desired
outcome. He normally receives a monthly salary of 300,000 Ugandan shillings with no additional
allowances. However, the company will start deducting taxes from his pay every month, and he
needs to determine his new take-home pay. The company will follow the tax brackets shown
below: Monthly taxable income (Ugx.) Rate (%) 0–100,000 0 100,001–200,000 5 200,001–
300,000 10 He needs to deliver the gift box to the customer and charges a delivery fee based on
the amount of fuel used. His motorcycle uses 0.035 liters of fuel per kilometer, and he travels at
an average speed of 20 meters per second. With fuel priced at 5,000 Ugandan shillings per liter,
he wants to calculate the delivery fee. According to the customer, the journey will take 45
minutes.
TASK:
a) Help Moses develop a sketch outlining the specifications he needs.
b) What is the new salary Moses will be receiving monthly?
c) How much will Moses charge the customer for delivery

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