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)

• Capitalization
• Punctuation
  • End Marks
  • The Comma
  • The Semicolon and the Colon
  • The Hyphen
  • The Apostrophe
  • Quotation Mark
• Question Tags

Capitalization

Capitalization is the writing of a word with its first letter as an upper case and the remaining letters in lower case. The following are the cases when capitalization is used:

1. Abbreviations
   Abbreviations begin with a capital letter.
   1. Title of persons
      Examples
      Prof George Saitoti
      Mr. Stephen Kiama
   2. Words used as addresses 
      Examples:
      St. (street)
      Ave. (Avenue)
      Note that all the above abbreviations end with a period. Miss is not an abbreviation, so it doesn’t end with a period.
   3. Words used as business
      Examples:
      Co. (Company) 
      Corp. (Corporation)
   4. Some abbreviations are written in all capital letters, with a letter standing for each important word.
      Examples:
      P.O. (Post Office)
      USA (United States of America)
   5. Initials of names of persons
      Examples:
      E.W. Gichimu
      D.M. Weyama
2. Titles of books, newspapers, magazines, TV shows and movies.
   Examples:
   The Minister’s Daughter (book)
   Tahidi High (TV show)
   The Daily Nation( newspaper)
   Capitalize the first and last words only. Do not capitalize little words such as a, an, the, but, as, if, and, or, nor etc
3. Titles of shorts stories, songs, articles, book chapters and most poems.
   Examples:
   Half a Day (short story)
   Kigeugeu (song)
4. Religious names and terms
   Examples:
   God
   Allah
   Jesus
   the Bible
   Do not capitalize the words god and goddess when they refer to mythological deities.
5. Major words in geographical names
   Examples:
   Continents – Africa, Asia, Europe, Australia
   Water bodies – the Indian Ocean,
6. Names of organisations and institutions
   Examples:
   Kianjege West Secondary School, United Nations, University of Nairobi, Nairobi Women’s Hospital
   Note that here you capitalize only the important words. Do not capitalize such words such as a, in, and of. Do not capitalize such words as school, college, church and hospital when they are not used as parts of names.
   Example: There will be a beauty contest at school
7. Months, days and holidays
   Examples:
   June
   Tuesday
   Labour Day
   Do not capitalize names of seasons: autumn, summer, winter, spring
8. Languages, races, nationalities and religions
   Examples:
   Chinese
   Kikuyu
9. The first word of every sentence
   Example:
   What an exciting day it was!
10. The pronoun I
    Example: 
    What should I do next?
11. Proper Nouns
    Examples:
    Lang’ata Cemetery
    Ann Pauline Nyaguthii Kangaita
    Women’s Group
    Muhigia Teachers Sacco
12. Proper Adjectives
    Examples:
    We ate at an Italian restaurant
    She is a German
13. The first word in greetings and the closing of a letter.
    Examples:
    Dear Mark,
    Yours sincerely,
14. Quotations
    Examples:
    Jamlick exclaimed, “This book would make a great movie!”
    “Where,” asked the stranger, “is the post office?”
    “It’s late,” Billy said. “Let’s go home!”
15. First word of each main topic and subtopic in an outline
    Examples:
    1. Parts of speech
       1. Nouns
          1. Proper nouns

Punctuation

Punctuation is the system of symbols that we use to separate sentences and parts of sentences, and to make their meaning clear. Each symbol is called a punctuation mark. For example (. , ! - : etc)
Punctuation marks can be grouped into:

1. End marks
2. The comma
3. The semicolon and the colon
4. The hyphen
5. The apostrophe
6. Quotation mark

1. End Marks

   There are three kinds of end marks: the full stop (.), the question mark (?), and the exclamation mark (!). End marks show where sentences end

   1. The full stop (.)

      A full stop is used to end a complete sentence. We use a full stop to end:
      • a declarative sentence- a sentence that makes a state
        Example:
        The highest skyscraper in Nairobi is Times Tower.
      • an imperative sentence – a sentence that makes a request or tells someone to do something.
        Example:
        Please climb the stairs carefully.
        Note: An imperative sentence is followed by an exclamation mark when it expresses a
        strong emotion.
        Example:
        Be careful!
      • at the end of an indirect question – one that tells what someone asked, without using the person’s exact words.
        Other uses of the full stop
        Full stops are also used:
      • after initials and after most abbreviations
        Examples:
        L.L. Coo J.
        Mr. Sammy Njagi
        11:00 A.M.
        Note that some abbreviations do not require full stops:
        Examples:
        M (metres) FM (frequency modulation) Km kilometres)
      • after each number or letter that shows a division of an outline or precedes an item in a list.
        Examples:
        Outline                       List
        1.Parts of speech      1. Water – borne diseases
          A. Nouns                  2. Air-borne disease
      • between numerals representing dollars, cents, before a decimal and in percentages
        Examples:
        $ 25.65
        165.42
        25.3%

   2. The question mark (?)

      The question mark is used at the end of an interrogative sentence (a sentence that asks a question).
      Examples:
      When was the Times Tower built?
      Who built it?

   3. The Exclamation mark (!)

      The exclamation mark is used at the end of the exclamatory sentence and after an interjection. (An exclamatory sentence expresses strong feeling, emotion or emphasis. An interjection is a word or group of words that expresses strong feelings).
      Examples:
      Exclamatory sentence: Oh, what a tall building it is!
      Interjections: Superb! Fantastic! Impressive!
      An exclamation mark can also be used at the end of an imperative sentence that expresses strong feeling.
      Example:
      Sit! And stay in that chair if you know what’s good for you!

2. The Comma

   There are a number of uses of the comma in English. A comma generally tells the reader where to pause. They are used:
   
   • to separate words in a series except the last
     The three or four items in a series can be nouns, verbs, adjectives, adverbs, phrases, independent clauses, or other parts of sentences.
     Examples:
     Nouns: John, Jim, Jack walk to school everyday.
     Verbs: He located, patched, and sealed the leak in the tyre.
     Adverbs: She walked quickly, steadily, and calmly.
     Prepositional phrases: He walked through the park, over the bridge, and onto the streets.
     Independent clauses: The match was over, the crowd cheered, and Barcelona received the first- place trophy.
     Adjectives: The fresh, ripe fruit was placed in a bowl.
     Note in the above examples that a comma must be used just before the conjunction.
   • Before the conjunction in a compound sentence
     Some students were taking their lunch, but others were studying.
     Marto photographed the accident scene, and he sold the pictures to the newspaper.
     Example:
     Would she be a lawyer, or would she be a doctor?
     Note: A comma is not required in very short compound sentence in which the parts
     are joined by and. However, always use a comma before the conjunctions but and or.
     Examples:
     Marto photographed the accident scene and Toni reported it.
     Marto photographed the accident scene, but Toni reported it.
     Note also: A comma is not required before the conjunction that joins the parts of a compound verb unless there are more than two parts.
     Examples:
     Mary entered and won the beauty contest.
     That camera focuses, flashes, and rewinds automatically.
   • after introductory words phrases or clauses
     Special elements add specific information to a sentence, but they are not essential. A comma is used to separate a special element from the rest of the sentence.
     Examples:
     Word: Cautiously, he entered the building
     Phrase: After his failure, he disappeared from the public scene.
     Clause: Because he had practised daily, he presented his new song perfectly.
     Note: If the pause after a short introductory element is very brief, you may omit the comma.
     Examples:
     At first he was unsure of his singing ability.
     Finally it was his turn.
     Commas are also used after introductory words such as yes, no, oh and well when they begin a sentence.
     Examples:
     Well, it’s just too cold out there.
     No, it isn’t seven yet.
     Oh, you have spilled the milk.
   • with interrupters
     Interrupters are words that break, or interrupt the flow of thought in a sentence. The commas are used before and after the interrupter to indicate pauses.
     Examples:
     I didn’t expect, however, to lose the job.
     So many people, assumed, sing as well as he does.
     He was chosen, nevertheless, as the new band leader.
   • to set off nouns of direct address
     Examples:
     Yes, Kamau, you can borrow my book.
     Serah, do you know where I kept my phone?
     How is your leg, grandpa?
   • to set off the spoken words in a direct sentence or quotation from the speech tag
     Examples:
     Jackson said, “After my injury I had to learn to walk again.”
     “The therapists urged me to keep trying,” he continued.
     If the speech tag interrupts the spoken words commas are used after the last word of the first part of the spoken words and after the last word in the speech tag.
     Example:
     “After a while,” he added, “I was walking without a cane”.
     Note: When a sentence is indirect or reported, no commas are used.
     Example:
     He added that after a while he was walking without a cane.
   • when writing dates
     Place a comma after the day of the month.
     Examples:
     July 3, 1965 December 12, 2010
   • when referring to geographical location.
     Place a comma between the name of the town or city and the name of the state, district, or country.
     Examples:
     Kibingoti, Kirinyaga County   
     Mombasa, Kenya
   • after the closing of a friendly or business letter.
     Examples:
     Dear Rose,
     Yours sincerely,

3. The Semicolon (;) and the Colon(:)

   1. The semicolon (;)
      The semicolon is used:
      • to separate the parts of a compound sentence when no conjunction is used.
        Example:
        Mountain climbing is exciting; it can also be dangerous.
        Note that the semicolon replaces the comma and the coordinating conjunction. Conjunctions that are commonly replaced by semicolons are and, but, or, for, and nor.
      • before a conjunctive adverb that joins the clauses of a compound sentence (conjunctive adverbs are words like therefore, however, hence, so, then, moreover, nevertheless, yet, consequently, and besides).
        Example:
        The competition takes place in July; however, I prefer August.
      • to separate the parts of a series when commas occurs within the parts.
        Example:
        Last year I flew to Johannesburg, South Africa; Cairo, Egypt; and Kingston, Jamaica.

4. The Colon (:)

   The colon is used:
   • to introduce a list of items
     Example:
     My school bag contains the following items: exercise books, text books, pencils, pens, a geometrical set, and a packet of crayons.
   • after the greeting of a business letter
     Example:
     Dear Mr. Mututho:
   • between numerals that represent hours and minutes and between chapter and verse in a biblical reference
     Examples:
     9:00 A.M.
     6:00 P.M.
     Exodus 2:1-3

5. The Hyphen (-)

   The hyphen is used:
   • to divide a word at the end of a line of writing.
     Note that only words with two or more syllables may be divided at the end of a line and words should be divided only between syllables.
     Example:
     When walking along the streets of Naivasha Town, he met his friend, Wainaina.
     Never divide a word of one syllable and do not divide words to leave a single letter at the end or beginning of a line.
     Incorrect: a-ttraction
     Correct: attra-ction.
   • in compound adjectives that come before the nouns they modify and in certain compound nouns.
     Examples:
     Samuel Wanjiru was a world-famous athlete.
     She is my sister-in-law.
   • in compound numbers from twenty-one through ninety-nine and in fractions.
     Examples:
     seventy-three relatives one-quarter full

6. The Apostrophe (’)

   The apostrophe is used:
   • to form the possessive of a singular noun
     Add an apostrophe and an s.
     Examples:
     the baby’s cot
     James’s car
   • to form the possessive of a plural noun that does not end in s
     Add an apostrophe and an s.
     Examples:
     children’s
     men’s 
   • to form the possessive of a plural noun that ends in s – Add only the apostrophe.
     Examples:
     tricksters’
     tenants’
   • to form the possessive of an indefinite pronoun
     Use an apostrophe and an s.
     Examples:
     everybody’s
     somebody’s
     nobody’s
     Note: Never use an apostrophe with a possessive pronoun like our, yours, hers, theirs.
   • in names of organisations and business,
     Show possession in the last word only
     Example:
     the United Nations’ brochure
   • in hyphenated terms
     Show possession in the last word only.
     Example:
     My mother-in-law’s photograph album
   • in cases of joint ownership
     Show possession in the last word only.
     Example:
     Peter and Patrick’s Limousine
   • in forming contractions
     In contractions, apostrophes replace omitted letters.
     Examples:
     she’s = she is
     aren’t = are not
     I’m = I am
   • To show that part of a date has been omitted
     Examples:
     The tribal clashes of ’08 (the tribal clashes of 2008)
     The’82 coup attempt (the 1982 coup attempt)

7. Quotation Marks (“ “)

   The quotation marks are used:
   • to enclose the spoken words in a direct sentence. Indirect sentences need no quotation marks.
     Example:
     Direct speech: The presidential candidate promised, “Creating new jobs for the youths will be my first priority.”
     Indirect speech: The presidential candidate promised that creating new jobs would be his first priority.
     Note:
     1. Always begin a direct quotation with a capital letter.
        Example:
        The minister said, “You must conserve our environment.”
     2. When the spoken words are divided by the speech tag, begin the second part of the quotation with a small letter.
        Example:
        “Bring me the money,” said the moneylender, “before the end of the day.”
        If the second part of the quotation is a complete sentence, the first word of this sentence is capitalized.
        Example:
        “I am scared,” said the borrower. “That money lender is a brute.”
     3. Place commas and fullstops inside quotation marks
        Place semicolons and colons outside quotation marks.
        Examples:
        “Last month,” the borrower explained, “I borrowed some money from the moneylender.”
        Carol said to the borrower, “And you refused to repay back on time”; however, the borrower did not agree.
     4. Place question marks and exclamation marks inside quotation marks if they belong to the quotation. Place them outside if they do not belong to the quotation.
        Examples:
        Carol asked, “How much money did you borrow?”
        Did the borrower say, “I can’t remember”?
        “You are a fool!” exclaimed Carol.
     5. Use single quotation marks to enclose a title or quotation within a quotation.
        Example:
        “Carol heard the borrower say, ‘I can’t remember’ before she lost her temper.”
        If the tile or quotation within the quotation ends the sentence, use both the single and the double quotation marks after the last word of a sentence.
        Example:
        “Carol heard the borrower say, ‘I can’t remember.’”
     6. In a quotation of more than one paragraph, use quotation marks at the beginning of each paragraph and the end of the final paragraph.

Question Tags

A question tag or a tag question is a phrase that is added at the end of a statement to turn into a question. When a speaker uses a question tag at the end of a statement, he/she is seeking for approval, confirmation or correction.

Examples:
APPROVAL: I look smart today, don’t I? Yes you do.
CORFIRMATION: These are the new students, aren’t they? Yes they are.
CORRECTION: I paid your money yesterday, didn’t I ? No you didn’t

Many learners face a problem of supplying the correct question tags to sentences. This is because they fail to observe the following rules of question tags:

1. A comma must be put to separate the statement with the question tag. A question mark must be placed at the end of the question tag.
   Examples:
   Rufftone has released a new album, hasn’t he?
   He is pushing for a decision by tomorrow, isn’t he?
2. The auxiliary verb in the statement must be repeated in the question tag
   Examples:
   Neson Mandela was in prison for 27 years, wasn’t he?
   The people of South Africa have lost a great hero, haven’t they?
3. When there is no auxiliary verb in the statement, the appropriate form of the auxiliary verb Do must be used in the question tag
   Examples:
   Mark Francis wakes up very early, doesn’t he?
   Peter Bryan bought an I-pad phone, didn’t he ?
4. The subject in the statement must be repeated in the question tag. If it is a noun in the statement, it changes to the appropriate pronoun. If it is a pronoun in the statement, it remains a pronoun in the question tag.
   Examples:
   Fatou Bensouda is prosecutor in ICC, isn’t she?
   She does her work meticulously, doesn’t she?
5. When the statement is positive ( i.e. It does not have the word not in it), the question tag must be negative ( i.e. must use the negative word not) and visa versa.
   Examples:
   David Rudisha has broken another record, hasn’t he?
   Cathrerine Ndereba hasn’t been very active, has she?
   Douglas Wakiihuri does not run any more, does he?
   Ezekiel Kemboi entertains the audience after winning, doesn’t he?
   You will note form the above examples that the auxiliary verb is usually contrated (joined) with the negative indicator not when using question tags.
   However, this does not apply when using primary auxiliary verb am and the modal auxiliary verbs will and shall. Am does not allow contraction with not, will and shall usually change their forms to allow contraction.
   Examples:
   WRONG : am the next speaker, amn’t I?
   CORRECT: I am the nest speaker, am I not?
   WRONG: They will be late for church, willn’t they?
   CORRECT: They will be late for church, won’t they?
   WRONG: We shall attend the Memorial service, willn’t we?
   CORRECT: we shall attend the memorial service, shan’t we?
6. Whereas there is no inversion in the statement, inversion must occur in the question tag i.e. the auxiliary verb comes before the subject
   Examples:
   President Uhuru Kenyatta has won the case, hasn’t he?
   Subject verb                             verb                 verb subject
   He can now relax and attend to his duties, can’t he?
   Subject verb                                                 verb subject
7. For sentences that are inform of requests and commands, the question tags will commonly take the auxiliary verb will or shall followed by the appropriate pronoun.
   Examples:
   Please help me with your pen, will you?
   Let us go for a swim, shall we?
   Bring me that chair, will you?
   Stop that noise, will you?
   Kneel down right away, will you?

Those are the rules that govern question tags and if followed well, the learners will not have any problems with question tags.

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)

• Subjects and Objects
  • Subjects and Predicates
  • Objects
• Complements
  • Subject Complements
  • Object Complements
• Direct and Indirect Objects
• Preparatory It and There
• Phrases and Clauses
• Sentence Types
• Direct and Indirect Speech
  • Direct Speech
  • Indirect Speech

A sentence is a group of words that expresses a complete thought. A complete thought is clear. A sentence always begins with a capital letter. It ends with a full stop (.), a question mark (?) or an exclamation mark (!).

Examples:

• Ted sent me a letter.
• Jane slept soundly.

Subjects and Objects

Subjects and Predicates

The two fundamental parts of every English sentence are the subject and the predicate. A subject can be described as the component that performs the action described by the predicate. It tells who or what does or did the action. It may also name the topic.

The predicate tells about the subject. It tells what the subject does or is.
Examples:
Subject                                    Predicate
(Who or what)                         (What is said about the subject)
The antelope                           jumped over the high fence.
Pigs                                         eat anything is sight when hungry.

In a sentence, a few key words are more important than the rest. These key words make the basic framework of the sentence. The verb and its subject are the key words that form the basic framework of every sentence. The rest of the sentence is built around them.
Examples:
Sentence                                                             Key words
The young kids jumped playfully.                         kids, jumped
Their faces shone brightly.                                   faces, shone

To find out the subject, ask who or what before the verb.
Examples:

• Who jumped playfully? – kids
• What shone brightly? – faces

To find out the verb, ask what after the subject.
Examples:

• The young kids did what? – jumped
• Their faces did what? – shone

The key word in the subject of a sentence is called the simple subject. For example, kids, faces. The complete subject is the simple subject plus any words that modify or describe it. For example, The young kids, Their faces.

The key word in the predicate is called the simple predicate. For example, jumped, shone. The complete predicate is the verb plus any words that modify or complete the verb’s meaning. For example, jumped playfully, shone brightly.

The simple subjects and predicates may sometimes be more than one word. For simple subjects, it may be the name of a person or a place.
Examples:

• Barrack Obama won the US presidential race.
• South Africa is the home of many bats.

The simple predicate may also be more than one word. There may be a main verb and
a helping verb.

• Tanya has acted in many TV shows.
• She will be performing again tonight.

Objects

An object in a sentence is a word or words that complete the meaning of a sentence. It is involved in the action but does not carry it out. The object is the person or thing affected by the action described in the verb. It is always a noun or a pronoun and it always comes after the verb.
Example:

• The man climbed a tree.

Some verbs complete the meaning of sentences without the help of other words. The action that they describe is complete.
Examples:

• It rained.
• The temperature rose.

Some other verbs do not express a complete meaning by themselves. They need to combine with other words to complete the meaning of a sentence.
Examples:

• Christine saw the snake.
• Rose wears goggles.
• He opened the door.

In the above examples, the snake, goggles and the door are the objects as they are the things being affected by the verbs in the sentences.

Complements

Some sentences do not take objects or adverbs (or adverbial phrases) after the verbs. Instead, they take complements. A complement is the part of the sentence that gives more information about the subject (subject complement) or about the object (object complement) of the sentence.

Subject Complements

Subject complements normally follow certain verbs like be, seem, look, etc.
Examples:

• He is British. (British gives more information about he)
• She became a nurse. (nurse gives more information about she)

Object Complements

Object complements follow the direct objects of the verb and give more information
about those direct objects.
Examples:

• They painted the house red. (red is a complement giving more information about the direct object house)
• She called him an idiot. (an idiot is a complement giving more information about the direct
  object he).

The complement often consists of an adjective (e.g. red) or a noun phrase (e.g. an idiot) but can also be a participle phrase.
Example:
I saw her standing there. (standing there is a complement telling more about her).

Direct and Indirect Objects

Objects come in two types, direct and indirect:

Direct Objects

The direct object is the word that receives the action of a verb.
Examples:

• Christine saw a snake. ( a snake receives the action of saw)
• Rose wears goggles. (goggles receives the action of wears)

Sometimes the direct object tells the result of an action.
Examples:

• Tecla won the race.
• She received a trophy.

To find the direct object first find the verb. Then ask whom or what after the verb.
Examples:

• Christine saw a snake.
  Verb: saw
  Saw what? a snake
• Rose ears goggles
  verb: wears
  wears what? goggles
• Tecla won the race
  Verb: won
  Won what? the race
• She received a trophy
  verb: received
  received what? a trophy

Remember, we said earlier that a verb that has a direct object is called a transitive verb and a verb that does not have an object is called an intransitive verb. We also said that a verb may be intransitive in one sentence and transitive in another. Other verbs are strictly intransitive like disagree.

Indirect Objects

The indirect object refers to a person or thing who receives the direct object. They tell us for whom or to whom something is done. Others tell to what or for what something is done.
Examples:
I gave him the book.
He is the indirect object as he is the beneficiary of the book.

Direct object or adverb?

Direct objects are sometimes confused with adverbs. The direct object tells what or whom as we have seen earlier. Adverbs on the other hand tell how, where, when or to what extent. They modify the verbs.
Examples:
Brian Swam slowly. (slowly is an adverb telling how)
Brian Swam a tough race. (race is a direct object telling what).

Verbs can also be followed by a phrase that tells how, when, or where. This kind of a phrase is never a direct object but an adverbial phrase.
Example:
Brian swam across the pool. (a cross the pool tells where Brian Swam).

Therefore, to decide whether a word or a phrase is a direct object or adverb, decide first what it tells about the verb. If it tells how, where, when or to what extent, it is an adverb. If it tells what or whom, it is a direct object

Preparatory It and There

The preparatory It is used to show opinion or condition (especially concerning time, distance, and weather). The preparatory It acts as a dummy subject and is usually followed by the verb be (or a modal + be). The logical subject in sentences beginning with It is often a to-infinitive phrase or a noun clause.

• It is nice to meet you.
• It would be fun to live on a sailboat.
• It is important that we not litter in the park.
• It is 3:30 p.m. right now.
• It never snows in July around here.
• It is believed that he will arrive next week.

The preparatory There often begins sentences that show location or existence, especially when the existence of something or someone is mentioned for the first time. It is usually followed by the verb be (or a modal + be).

• Look! There’s a bear.
• There’s a shooting star in the sky.
• There will be a party on Saturday.
• There is a mosquito in my bedroom.
• There was a new girl at school today.

Phrases and Clauses

Phrases are groups of related words that can include either a subject or a tensed verb.

Prepositional phrases have a preposition and an object of the preposition.

• There was a delicious smell coming from the kitchen.
• The dog barked at the stranger.

Gerund phrases have a gerund and can function the same way as a noun. They often appear as the object of a preposition.

• Thank you for coming to my house.
• Walking alone late at night is dangerous.

Infinitive phrases have an infinitive and can function as a noun, adjective, or adverb.

• Lisa is going to university to study economics.
• To see the Eiffel Tower is a dream of mine.

Participial phrases have a participle and function as an adjective. They are set off from the rest of the sentence by commas.

• Having seen the play three times, she didn’t want to see it again.
• Janice, not used to ice skates, fell down and hurt her knee.

Clauses are groups of related words that include both a subject and a tensed verb

Independent clauses can stand alone as a sentence. Two independent clauses are often connected with a coordinating conjunction.

• Maria is afraid of animals, so she doesn’t go near them.
• We are going swimming, but they are going shopping.

Dependent clauses cannot stand alone as a sentence. They need an independent clause to form a complete sentence. When a dependent clause begins a sentence, a comma is used to separate it from the independent clause.

• We are going swimming since it is so hot outside.
• Since it is so hot outside, we are going swimming.

Sentence Types

Simple sentences have just one independent clause.

• We celebrated Grandpa’s eightieth birthday yesterday.
• Amy loves peanut butter and jelly sandwiches.

Compound sentences have more than one independent clause.

• He finished all of his homework, but he forgot to bring it to school.
• Sue was late for swimming practice, and she left her goggles at home.

Complex sentences have one independent and one dependent clause.

• She didn’t eat because she wasn’t hungry.
• Although he sprained his ankle, he finished the race.

Compound-complex sentences have more than one independent clause and at least one dependent clause.
Before the plane took off, Sarah called her dad to say good-bye, but he didn’t answer the phone.
I like this class; though early in the morning, it’s very interesting.

Direct and Indirect Speech

Direct Speech

Direct speech is used to give a speaker’s exact words. It is also referred to as direct quotation.
Direct speech is always enclosed within quotation marks.
Examples:

• Hemedi announced, “My aunt works in a biscuit factory ”
• “Creating jobs will be my first priority” the governor said.

A comma always separates the quoted words from the speaker’s name, whether the name comes before or after the quotation
Examples:

• Jim asked “Who are you voting fir?”
• “I don’t know yet” answered Carol.

A direct quotation always begins with a capital letter
Example:

• Senator Karabba said, “You must believe in the new constitution”.

When a direct quotation is divided by speech tags, the second part of the quotation must  begin with a small letter.
Example:

• “Register to vote,” said the senator, ‘before the end of the day”.

If the second part of the quotation is a complete sentence, the forst kword of this sentence is capitalized.
Example:

• “I did register,” said Carol. “It took only a few minutes”

Commas and full stops are placed inside quotation marks
Example:

• “Last night,” said Joyce,” I listened to a debate”

Quotation marks and exclamation marks are placed inside a quotation mark if they belong to the quotation. If they do not, they are placed outside the quotation.
Examples:

• Joyce asked, “Whom are you voting for?
• Did Carol say, “I don’t know yet’?
• I can’t believe that she said, “I don’t know yet’!

Speech tags may appear before, in the middle or at the direct speech.
Examples:

• He said, “You know quite well that you have to vote”
• “You know quite well, he said, “that you have to vote”.
• “You know quite well that you have to vote,” he said.

Indirect Speech

Indirect speech is used to refer to a person’s words without quoting him or her exactly. It is also referred to as indirect quotation or reported speech. The original spoken words are not repeated.

The exact meaning is given without repeating the speaker’s words.
Example:
Direct speech: The governor said, “Creating new jobs will be my first priority”
Indirect speech: The governor said that creating new jobs would be his first priority.

Several changes do occur when changing a sentence from direct to indirect speech

1. Quotation marks
   Quotation marks are left out when writing a sentence in direct speech.
   Example:
   Direct: Hemedi announced, “My aunt works in a biscuit factory”
   Indirect: Hemedi announced that his aunt worked in a biscuit factory.
2. Tense - The tense of a verb in the direct sentence will change in indirect speech
   Examples:
   1. Simple present changes to past simple
      Direct: John said, “She goes to school early”
      Indirect: John said that she went to school early.
   2. Simple past changes to past perfect
      Direct: John said, “She went to school early”
      Indirect: John said that she had gone to school early.
   3. Present progressive changes to past progressive
      Direct: “The baby is eating a banana,” the nurse said.
      Indirect: The nurse said that the baby was eating a banana.
   4. Present perfect changes to past perfect
      Direct: “South Sudan has become a republic,” the new president declared.
      Indirect: The new president declared that South Sudan had become a republic
   5. Past progressive changes to past perfect progressive
      Direct: “ I was dreaming when the fire started,” the boy said.
      Indirect: The boy said the he had been dreaming when the fire started.
   6. Future simple changes to modal
      Direct: “I will visit you tomorrow,” my desk mate said.
      Indirect: My desk mate said the he would visit me the following day.
   7. May changes to might
      Direct: : I may also visit you too,” I replied.
      Indirect: I replied that I might also visit him too.

Sometimes the verb in indirect speech does not change tense. This occurs in sentences that are universal truths
Direct: Our Geography teacher said “The earth rotates round the sun”
Indirect: Our Geography teacher said that the earth rotates round the sun

Words referring to place also change
Examples:
Direct: “I live here,” retorted the old man.
Indirect: The old man retorted that he lived there
Direct: “This place stinks,” noted the boy.
Indirect: The boy noted that that place stunk.

Words referring to time also change
Examples:
Direct: “I will visit you tomorrow,” he shouted.
Indirect: He shouted that he would visit me the following/next day

Direct : “ He died last year,” the policeman reported.
Indirect: The policeman reported that he had dies the previous year/ the year before.

Demonstrative pronouns also change:
Examples:
Direct: “This book is mine,” Jane claimed.
Indirect: Jane claimed that that book was hers.

Direct: “These are hard times,” observed the president.
Indirect: The president observed that those were hard times.

Pronouns also change when rewriting a sentence from direct to indirect speech.
Examples:
Direct: “My car is better than yours,” the teacher bragged.
Indirect: The teacher bragged that his/her car was better that his/hers/theirs.

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)

• Coordinating Conjunctions
• Subordinating Conjunctions
• Correlative Conjunctions

A conjunction is a word that connects words or groups of words. Like prepositions, conjunctions show a relationship between the words they connect. But, unlike prepositions, conjunctions do not have objects.
There are 3 main categories of conjunctions;

1. Coordinating conjunctions
2. Subordinating conjunctions
3. Correlative conjunctions

Coordinating Conjunctions

Coordinating conjunctions connect related words, groups of words, or sentences. There are three coordinating conjunctions: and, but and or. And is used to join words, groups of words, or sentences together. But shows contrast while or shows choice.
Examples:

• The bull and the cart are inseparable. (connects two subjects).
• The cart carries the farmer and his tools. (connects two direct objects).
• The food was hard and tasteless. (connects two predicate adjectives).
• Each night, the dancers danced in a circle or in several other patterns. (connects two prepositional phrases).
• Some people died in the fracas, but most managed to escape, alive. (connects two sentences).

Subordinating Conjunctions

Subordinating conjunctions connect two or more clauses to form complex sentences. Subordinating conjunctions introduce subordinate clauses. They include because, since, if, as, whether, and for.
Examples:

• If I go home, my dog will follow me.
  The subordinating conjunction if connects the subordinate clause I go home with the main clause my dog will follow me.
• The stayed inside the church because it was raining.
• He was always rude since he was a child.
• The rain fell as they entered the building.
• The pastor asked the congregation whether they were happy.
• The man rejoiced for he had won a prize.

Correlative Conjunctions

Correlative conjunctions are conjunctions that are used in pairs to connect sentence parts. These include either ….. or, neither ….. nor, not only……. but also, whether ……. or and both …… and.
Examples:

• Both boys and girls attended the conference.
• People brought not only food but also clothes for the victims of the floods.
• The students ride either on bicycles or motorbikes.
• The sailor had to decide whether to sail on or head back when the weather changed.
• Neither John nor James was moved by the shocking news.