Ratio And Proportion

Worked Exercise

1. Muraya and Dan each made 126kg of a mixture of maize and beans. Muraya mixed maize and beans in the ratio 4:3 while Dan mixed maize and beans in the ratio 4:3 while Dan mixed maize and beans in the ratio of 5:4. How many more kilograms?
   
   1. 2
   2. 3
   3. 4
   4. 6
      Working
      In Muraya’s mixture, maize: beans = 4:3
      maize = ^{Ratio of maize}/_{Total ratio} x No of kg
      =4/₇ x 126
      72kg
      In Dan’s mixture, maize: beans = 5: 4
      Maize = ^{Ratio of maize}/_{Total ratio} x No of kg
      =⁵/₇ x 126
      = 70kg
      Difference = ( 72 - 70 )kg
      = 2kg
      The correct answer is A (2kg)
2. A rectangular plot measures 12m by 10m.The length of the plot is increased in the ratio 3:2 , while the width is decreased in the ratio 4;5.By what ratio is the area of the plot decreased?
   1. 4:3
   2. 5:4
   3. 6:5
   4. 3:2
      Working
      New length after increase
      ³/₂ x 12 = 18m
      New width after decrease
      ⁴/₅ x 10 = 8m
      Original area before the increase/decrease
      = (12 x 10) m²
      = 120m²
      New area = (18 x 8) m²
      = 144 m²
      New ratio = 144: 120
      = 6 : 5
      The correct answer is C (6: 5).
3. Gladys keeps hens, ducks, and turkeys. The ratio of hens to ducks is 5:2. The number of turkeys is 35 less than the number of hens. How many turkeys are there if there are 30 ducks?
   1. 105
   2. 40
   3. 75 
   4. 65
      Working
      Hens : ducks = 5 : 2
      Total ratio = 7
      Ducks = ²/₅
      Hens = ⁵/₇ of x
      Therefore ²/₇ of x = 30
      x = 30 x  ⁷/₂
      x = 105 (Total hens and ducks)
      Hens =⁵/₇ x 105
      = 75
      Turkeys = (75 – 35)
      The correct answer is B (40)
4. Elijah and Paul shared some money in the ratio 5:8.Elijah got sh120 less than Paul. How much money did Paul get?
   1. Sh520
   2. Sh320
   3. Sh200
   4. Sh192
      Working
      Let the amount Paul got be t
      Therefore Elijah got (t-120)
      Therefore t -120 : t = 5 : 8
      ^{t – 120}/_t = ⁵/₈
      8t – 960 = 5t
      8t – 5t = 960
      3t = 960
      t = sh320
      Therefore Paul got = sh 320
      The correct answer is B (sh320)
5. A contractor employed 60 men to complete a piece of work in 150 days. How many more days would 50 men take to complete the same work?
   1. 180
   2. 30
   3. 40 
   4. 50
      Working
      60 men take 150 days
      1 man take (150x60) days
      Therefore 50 men will take ^{150 x 60}/₅₀ days
      = 180 days
      How many more? (180 – 150) days
      = 30 days
      The correct answer is B (30 days)
6. Eighteen men can finish to dig a piece of land in 45 days. How many days would 15 men take to finish the same piece of land?
   1. 54
   2. 2
   3. 7½
   4. 9
      Working
      18 men take 45 days
      1 man takes (45 x 18) days
      Therefore 15 men will take ^{45 x 18}/₁₅
      = 54 days
      The correct answer is A (54 days)

Scale Drawing

Worked Exercise

1. The distance between two villages is 6.4KM. On the map of the region this distance is represented by a line 1.6 cm long. What is the scale of the map?
   1. 1: 400000
   2. 1: 40000
   3. 1: 4000 
   4. 1: 4
      Working
      Actual distance = 6.4 km
      Drawing length = 1.6cm
      1 cm on the map = ^6.4/_1.6
      = 4km on the ground
      But 1 km = (1000 x 100) cm
      4km = 4 x 1000 x 100
      = 400000cm
      Therefore scale = 1: 400000
      The correct answer is A (1: 400000)
2. A rectangular field measures 105 m. On a scale drawing of the field, the longer side is 7cm.What is the measurement of the width on this scale drawing?
   1. 5 cm
   2. 50 cm
   3. 500 cm
   4. 7.35cm
      Working
      7 cm represents 105 m
      1 cm represents (105 ÷ 7)
      Scale used,
      1 cm represents 15m
             ?                  75m
      = ^{(75 x 1)}/₁₅
      5cm
      The correct answer is 5cm (A)
3. The scale on a plan is 1:20. How many cm will represent 1m on this plan?
   1. 50 cm
   2. 50 cm
   3. 5 cm
   4. 2 cm
      Working
      1 : 20 scale means 1 cm on a drawing represents 20 cm on the ground.
      1 cm represents 20cm
         ?                      100cm
      =¹⁰⁰/₂₀
      = 5cm
      The correct answer is C (5cm)
4. A rectangular field measuring 720m by 550m is to be represented on a scale drawing using the scale 1: 10,000. What is the perimeter of the drawing in centimetres?
   1. 0.254
   2. 2.54
   3. 25.4
   4. 254
      Working
      1: 10000 means 1 cm on the map represents 10000 cm on the ground
      If 1 cm = 10,0000cm
      ? = 72,000cm
      = (⁷²⁰⁰⁰/₁₀₀₀₀) cm
      = 7.2cm
      If 1cm = 10,000cm
      = 55,000cm
      = ( ^55,000/_10,000 ) cm
      = 5.5cm
      Perimeter
      
      P = 2(L+W)
      = 2 (7.2 + 5.5)
      = 25.4 cm
      The correct answer is (C)
5. The scale of a map is 1: 50000. What is the length of this map of a road 20km long?
   1. 40cm 
   2. 400 cm 
   3. 4000 cm
   4. 4 cm
      Working
      Scale 1 : 50000 means 1cm on the map represents 50000 cm on the ground.
      20 km = 20 x 1000 x 100
      = 2000000 cm
      50,000cm represents = 1 cm
      2,000,000 represents = 2,000,000 ÷ 50,000
      = 40cm
      The correct answer is A (40 cm)

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.