Open Source Content Management

Algebra

Worked Exercise

1. What is the value of x in the equation?
   2(3x – 2) = 3x + 8
   1. 12 
   2. 3
   3. 5
   4. 4
      Working 2 (3x – 2 ) = 3x + 8
      Step 1: Open brackets 6x – 4 = 3x + 8
      Step 2: Collect like terms and simplify 6x – 3x = 8 + 4 3x = 12 x = 4
      The correct answer is D (4)
2. Francis has r shillings. John has s shillings. Ouma has sh.150 less than the total money of both Francis and John. Which one of the following expressions gives the total amount of money do the three men have?
   1. 2r + 2s – 150
   2. r + s – 150
   3. 2r + 2s + 300
   4. r + s + 300
      Working
      Francis = r
      John = s
      Ouma = r + s - 150
      Total money = r + s + r + s – 150
      = 2 r + 2s – 150
      The correct answer is A (2r + 2s – 150)
3. If x = 2, y = z - x and z = 3, What is the value of
   3x – 4y + 2z  
   2 (x + 2y – z)
   1. 8
   2. 5
   3. 7
   4. 4
      Working
      Substitute the values of x, y, and z
      = (3x2) – (4x1) + (2x3)
             2 (2+2 x 1 - 3)
      = ⁸/₂ = 4
      The correct answer is D (4)
4. In a meeting there were 30 women than men and three times as many men as children.If there were 1,360 people altogether. What was the number of children in the meeting?
   1. 220
   2. 190
   3. 600
   4. 570
      Working Men 3x
      Children x
      Women 3x + 30
      Total 7x + 30 = 1360
      7x = 1360- 30
      7x = 1330
      x = 190
      Children are 190
      The correct answer is B (190)
5. What is the value of p in the equation?
   ¾(8p- 4) = 4p +7
   1. 2
   2. 5 
   3. 5½
   4. 2 ³/₈
      Working
      ¾(8p - 4) = 4p +7
      6p – 3 = 4p + 7 (opening brackets)
      6p – 4p = 7 + 3 (collecting like terms)
      2p = 10
      p = 5 (Simplifying)
      The correct answer is B (5) 
6. Omammo is two years older than Temo and three years younger than Mbeti. The sum of their ages is 64 years. If Omamo’s age is m, which of the following equations below can be used to find Omamo’s age?
   1. 3m + 1 = 64 
   2. 3m – 1 = 64
   3. 3m – 5 = 64
   4. 3m + 5 = 64
      Working
      Omamo = m
      Temo = m- 2
      Mbeti = m + 3
      Total age = 64
      X + m – 2 + m + 3 = 64
      m + m + m – 2 + 3 = 64
      3m + 1 = 64
      The correct answer is A (3m + 1 ) = 64
7. What is the simplified form of 5x + (8x – 2y)
   1. 37x – 8y
   2. 7x –
   3. 28x – 2y
   4. 7x – 2y
      Working
      5x + ¼(8x – 2y ) open brackets
      5x + 2x – ½y simplify
      = 7x – ½y
      The correct answer is B (7x – ½y)

Geometry

Worked Exercise

1. Find the value of x in the following.
   
   Working
   X+45+50=1800 (Angles on a straight lines are supplementary i.e. add up to 180º )
   X+95=180º
   X=85º
   The value of x =85º
2. Find the sum of angle “a” and angle “b” in the figure below.
   
   Working
   Lines AB and C D are transversals  are Therefore 90+b = 1800
   Co-interior angles - supplementally
   Therefore b=180-90
   B = 90º
   Angle a = 120º - (Corresponding angles)
   Therefore a = 120º
   Sum of a and b
   =120 + 90
   = 210º
3. Find the size of angle marked A B D in the figure below.
   
   X+4x+x+30=180º (angles on a straight line are supplementary)
   = 6x+30=180
   6x=180-30
   6x = 150
   X = 25
   Angle A B D =x + 4X
   But x = 25
   Therefore 25 + (4 x 25)
   = 25 + 100
   = 125º
4. Draw an equilateral triangle A B C where Line AB = 6cm.
   Draw a circle touching the 3 vertices of the triangle. What is the radius of the circle?
   Working
   Steps:
   1. Draw line A B = 6cm
   2. With A as the Centre with the same radius 6cm, mark off an arc above line A B.
   3. With B as the Centre with the same radius 6cm, mark off an arc above line A B to meet the arc in (II) above. Call the point of intersection point C
   4. Join C to A and C to B
   5.  Bisect line A B and B C and let the bisectors meet at point X.
   6. With X as the Centre, draw a circle passing through points A, B and C.
   7. Measure the radius of the circle.
      
5. Construct a triangle P Q R in which Q P = 6cm. Q R = 4cm and P R =8cm. Draw a circle that touches the 3 sides of the triangle, measure the radius of the circle.
   Working
   1. Draw line Q P 6cm
   2. With Centre Q, make an arc 4cm above line Q P.
   3. With Centre P, make an arc 8cm above line Q P and let the arc meet the one in (II) above. Label the point of intersection as R.
   4. Join R to P and R to Q.
   5. Bisect any two angles and let the bisectors meet at point Y.
   6. With Y as the Centre, draw a circle that touches the 3 sides of the triangle.
      
      Construction
      R = 3.5cm
6. A rectangle measures 6cm by 2½ cm. What is the length of the diagonal?
   Working
   
   AC² = AB² + BC² [ Pythagoras Theorem]
   AC² = 6² + 2 ½²
   AC² = 36 + 6.25
   AC² = 42.25
   AC = √42.25
   = 6.5 or 6 ½
   NB: The Pythagoras theorem states
   H² =B² +h²
   h² = H² – b²
   b² = H² –h²
7. In the figure below, A B C is a straight line and B C D E is a quadrilateral. Angle CBD = 620 and lines EB = BD = DC. Line EB is parallel to DC.
   
   What is the size of angle BDE?
   Working
   Consider triangle BCD (isosceles triangle)
   Therefore base angles are equal
   CBD = 62º
   BCD = 62º
   Therefore, BDC = 180 – 124 = 56º
   Angle CDB = angle EBD [Alternate triangle]
   Therefore EBD = 56º
   Angle BDE =^{180 - 56}/₂
   = 62º
   Therefore, BDE = 62º
8. Find the size of the largest angle from the following triangle.
   
   Working
   4X – 10 + x – 20 + 3x + 10 = 180 [Angle sum of a triangle]
   8x – 20 = 180
   8x = 200
   X = 25
   4x – 10 = (100 – 10)º
   = 90º largest angle.

Time, Speed and Temperature

Worked Exercise

1. An airplane took 4½ hours to fly from Cairo to Zambia. If it landed in Nairobi at Nairobi at 0215 h on Saturday, when did it take off from Cairo?
   1. Friday 2145 h
   2. Saturday 2245h 
   3. Friday 2245h
   4. Saturday 2145 h
      Working
      The time the aeroplane took from midnight to 0215h of Saturday = 2h 15min
      The difference (4h 30min – 2h 15min) is the time the aero plane took on Friday night.
      Time on Friday night
        h         min
        4         30
      - 2         15
        2         15
      = 2h 15min before midnight
      Time of takeoff from Cairo
      h      min
      24     00
      - 2     15
      21     45 on Friday
      The correct answer is A (Friday 2145 h)
2. A train let Mombasa on Monday at 2125 h and took sixteen and half hours to reach
   Kisauni. When did the train reach Kisumu?
   1. Tuesday 1.55 a.m
   2. Tuesday 1.55 p.m
   3. Wednesday 1.55 p.m
   4. Monday 1:55 a.m
      Working
      Monday: from 2125h to midnight = 2400h - 2125h
      = 2h 35min
      Tuesday: Number of hours traveled from midnight
      = 16h 30min - 2h 35 min
      = 13h 55min
      The train arrived at Kisumu on Tuesday at 1355h
      This is the same as 1.55p.m
      The correct answer is B (Tuesday 1.55pm)
3. A meeting started at quarter to noon. If the meeting lasted for 2 h 35min, what time in 24-h clock system did the meeting end?
   1. 1320h 
   2. 1420h
   3. 1310h
   4. 1410h
      Working
      The meeting started at 11.45
      Add the meeting time
         h       min
        11       45
      + 2       35  
        14      20  
      The meeting ended at 1420h
      The correct answer is B (1420 h)
4. A wall clock gains 3 seconds every one hour. The clock was set correct at 1pm on Tuesday. What time was it showing at 1pm on Friday on the following week?
   Working
   The number of days from Tuesday 1 pm to Friday 1pm the following week = 10days.
   Number of hours = (24 x 10) = 240 hrs.
   The clock gains 3 seconds after every hour in ten days.
   240 x 3 = 720 seconds
   Min = ⁷²⁰/₆₀ = 12 min
   Hence it will show 1 p.m. + 12 min = 1.12 pm
   In 24 h clock system
   = 1312h
   The correct answer is B (1312h)
5. A cyclist traveled from Nairobi to Nyeri for 4h 30min at a speed of 80km/h. He drove back to Nairobi taking 4 hours. What is his speed, in km/h?
   1. 90
   2. 72
   3. 80
   4. 100
      Working
      Distance = speed x time
      = 80 x 4½
      = 360 km
      From Nyeri - Nairobi distance = 360km
      Time taken = 4hrs
      Therefore speed = ^Distance/_Time
      = 90km/h
      The correct answer is A (90km/hr)
6. A motorist crosses a bridge at a speed of 25m/s. What is his speed in km/hr?
   1. 80
   2. 90
   3. 60
   4. 30
      Working
      When working out this kind of question we use a relationship,
      If 10 m/s = 36 km/h
      25m/s = ?
      = ( x 36) km/h
      = 90 km/h
      The correct answer is B (90km/h)
7. The distance between Mombasa and Mtito Andei is 290km. A bus left Mombasa at 1035h and traveled to Mtito Andei at a speed of 50km/h. At what time did it arrive at Mtito Andei?
   1. 1623h
   2. 1523h
   3. 1423h
   4. 1723h
      Working
      Time = ^Distance/_Speed
      = ²⁹⁰/₅₀
      = 5 ⁴/₅hours or 5h 48min
      Arrival time = Departure time = Time taken + Time taken
       h       min
      10      35
       +5     48
      16      23
      The arrival time 1623 h
      The correct answer is A (1623h)
8. Kamau drove from town M to town N a distance of 150 km. He started at 9.30 am and arrived at town N at 11.00 am. He stayed in town for one hour and 50 minutes. He drove back reaching town M at 2.30pm. Calculate Kamau’s average speed for the whole journey.
   1. 90km/h
   2. 100km/h
   3. 60km/h
   4. 150 km/h
      Working
      Total distance from M to N and back
      = 150 x 2
      = 300 km
      Total time taken
      From 9.30 - 11.00 = 1 h 30 min
      Time spent in town
      = 1 h 50 min
      Time taken from N to M
      = 1430h – 1250h
      = 1h 40min
      Total time = 5 hours
      Average speed = ^{Total distance}/_{Total time taken}
      =(60km/h)
      The correct answer is C (60km/h)
9. The temperature of an object was 20º C below the freezing point. It was warmed until there was a rise of 40º in temperature. What is the reading in the thermometer?
   1. 60 Cº
   2. 40Cº
   3. 20Cº
   4. 20Cº
      Working
      Below freezing point means; - 20
      Rose by 40º
      Therefore - 20º + 40 = 20 C
      The correct answer is C (20º C)

Money

Worked Exercise

1. Mutiso paid sh.330 for an item after the shopkeeper gave him a 12% discount. What was the marked price of the radio?
   1. sh300
   2. sh369.60
   3. sh375
   4. sh350
      Working
      Marked price = 100%
      Discount = 12%
      S.P = 100% - 12%
      = 88%
      If 88 % = 330
      100% = ?
      ^{100 x 300}/₈₈ = Sh375
      The correct answer is C (375)
2. Olang’ borrowed sh.54000 from a bank which charged interest at the rate of 18% p.a. He repaid the whole loan after 8 months .How much did he pay back?
   1. sh6480
   2. sh60, 480
   3. sh14580
   4. sh77760
      Working
      I = ^PRT/₁₀₀
      = ^{54000 x 18 x 8}/_{100 x 12}
      = sh6480
      Amount = P + I
      = (54,000 + 6,480) shillings
      = Ksh 60, 480
      The correct answer is B
3. The cash price of a microwave is sh. 18000. The hire purchase price of the microwave is 20% more than the cash price. Bernice bought it on hire purchase terms by paying 40% of the hire purchase price as the deposit and the balance equal monthly installments of sh1620. How many installments did she pay?
   1. 12
   2. 10
   3. 9
   4. 8
      Working
      Let the cash price be 100%
      Hire purchase = 100% + 20%
      = 120% of the cash price
      = ¹²⁰/₁₀₀ x 1800
      = sh.21, 600
      Deposit = 40% of HPP
      = ⁴⁰/₁₀₀ x 21,600
      = sh.8, 640
      HPP = D + MI
      I = ^{HPP - D}/_MI
      = ^{21600 – 8640}/₁₆₂₀
   5. = 8 Months
      The correct answer is D (8)
4. Salim deposited sh25000 in a bank which paid compound interest at the rate of 10% per annum. If he withdraws all his money after years, how much interest did his money gain?
   1. sh5250
   2. sh2500
   3. sh1375
   4. sh387
      Working
      Interest for year 1
      I = ^PRT/₁₀₀
      = ^{25000 x 10 x 1}/₁₀₀
      = Sh2500
      Amount = 25000 + 2500
      = 27,500
      Interest for 2nd year
      I = ^PRT/₁₀₀
      = ^{27,500 x 10 x ½}/₁₀₀
      = Sh13775
      Total interest (2,500 + 1,375)
      = Sh3875
      The correct answer is D (Sh 3875)
5. Kamaru bought bananas in groups of 20 at sh20 per group. He grouped them into smaller groups of 5 bananas each and sold them at sh10 per group. What percentage profit did he make?
   1. 40%
   2. 50% 
   3. 60 %
   4. 70%
      Working
      For every 20 bananas = sh 25
      One group produces 4 smaller groups of 5 bananas each
      S. P = 4x 10
      = sh40
      B.P price = sh25
      Profit = 40 – 25
      = sh15
      % profit = ^P/_BP x 100
      = 60%
      The correct answer is C (60).
6. A shopkeeper bought 3 trays of eggs at sh 150 per tray. On the way to the shop, he realized 20% of the eggs were broken. He sold the rest at sh 72 per dozen. How much loss did he make?
   1. sh450
   2. sh432
   3. sh18 
   4. sh28
      Working
      B.P for 3 trays = 3 x 150
      = sh450
      Number of eggs = 3 x 30
      = 90 eggs
      20% eggs broke = ²⁰/₁₀₀ x 90
      = 18 eggs broken
      Therefore remained = (90 - 18) eggs
      = 72 eggs
      1 dozen = 12 eggs
                 ? = 72 eggs
      = 6 dozens
      1 dozen = sh.72
      6 dozens = ?
      Loss = B.P – S.P
      = 450 - 432
      sh18
      The correct answer is C (sh18)
7. A Salesperson earns a basic salary of sh7500 per month. He is also paid a 5% commission on all sales above sh30, 000. In a certain month his total earnings were sh.14250. What was his total sales for that month?
   1. sh135000
   2. sh285000
   3. sh165000
   4. sh315000
      Working
      Commission = sh14250 – sh7500
      = sh6750
      5% = sh6750
      100% = ?
      = ¹⁰⁰/₅ x 6750
      = Sh. 135,000
      Total sales = (135,000 + 30,000)
      = sh165000
      The correct answer is C (sh 165,000)
8. Shiku bought the following items from a shop
   6kg of sugar @ sh45
   ½ of tea for sh90
   3 kg of rice @ sh30
   2kg of fat @ sh70
   If she used one thousand shillings to pay for the items, what balance did she receive
   1. sh410
   2. sh455
   3. sh590
   4. sh765
      Working
      Shiku’s Bill
      

      Item	Sh	ct
      6kg sugar @ sh45	270	00
      ½ kg tea for sh90	90	00
      3kg rice @ sh30	90	00
      2kg fat @ sh70	140	00
      Total	590	00

      Total expenditure = sh590
      Balance = sh1000 – sh590
      The correct answer is = sh410 (A)
9. Maranga paid sh4, 400 for a bicycle after he was given a 12% discount. James bought the same item from a different shop and was given a 15%. How much more than James did Maranga pay for the bicycle?
   1. sh250
   2. sh300
   3. sh750
   4. sh150
      Working
      Maranga B.P = 100% - 12%
      = 88%
      ⁴⁴⁰⁰/₈₈ x 100 = sh5000
      James B.P = 100% - 15%
      = 85%
      ^{85 x 100}/₄₄₀₀ = sh4,250
      How much more? = (5000-4250) shillings
      = sh750
      The correct answer is C (750)
10. The table below shows postal charges for sending letters;
    

    Mass of letter	Sh	ct
    Up to 20g	25	00
    Over 20g up to 50g	30	00
    Over 50g up to 100g	35	00
    Over 100g up to 250g	50	00
    Over 250g up to 500g	85	00
    Over 500g up to 1kg	135	00
    Over 1kg up to 2kg	190	00

    
    Namu posted two letters each weighing 95g and another one weighing 450g. How much did he pay at the post office?
    1. sh120
    2. sh135
    3. sh155
    4. sh240
       Working
       Two letters
       95g → Sh35.00
       95g → Sh35 .00
       Another 450g → Sh85.00
       The correct answer is C (sh155)

Volume, Capacity and Mass

Worked Exercises

1. A Jerry can contains 5 litres of juice. This juice is used to fill 3 containers each of radius 7 cm and height of 10cm. How many milliliters of juice are left in the jerry can?
   1. 38
   2. 480 
   3. 400
   4. 420
      Working
      Volume of container: = Πr² h
      =²²/₇ x 7 x 7 x 10
      = 1540 cm³
      Volume of 3 such containers
      = (1540x3) cm³
      = 4620 cm³
      Volume of juice in jerry can = (5 x 1000)
      = 5000cm³
      Volume of juice left = (5000-4620) cm3
      = 380 cm³
      = 380 ml
      The correct answer is A (380ml)
2. The diagram below represents a solid whose dimensions are shown.
   
   What is the volume in cm³?
   1. .30000
   2. 300000
   3. 3000
   4. 3000000
      Working
      Volume = Area of the Cross-section x length
      Volume of the top = (20 x 10 x 150)
      = 30,000cm³
      Volume of the bottom = 60 x 30 x150
      = 270,000cm³
      Whole solid = top + bottom
      = 30,000 + 270,000
      = 300,000cm³
      The correct answer is B (300 000)
3. In the month of October, a farmer delivered 48750kg of maize to a miller. In November the amount of maize delivered was 1850kg more than that of October. The amount delivered in December was 2450kg less than that of November. What was the total mass, in tonnes, was delivered by the farmer in the 3 months?
   1. 145.65
   2. 147.5
   3. 152.4
   4. 150.55
      Working
      October = 48750 kg
      November = (48750+1850) kg
      = 50,600 kg
      December = 50,600-2,450) kg
      = 48,150 kg
      Total mass = 48750+50600 +48150
      = (147500/1000) tonnes
      = 147.5 tonnes.
      The correct answer is B (147.5)
4. A rectangular tank measures 1.2m by 80cm by 50cm. water is poured into the tank to a height of 15cm. How many more liters of water are needed to fill the tank?
   1. 144
   2. 14.4
   3. 33.6
   4. 336
      Working
      Capacity of the tank = 120 x 80 x 50
      = 480,000cm³
      Convert to litres = ^480,000/¹⁰⁰⁰
      = 480litres
      Volume of the water poured = 120 x 80 x 50
      = 144000cm³
      Convert to litres = ¹⁴⁴⁰⁰⁰/₁₀₀₀
      = 144 litres
      Volume of water needed = 480 – 144 = 366litres.
      The correct answer is D (366)
5. The diagram below represents a solid triangular prism.
   
   What is the volume in cm³?
   1. 2400
   2. 2000
   3. 5200
   4. 576
      Working
      Apply Pythagorean relation in triangle ABC
      BC =√26² -10²
      =√576
      = 24cm
      Volume = Area of the Cross section x length
      = ½ x 24 x 10x 20
      = 2400cm³
      The correct answer is A (2400cm³)
6. A cylindrical tank has a radius of 2m and a height of 1.5m. The tank was filled with water to a depth of 0.5M. What is the volume of water in the tank, in litres? (П = 3.14)
   1. 6280
   2. 628 
   3. 9240
   4. 18840
      Working
      Volume = П r ²h
      = 3.14 x 2 x 2 x 0.5
      = 6.28 m³
      In litres = (6.28 x1000) litres
      = 6280 litres
      The correct answer A (6280)
7. When processed, 7kg of coffee beans produce 1kg of processed coffee. Processed coffee is then packed in 50kg bags. A farmer delivered 5.6 tonnes of coffee berries in one month. How many bags were obtained?
   1. 12
   2. 16
   3. 40
   4. 20
      Working
      Mass of coffee berries = 5.6tonnes
      = 5.6x1000
      = 5600kg
      Mass obtained = ⁵⁶⁰⁰/₇
      = 800kg
      Number of bags = 800 ÷ 50
      = 16 bags
      The correct answer is B (16)
8. A rectangular container whose base measures 40cm by 60cm has 30 liters of water when full. Find the height of the container in cm.
   
   1. 0125
   2. 1.25
   3. 12.5
   4. 125
      Working
      V = base area x height
      Height = ^volume/_{base area}
      Volume = 30 litres
      = 30x1000
      = 30,000cm³
      Height = ^30,000/₂₄₀₀
      = 12.5cm
      The correct answer is C (12.5)
9. A shopkeeper had 43 litres sand 5 litres and 5 dl of paraffin. He packed all the paraffin in 7.5 dl-containers. How many containers did he fill?
   1. 58
   2. 5.8
   3. 6
   4. 60
      Working
      Convert decilitres into litres
      1 dl =1/10 litres
      5 dl =5/10 litres
      7.5 dl =7.5/10 litres = 0.75 litres
      Hence 43 litres 5dl = 43.5 litres
      No of containers = ^43.5/_0.75 = 58 containers
      The correct answer is 58 (A)
10. The figure below shows a cylindrical solid of diameter 28cm and length 20 cm. A square hole of side 1.5 cm has been removed. What is the volume of the material in the solid, in 3cm³?
    
    1. 12320
    2. 4500
    3. 8400
    4. 7820
       Working
       
       Volume of solid = volume of a cylinder - volume of the square hole
       = ( x 14 x 14x 20) - (15 x 15 x 20)
       = 12320 - 4500
       = 7,820 cm³
       The correct answer is D (7,820cm³)

Length, Perimeter and Area

Worked Exercise

1. Tracy used a piece of wire m long to support tomato plants in the garden. The wire was cut into pieces of 28cm long. How many complete pieces were obtained?
   1. 85
   2. 30
   3. 20
   4. 30.10
      Working
      1 M = 100cm
      8½m = ?
      8½ x 100 = 850cm
      1 piece = 28 cm
                ? = 850cm
      = ⁸⁵⁰/₂₈
      = 30 complete pieces remainder 10cm
2. The figure below represents a flower garden
   
   What is the perimeter of the garden?
   1. 25m
   2. 38.5m
   3. 11m
   4. 44m
      Working
      P = ¼П d + r + r
      = ( ¼ x ²²/₇ x 14) + (7+7)
      = 11 + 14
      = 25 m
      The correct answer is A (25)
3. The parallel sides of a trapezium measure 10cm by 18cm respectively. If the distance between the parallel sides is 8cm, what is the area of the trapezium in cm²?
   1. 224
   2. 112
   3. 108
   4. 84
      Working
      Area of a trapezium = ½h (a + b)
      = ½ x 8 x (10+18)
      = ½ x 8 x 28
      = 112cm²
4. The figure below shows vegetable garden.
   
   What is the perimeter?
   1. 0.526m
   2. 5.26m
   3. 52.6m
   4. 526m
      Working
      Perimeter of semi-circle
      = ½П d(Circumference only)
      =½ x 2 x ²²/₇ x 7
      = 22m
      To get DC = √ 25 – √ 16
      = √ 9
      = 3m
      Length DE = AB – ED
      = 12.8 – 7
      = 5.8m
      Total length 12.8+ 5 + 3 + 5.8 + 22 + 4
      = 52.6 m
      The correct answer is (52.6)
5. What is the perimeter of the following shape?
   
   
   1. 88cm
   2. 44cm
   3. 176cm
   4. 56cm
      Working
      P = circumference of a circle of radius 7cm
      = 2Π r
      = 2 x ²²/₇ x 7
      = (44 cm)
6. The figure below shows a right angled triangle LMN in which LM = 7.5cm and LN = 19.5cm
   
   What is the area of the triangle in cm²?
   
   1. 18
   2. 67.5
   3. 27
   4. 34.5
      Working
      Apply Pythagoras relation in triangle LMN
      LN² = LM² + NM²
      Nm² = LN² – LN²
      = 19.5² – 7.5²
      = 380.25 – 56.25
      = 324
      NM = √ 324
      = 18 cm
      Area of triangle LMN
      = Base x height
      = ½ X 18 X 7.5
      = 67.5cm²
      The correct answer is B (67.5cm²)
7. The area of a right-angled triangle is 84cm². If the height of the triangle is 7cm, what is the length of the longest side?
   1. 25cm
   2. 24cm 
   3. 19cm
   4. 12cm
      Working
      
      The Pythagoras relationship states that
      H² = b² + h²
      But Area = ½bh
      84 =½ x b x 7
      84 x 2 = 7b
      24 = b
      H² = 24² + 7²
      H² = 576 + 49
      H² = 625
      H = 25
      Therefore the correct answer is 25cm (A)
8. What is the surface area of an open cylinder whose radius is 6.3cm and height of 25cm.
   1. 114.74cm²
   2. 1239.48cm²
   3. 3118.50cm²
   4. 619cm²
      Working
      Total surface area = Πr²+Πdh
      = ( ²²/₇x 6.3 x 6.3) + 2 x ²²/₇ x 6.3 x 25
      = 124.74 + 990
      = 1114.74 cm²
      The correct answer is 1114.74 cm² (A)
9. A Welder made a door with a design as shown below.
   
   What is its area? (Take Π =²²/₇ )
   1. 15.12m²
   2. 12.04m²
   3. 13.36m²
   4. 21.28m²
      Working
      Area of the semi- circle = ½Π r²
      = ½ x ²²/₇ x 1.4 x 1.4
      = 3.08m²
      Area of the rectangle = L x w
      = 3.2 x 2.8
      = 8.96 m²
      Total area = (3.08 + 8.96 )m²
      = 12.04 m²
      The correct answer is B (12.04m²)
10. The diagram below represents a plot with a diameter of 28 meters.
    
    The plot was fenced by erecting posts 4m apart. How many posts were used ? (Π = ²²/₇)
    1. 12
    2. 17
    3. 18
    4. 19
       Working
       Perimeter =½ П d + d
       = (½ x ²²/₇ x 28 + 28)
       = 72
       No of posts = ^Perimeter/_Interval
       =⁷²/₄
       = 18 posts
       The correct answer is C (18)

In this section you will need the following hints to solve the exercises:

• Place value of whole numbers
• Total value of whole numbers
• Multiplication of whole numbers/tables
• BODMAS
• LCM and GCD

Worked Exercise

1. What is four million seventy thousand and five hundred and thirty three?
   1. 4,070,353
   2. 4,070,533
   3. 4,007,533
   4. 4,700,533
      
      Working
      Using the place value table, the question can be solved as follows:
      

      Millions	Hundred Thousands	Ten thousands	Thousands	Hundreds	tens	Ones
      4	0	7	0	5	3	3

      The correct answer is B (4070533)
2. What is the square root of 7 ⁹/16
   1. 7 ¾
   2. 2 ¾ 
   3. 1 ³/₈
   4. ²¹/₁₆
      Working
      Step 1: Change the mixed fraction to improper Find the square root of both numerator and denominator.
      Step 2: Find the square root of both numerator and denominator
      = √121
          √16
      =¹¹/₄
      Step 3: Change the improper fraction to mixed fraction
      = 2¾
      The correct answer is B
3. What is 25% as a fraction?
   1. ¹/₅
   2. ¾
   3. ½
   4. ¼
      Working
      Step1: Express the percentage with 100 as a denominator.
      =²⁵/₁₀₀
      Step 2: Simplify
      =¼  correct answer is D
4. What is the value of of ¹/₃ of(½ + ¹/₉) ÷¹/₆
   1. ¹¹/₃₂₄
   2. ¹/₉₉
   3. 1²/₉
   4. ⁴/₁₁
      Working
      Step1: Using the order of operation, BODMAS, solve the brackets first.
      ¹/₂ + ¹/₉ = ¹¹/₁₈
      Step 2: Open brackets and calculate ‘of ‘
      =¹/₃ of (¹¹/1₈) ÷ ¹/₆
      =¹/₃ x (¹¹/₁₈) ÷ ¹/₆
      =¹¹/₅₄ ÷ ¹/₆
      Step3: Calculate the division part
      =¹¹/₅₄ ÷ ¹/₆
      =¹¹/₅₄ x ⁶/₁(multiply by the reciprocal of ¹/₆)
      =¹¹/₉
      Step 4: Change the improper fraction to mixed fraction.
      = 1 ²/₉
      The correct answer is C.
5. The price of radio is Sh1800. The price was reduced by 15% during an auction. How much is the price after the reduction?
   1. Sh270
   2. Sh2070
   3. sh1530
   4. sh1785
      Working
      Marked price = Sh1800
      Percentage decrease = 15%
      New price
      85% of Sh1800 (100% - 15%)
      = ^{85 x 1800}/₁₀₀
      = Sh1, 530
      The correct answer is Sh 1530 (C)
6. In a certain year a tea factory produced 2500 tonnes of tea leaves. The following year the tonnes increased to 4000. What is the percentage increase?
   1. 160%
   2. 62½ %
   3. 60%
   4. 37½ %
      Working
      First year = 2500 tonnes
      Second year = 4000 tonnes
      Increase = 1500 tonnes (4000-2500)
      % Increase = ^{Increase x 100}/<sub>Original
      
      = 60%</sub>
      The correct answer is C (60%)
7. What is the next number in the sequence below.
   6, 10, 19, 35, …..
8. 60
9. 84 
10. 71
11. 51
    Working
    
    The next difference is 5² = 25
    The next number is 35 + 25 = 60
    The correct answer is A (60)

• General Geometric Shapes
  • Square
  • Rectangle
  • Parallelogram
  • Rhombus
  • Trapezium
  • Right-angled Triangle
  • Equilateral Traingle
  • Isosceles Triangle
• Properties of Triangles and Parallel Lines
  • Triangle
  • Parallel Lines and Transversal
• Speed,Distance and Time

General Geometric Shapes

Square

• All sides are equal
• Opposite sides are parallel
• Each interior angle is a right angle (90º)
• The interior angles total up to 360º
• Diagonals bisect each other at right angles.
• Diagonals measure the same length and bisect interior angles.

Rectangle

• Each interior angle is 90º and they all add up to 360º
• Diagonals are equal
• Diagonals bisect each other but NOT at right angles

Parallelogram

• Opposite sides are equal and parallel
• Opposite angles are equal
• Diagonals bisect each other
• Diagonals are not equal
• Adjacent angles are supplementary (add up to 180º)

Rhombus

• All sides are equal
• Opposite sides are parallel
• Opposite angles are equal
• Diagonals bisect each other at 90º
• Diagonals bisect the interior angles

Trapezium

• The sum of the interior angles is 360º
• Has a pair of parallel lines which are not of the same length
• Has a perpendicular height joining the two parallel lines

Right-angled Triangle (Pythagorean relationship)

• H2 = b2 + h2
• b2 = H2 – h2
• H2 = H2 - b2
  Examples of relationships
  

  Base	Height	Hypotenuse
  3	4	5
  6	8	10
  5	12	13
  7	24	25
  8	15	17
  9	40	41

Properties of Triangles and Parallel Lines

Triangle

Exterior angles & interior angles

• Angles x, y, and z are exterior angles while a, b, and c are interior angles.
• Exterior angles add up to 360º while interior angles add up to 180º.
• Angles x, a; b, z; and c, y; are adjacent to each other and they add up to 180º (supplementary angles)

Parallel Lines and Transversal

1. Angles at a point e.g. a + b+ c + d = 360º
2. Vertically opposite e.g. a/d, b/c, f/g, e/h. They are equal
3. Corresponding angles e.g. b/f, a/e, c/g, d/h. They are equal
4. Alternate angles e.g. c/f, d/e are always equal.
5. Co-interior angles e.g. c/e, d/f, are always equal.
6. Co-interior/allied angles e.g. c/e, d/f are formed by parallel lines. They are supplementary.

Speed, Distance and Time

The formulae related to speed, distance and time can be derived from the following triangle.

• Length
• Mass
• Volume And Capacity
• Time
• Area
  • Rectangle
  • Square
  • Parallelogram
  • Rhombus
  • Triangle
  • Trapezium
  • Circle, half circle, quarter circle
• Surface Area
  • Surface area of a Cylinder
  • Surface area of a Cube
  • Surface area of a Cuboid
  • Surface area of a Triangular Prism
• Volume of Cylinder and Rectangular Shapes
  • Volume of a Cylinder
  • Volume of a Rectangular Shape
• Perimeter
  • Perimeter of a Rectangle
  • Perimeter of a Square
  • Perimeter of a Circle
  • Perimeter of a Half a Circle
• Expressing areas of Large Shapes

Length

The units of length that are used include the following:

• millimetre (mm)
• centimetre (cm)
• decimetre (dm)
• Metre (m)
• Dekametre (Dm)
• Hectometre (Hm)
• Kilometre (Km)
  
  
  

From the illustration:

• 10mm = 1cm
• 10cm = 1dm
• 10dm = 1m
• 10m = 1 Dm
• 10Dm = 1Hm
• 10 Hm = 1Km

The relationship between the units of lengths may be clearly seen if the units are written with a 10 between them.

So to find how many small units are equivalent to another, multiply the number of tens between the units, hence:

• Km 1
• Hm 10
• Dm 100
• M 1000
• dm 10000
• mm 1000000

Mass

• 1000 g = 1Kg
• 1000 Kg = 1Tonne
• 1000000 g = 1Tonne

Volume and Capacity

• 1 cm³ = 1 Ml (millilitre)
• 1000 cm³ = 1 L (litre)
• 100 cm³ = 1 dl (decilitre)
• 1 m³ = 1000 litre
• 1000000 cm³ = 1 m³
• 10 dl = 1 Litre
• 1000ml = 1 Litre

Time

• 60 Seconds = 1 Minute
• 60 Minutes = 1Hour
• 3600 Seconds = 1 Hour
• 24 Hours = 1 day
• 7 Days = 1 Week

Area

Rectangle

Area = Length x Width
A = L X W

Square

Area = Side x Side
A = S x S
A = S²

Parallelogram

A = base x Height
A = b x h

Rhombus

Area = base x height
A = b x h

Triangle

Area = base x height
A = b x h

Trapezium

Area = x sum of parallel lines x height
A = (a + b) x h
A = h (a + b)

Circle, half circle, quarter circle

Circle

Area = П x radius x radius

A = П x r x r

A = Пr2

Half circle

Area = Area of a full circle ÷ 2

A = П r2
       2

A =½ Пr2

Quarter circle

A = Area of the full circle ÷ 4

A = Пr2 ÷ 4

A = ¼Пr2

Note: П = 22/7 or 3.14 or 3 1/7

Surface Area

Surface Area of  a Cylinder

T.S.A = Area of circular ends + area of the curved surface

=2Пr2 + Пdh ( if closed both ends)

T.S.A = Пr2 + Пdh (if open one end )

T.S.A = Пdh (if open both ends/pipe)

Surface Area of  a Cube

T.S.A = Total area of all the six faces

= 6 x L x L

= 6L2 (if closed)

or

= 5L2 (if open one end)

Surface Area of  a Cuboid

T.S.A = Total area for the six faces

= 2 (L x w) + 2 (L x h) + 2 (w x h)

or

= (L x w) + 2 (L x h) + 2(w x h) ( if open on top)

Surface Area of  a Triangular prism

T.S.A = Area of all the 5 faces of the prism

Volume of Cylinder and Rectangular Shapes

Volume of a Cylinder

Volume = Base area x height

= Пr2 x height

= Пr2h

Volume of a Rectangular shape

Volume = Base area x height

V = L x w x h

Note: Depending on the cross-section, the volume of any shape / solid is given by.

V = Area of cross-section x height/length

Perimeter

Perimeter of a Rectangle

P = Length + Length + Width + Width

= L + L + W + W

= 2L + 2W or 2(L + W)

Perimeter of a Square

P = L + L + L + L

= 4L

Perimeter of a Circle

C = П x diameter

= Пd or 2Пr

Note: Perimeter of a full circle is called circumference

Perimeter of a Half a Circle

Perimeter = circumference + diameter

P = ½Пd + d

Note: For triangles and irregular shapes, JUST ADD THE DISTANCE ALL ROUND.

Expressing area of large shape

Hectare – A shape that measures 100m by 100m

Therefore 1ha = (100 x 100)m2

1 ha = 10000m2

Are – a piece / shape that measures 10m by 10m

Therefore 1 are = 10 x 10

1 are = 100m2

Hence:

1 ha = 10000m21are = 100m2

1ha = 100ares

• The school
  • School management
    • The teacher
    • The Deputy Headteacher
    • The Senior Teacher
    • Head prefects
    • Importance of School Administration
  • The school Motto
• The family
  • Main types of Families
  • Needs of Family Members
    • Types of needs
  • Responsibilities of Family Members
• Marriage
  • Religious Marriage
  • Customary Marriage
  • Civil Marriage
  • Importance of Marriage
  • Rights and Responsibilities of Spouses in Marriage
• Succession and Inheritance
• Resources and Economic Activities

The School

• A school is a centre where learning takes place both formally and informally.
• It is also a place where the learners acquire knowledge.

School Management

• The public schools are run by the government through the ministry of education.
• At the district level, the schools are managed by:

1. the District Education Board. (D.E.B)
2. The chairman of the district education board is the district commissioner.
3. The secretary of the District education board is the district education officer (D.E.O)The D.E.O is in charge of all the education matters in the district. He/She:-
   1. Ensures there are teachers in the school.
   2. Inspects schools to ensure standards are set and maintained.
   3. Organizes co-curriculum activities.
   4. Assigns teachers responsibilities by posting and transferring them.

The Teacher

1. He/She is the secretary to the school committee.
2. He/She signs duties and responsibilities to all the teachers in the school
3. Receive information from the ministry and pass them on to teachers.
4. Ensures that the school is stable and runs smoothly.
5. He supervises the work of teachers, pupils and school workers.
6. He ensures that good academic standards are made and maintained.
7. He writes minutes during school committee meetings.
8. He maintains discipline among pupils.

The Deputy Headteacher

1.  He is the principal assistant of the head teacher.
2. He attends to lessons by planning and teaching.
3. He acts in the absence of head teacher.
4. He is in charge of discipline.

The Senior Teacher

1. Ensures all lessons are attended to.
2. Acts in the absence of head teacher and the Deputy head teacher.
3. Plans and teaches the pupils.

Head Prefects (head boy and head girl)

1. They co-ordinate the activities of other prefects.
2. Ensures the pupils are orderly.
3. Prefects act as the eyes of teachers on other pupils.

Importance of School Administration.

1. It promotes high academic standards.
2. It promotes high standards of discipline.
3. It helps in maintaining school facilities like chairs tables and desks.
4. It co-ordinates the daily academic activities in the school.
5. It maintains proper school records.
6. It organizes and promotes co-curricular activities in the schools e.g. games, music, athletics, drama.
7. It acts as the link between the community in school.
8. It ensures that the school maintains cleanliness.
9. It acts as a link between the school and the government education agents like:
   1. Assistant education officers (AEO)
   2. District Education Officers
   3. Provincial director of education
10. It allocates teachers their teaching subjects and other duties.

The School Motto

• It is a phrase that expresses the beliefs of a school.
• It describes the goals that a school intends the school learners to achieve by the time they leave the school.
• School routine is the program of activities in the school either on daily or weekly basis.
• The school timetable forms a major part of the school routine.

The Family

• Family is a group of people who are related by blood or marriage.

Main Types of Family

1. Nuclear family - father, mother and child/children
2. Extended family - nuclear and other relatives.
3. Single parent family - One parent and child/children

Needs of Family Members

• Needs are requirements that are necessary for people to live.

Types of Needs

1. Basic needs - things we cannot do without.
2. Secondary needs - Things that add comfort to our lives but we can do without them.

Resonsibilities of Family Members

• Roles and duties in a family are well defined.

EXAMPLES:

Responsibilities of Parents

1. Providing basic needs for the family.
2. Providing security in the family.
3. Providing medical care for the family.
4. Installing good morals in the children.
5. Providing financial assistance.
6. Teaching religious values.
7. Providing love for the family members.

Marriage

• Marriage is a permanent union between adults.
• Marriage systems recognized in Kenya are:
  1. Religious marriage.
  2. Customary marriage.
  3. Civil marriage.

Religious Marriage

• It is usually conducted in a church , mosque , or a temple.
• A wedding ceremony is conducted.
• Christians and Asian marriages are monogamous (one man and one wife)while Muslim marriages are polygamous (more than one wife).
• Couple exchange marriage vows.
• A marriage certificate is issued.

Customary Marriage

• Conducted according to the African customs and beliefs.
• Polygamy is allowed.
• Bride wealth is given before the wedding ceremony.
• It is usually conducted by the clan elders.

Civil Marriage

• It is presided over by a magistrate or an authorized government officer.
• A couple intending to marry must issue a 21 day notice to the district commissioner or the district registrar of marriage.
• The marriage partners pay a marriage fee.
• A marriage certificate is issued.
• Divorce or separation is granted by a court of law.

Importance of Marriage

1. It provides companionship.
2. It ensures the continuity of the family name and culture.
3. It unites different families hence promotes unity and harmony in the society.
4. It ensures good upbringing of the children.
5. It provides security and legal rights to the children, wife, and the husband.
6. It helps to regulate social behavior of the couple.
7. It helps to enrich culture especially when man and wife are responsible.

Rights and Responsibilities of Spouses in Marriage

1. To be loyal and faithful to each other.
2. To stand by each other as a source of comfort and strength.
3. To love one another.
4. To give each other emotional and physical security and protection.
5. To earn an income to support the family.
6. To promote the family’s standard of living.
7. To discuss the decisions regarding the family matters.
8. They should be caring to the children.

Succession and Inheritance

• Succession means taking over property after the owner dies or give up ownership.
• Inheritance is receiving property left behind when the owner dies.
• People succeed or inherit the estate (belonging of the deceased ) through:
  1. customary laws
  2. written wills
  3. parliamentary acts (law of succession).
• The property of the diseased is called an estate.
• A written document that shows how the property of the deceased should be shared out is known as the will.
• The person who inherits the estates of the deceased is known as an heir.
• The distribution of the estate of the deceased is done by:
  1. Court of law.
  2. The public trustee.
  3. The bearer of the letter of administration or the grant of probate.
• The authority to manage the estate is granted by a court of law.
• The following are entitled to the estate of the deceased :
  
  1. wife or wives
  2. former wife ( in case of a divorce in a court of law).
  3. sons
  4. daughters
  5. parents
  6. Any other person with proof that they depend on the deceased.
• Where both the parents have died, the adult first born child should apply to get a letter of administration, if the parent did not leave a will or a grant of probate, if the parent left a will.
  NB: daughters of the deceased whether married or not have the right to benefit from the property of the deceased.

Resources and Economic Activities.

1. Resources are the things that are useful to human beings e.g. Soil, water, money, land, forest, mineral, wildlife, domestic animals.
2. Economic activities are the different ways that we use the resources to earn income.
3. The main economic activities in Kenya include:
   1. Transport and communication.
   2. Livestock keeping.
   3. Wildlife and tourism.
   4. Fishing.
   5. Mining.
   6. Crop farming.
   7. Forestry.
   8. Manufacturing.