Commercial Arithmetic 2 Questions and Answers - Form 3 Topical Mathematics

Questions

1. Chepkemoi bought a new washing machine for Kshs.420,000. Its value depreciated over
   the next 5years at the following rates; 15%, 13%, 12%, 9% and 7%. For the next 6 years,
   the rate of depreciation remained constant at 5% then the rate of depreciation remained at 4%
   each. How long did it take for the value of the washing machine to be 1/3 of its original value?
2. The table below shows income tax rates for the year 2006
   

   Taxable income in shs. Pa	Rate of tax in %
   1 – 120,000	10
   120,001 – 240,000	15
   240,001 – 360,000	25
   360,001 – 480,000	35
   Over 480,000	50

   Nafula is married and claims a tax relief of shs.1,120 per month. She stays in a company house For which she pays a nominal rent of shs.1200 per month. She found that in a particular month, her employer deducted shs.4830 as tax. If she is entitled to a maximum insurance policy; relief of shs.600 per month. Calculate her monthly salary. (10mks)
3. The figure below represents two pulley wheels, centres A and B with a rubber band CDEFGHC stretched round them.
   
   Radius of wheel centre A = 16cm, AB = 30cm. CD, GF are tangents to the circles ∠ CAB = 86.3°
   1. calculate the length of the belt CD
   2. Find the angle ABD
   3. Find the length of the belt that would go round the pulleys (CDFGHC)
4. In the figure below, ABCD is a cyclic quadrilateral and BD is a diagonal. EADF is a straight line, ∠CDF = 68^o, ∠BDC = 45^o and ∠BAE = 98^o.
   
   Calculate the size of:
   1. ∠ABD.
   2. ∠CBD
5. A customer deposited Ksh.15,500 in a savings account. Find the accumulated amount after 3½ years if interest was paid at 16% per annum compounded semi-annually
6. A retailer mixes three types of rice, Bismatti costing shs.120 per tin with Pishori costing shs.150 per tin and Ahero rice costing shs.80 per tin in the ratio x : 1 : 2 respectively. If he sells the mixture at shs.137.50 per tin making a profit of 25%. Calculate the value of x.
7. Ashanti is a saleswoman and earns a commission on sales based on the monthly rates shown in the table below:- 
   

   Sales (Kshs)	Commission rate % of sales
   The first 5000	10%
   The next 3000	15%
   Sales above 8000	20%

   In addition, she earns a basic monthly pay of Kshs.6700. During a certain month, she earned a total salary amounting to Kshs.8368. How much worth of sales did she make?
8. The table below shows the annual income tax rates for a certain year.
   

   Total income per month in Kshs.	Rates in Kshs. Per £
   1-10164 10165 – 19740 19741 – 29316 29317 – 33892 388983 and above	2   3   4   5  6
   Automatic personal relief shs.1162

   Kiptoo earns a monthly salary of Kshs.25000. He is entitled to house and medical allowances of Kshs.12000 and Kshs.3000 respectively
   Calculate:
   1. His taxable income per month
   2. His monthly tax payable
   3. His annual tax payable
9. A company employee earns a basic salary of Kshs.25,000 and is also given taxable allowances amounting to Kshs.10,480. Using the table of taxation above:-
   1. Calculate the employee’s taxable income
   2. If the employee is entitled to a personal tax relief of Kshs.800 per month, determine the net tax
   3. If the employee was given 40% increase in his income, calculate the percentage increase in his income tax
10. A certain amount of money was invested at compound interest of 10% compounded every two years for ten years. Given that the investor invested a total of 500,000/= at the end of the ten years, find the amount of money invested to the nearest shillings
11. The cash price of a T.V set is Ksh. 26,000. Linda bought the set on hire purchase terms by paying a deposit of Ksh. 6,000 and the balance by 24 equal monthly installments of Kshs. 1,045.30. Find the compound rate interest per year.
12. What would Kshs. 15000 amount to after 3 years at 16% per annum compounded quartely?
13. Income rates for income earned were charged as follows:
    

    Income in Kshs. p.m.	Rate in Kshs per sh.20
    1 - 8400  8,401 - 18,000  18,001 - 30,000  30,001 - 36,000  36,001 - 48,000  48,001 and above	2  3  4  5  6  7

    A civil servant earns a monthly salary of Ksh.19,200. His house allowance is Ksh12,000 per month. Other allowces per month are transport Ksh.1300 and medical allowance Ksh.2300.
    He is entitled to a family relief of Kshs. 1240 per month.
    Determine:
    1. 1. His taxable income per month.
       2. Net tax.
    2. In addition, the following deductions were made
       NHIF                                      shs. 230
       Service charge                      Kshs. 100
       Loan repayment                    Kshs. 4000
       Co-operative shares of           Kshs. 1200
       Calculate his net salary per month
14. Use the taxation rates in the table below to answer the questions that follow;-
    

    Taxable income in K £ p.a	Rate % per K£
    1-4500 4501-7500 7501 – 10500 10501 – 13500 13501 – 16500 0ver 16500	10 15 20 25 30 35

    The manager of a certain company is entitled to a monthly personal relief of shs.3000 and her tax (PAYE) is kshs.9000 per month she is also deducted NHIF shs.350 per month, WCPS shs.800 per month and cooperative shares shs.1200 per month, calculate
    1. The managers total deductions per month
    2. Total tax per month
    3. The manager’s annual gross salary
    4. The manager’s monthly basic salary if her monthly allowance and medical allowances are 10000 and 2000 shillings
15. The table below shows the income tax for a certain year;
    

    Monthly taxable income (Kshs.)	Tax rates (%)
    1- 9680 9681- 18800 18801 – 27920 27921 – 37040 37940 and above	10% 15% 20% 25% 30%

    In that year, Odero paid a net tax of Kshs.5,512 per month. His total monthly taxable allowances amounted to Kshs.15,220 and he was entitled to a monthly personal relief of kshs.1,162. Every month the following deductions were made;
    N.H.I.F                             Kshs.320
    Union dues                       Kshs.200
    Co-operative shares          Kshs.7,500
    1. Calculate Odero’s monthly basic salary in Kshs
    2. Calculate his monthly salary
16. 1. A car is worth shs.800,000 when new. During the first year it depreciates by 20% of its value and in the second it deprecates by 5% of its value at the start of the year. During the third, fourth and fifth year, depreciation rate is 10%. How much less will it cost at the end of the fifth year?
    2. Find by how much the compound interest will exceed simple interest on shs.3,000 for two years at 15% per year
17. The table below shows the income tax rates:
    
    Income per month (K£)         Rate in Kshs per £
    1 - 325                                                2
    326 – 975                                            3
    976 - 1300                                           5
    1301 – 1625                                         6
    Over 1625                                            7.50
    Mr. Misoi is a public servant who lives in a government house and pays a nominal rent of Kshs.1,220 per month. He earns a basic salary of Kshs. 24,800 and a house allowance of Kshs.12,000 per month. He is entitled to a monthly relief of kshs.1620.
    1. Calculate his monthly;
       1. Taxable income in K£
       2. Tax payable without relief
       3. Tax after relief
    2. Apart from the income tax. The following monthly deductions are made from his earnings
       -HELB loan repayment Kshs.2400
       - Health insurance fund Kshs.1200
       - 2% of Basic salary union fee
       Calculate:-
       1. the total monthly deduction made on Mr. Misoi’s income
       2. Mr. Misoi’s net income per month
18. Joseph bought a camera on hire purchase (H.P) term by paying a deposit of shs.7200 and cleared the balance in 24 equal monthly installments each of 1250.
    1. find the hire purchase price of the camera
    2. the hire purchase price of the camera is 24% higher than the cash price. Find the cash price of the camera
    3. Kangara took a loan from a financial institution and bought the camera with cash. He repaid the loan at 18% p.a compound interest at the end of the two years. Find the total interest paid by Kangara.
19. Income tax for all the income earned was charged at the rates shown.
    

    Total Income p.a (K.£)	Rate in sh per K£
    1 – 1980  1981 – 3960 3961 – 6440 6441 – 7920 7921 – 9900 Excess of 9900	2   3   5   7   9  10

    1. Wanyonyi earned a salary of Kshs.10,500 per month. In addition he was given a house allowance of Kshs. 6500 per month. He got tax relief of Kshs. 300 per month.
       Find ;
       1. His taxable income p.a
       2. Income tax he pays per month.
    2. A part from income tax the following deductions are made per month. NHIF of Kshs

Answers

1. After 1st year = (⁹⁵/₁₀₀ x 4200000)
   = Shs.357,000
   After 2nd year = ( ⁸⁷/₁₀₀ x 357000)
   = sh310590
   After 3rd year = (⁸⁸/₁₀₀ x 310590
   = shs.273319.20
   After 4th year = ( ⁹¹/₁₀₀ x 273319.20)
   =shs.248720.50
   After 5th year = (248720.50 x ⁹³/₁₀₀)
   The next 6 years = shs. 231310
   A = 231310 (1- 0.05)6 = 170034.10
   Then 140000 = 170034.10 (1-0.04)ⁿ
   (0.96)ⁿ = ¹⁴⁰⁰⁰⁰/_170034.10 = 0.8234
   n = ^{log 0.8234}/_{log 0.96}
   = ^0.0844/_0.01773 = 4.76yrs
   Total no. of years = 5 + 6 + 4.76yrs = 15.76years
2. Gross tax = 4830 + 1120 + 600 = sh 6550 per month
   Annual gross tax = 6550 X 12= 78,600
   10 X 120,000 = sh.12,000
   100
   15 X 120,000 = sh.18,000
   100
   25 X 120,000 = sh. 30,000
   100
   Re. tax = 78600 – (12000 + 18000+30000)
   =78600 – 60,000 = 18,6000
   35 X x =18,600
   100
   0.35x = 18,600
   x = sh 53142.86
   Taxable income p.a = 36,000+53142.86 =sh.412142.86
   Monthly salary = ^413142.86/₁₂ + 12,000
   = 34428.57 + 1200 = Sh 35628.57
3.  
   
   1. Sin 86.3° = ^XB/_AB
      Sin 86.3°= ^XB/₃₀
      XB = 30sin86.3°
      XB = CD = 29.93746855 cm
   2. ∠ ABX = 90° - 86.3°
      = 3.7°
      ∴∠ ABD = 3.7° + 90°
      = 93.7°
   3. ∠ DBF obtuse = 360° - 187.4°
      = 172.6°
      Arc DEF = ^ø/₃₆₀ πD or ^ø/₃₆₀ x2πr
      But cos 86.3° = ^AX/_AB
      Cos 86.3° = ^AX/₃₀
      AX = 1.935969248 cm
      DB = 16 – 1.935969248 = 14.06403075 cm
      ∴ Arc DEF = ^172.6°/_360° x ²² /₇ x 14.06403075
      = 106807.8751
                2520
      = 42.38407742 cm
      Arc CGH
      ∠reflex CAG = 360° - (2x 86.3°)
      = 187.4°
      ∴Arc CGH =187.4°/ 360° x2 x22 /7 x 16
      = 131,929.6
            2520
      = 52.35301587 cm
      Total length of belt to go round the belt
      = CD + DEF + GF + CHG
      = 29.93746855 + 42.38407742 + 29.93746855 + 52.35301587
      = 154.6120304 cm
      
4. ∠ABD = 31⁰
   ∠ CBD = 37⁰
5.  A = 15,500(1 + ⁸/₁₀₀)⁷
   = Ksh.25,707
6.  
   
7. Commission earned Kshs. (8368 – 6700) = Kshs. 1668/= 
   let sales in 3rd bracket be y
   ¹⁰/₁₀₀ x 5000) + (¹⁵/₁₀₀ x 3000) + (²⁰/₁₀₀ x y) = 1668
   500 + 450 + 0.2y = 1668
   0.2y = 1668 – 950 = 718
   y =⁷¹⁸/_0.2 = 35%
   Total sales = (8000 + 3590)
   = shs.11590
8.  
   
   1. Taxable income = (25000 + 12000 + 3000)= 40000
   2. Income tax
      10164 x ²/₂₀ = Shs.1016.40
      10164 x ³/₂₀ = Shs.1524.60
      10164 x ⁴/₂₀ = Shs.2032.80
      Remaining :
      9508 x ⁵/₂₀ = Shs.2377
      Total tax payable p.m = 6950.8 – 1162 = Shs.5788.80
   3. Annual tax payable = 5788.80 x 12 = Shs.69465.60
9.  
   
   1. taxable income = Kshs. 25000 + Kshs.10480
      = Kshs. 35480
   2. tax charged:
      1st 4350 = 4350 x ²/₂₀ = 683.25
      2nd 4555 = 4555 x ³/₂₀ – 683.25
      3rd 4555 = 4555 x ⁴/₂₀ – 911
      4th 4555 = 4555 x ⁵/₂₀ - 1138.75
      Rem. 17465 = 17645 x ⁶/₂₀ – 5239
      Total tax = 8407.5 - 800.00 = 7607.50
   3. ⁴⁰/₁₀₀ x 35480 – 14.192 = 49672
      New income = 35480 + 14192 = 49672
      Remainder = 49672 – 18015 = 31657
      Tax charged = 31657 x 6/20 =12665.1
      Total tax = 12665.1
      % increase in income ax = 4257.6 x 100
      7607.5 = 55.97%
10. A = P(^HR/₁₀₀)ⁿ
    500000 = P( 1+ ²⁰/₁₀₀)⁵
    500,000 = (¹²⁰/₁₀₀)⁵
    500,000 = P
    (1.2)⁵
    P = Shs.200,938.786 ≈shs. 200,939
11. Principal = 26, 000 - 6,000 = 20,000
    Total H.P instalments = 1045.3 x 24= 25087.20
    25087.20 = 20,000 [¹/₁₀₀ + r]²
    1.254 = [1+ ^r/₁₀₀]²
    1.120 = ¹/₁₀₀ + r
    ^r/₁₀₀ = 0.12 or 12%
12. No. of periods =12
    r = 4% per period
    A = 1.0412 x 15000
    = 24015.5
13.  
    
    1. Taxable income = 19200 + 12000 + 1300 + 2300 = 34800
    2. Net tax
       8400 x ²/₂₀ = 840
       9600 x ³/₂₀ = 1440
       12000 x ⁴/₂₀ = 2400
       4800 x ⁵/₂₀ = 1200/5800
       Net tax = 5800 – 1240
       = 4560
    3. Net salary = 34800 – (4560 + 5530)
       = 24710
14.  
    
    1. 9000 + 350 + 800 + 1200 = 11350
    2. 9000 + 3000 = 12000
    3. Total taxes = 12000 x 12
       = shs.144000p.a
       Taxes
       450 x 2 = shs.9000
       3000 x 3 = shs.9000
       3000 x 4 = shs.12000
       3000 x 5 = shs.15000
       3000 x 6 = shs.18000
                        Shs.63,000
       144000 – 63000= shs.81000
       7y = 81000y = 11571
       Taxable income= 4500 + 3000 x 4 + 11571 = K 28071p.a
       Gross salary = shs. 561420p.a
    4. Total allowances = 12000 x 12
       = 144,000
       Basic salary = 561420 - 144000
       = Shs.417,420
       Monthly basic pay = shs. 34785
15.  
    
    1. Net tax         5512
       Add relief      1162
       Tax payable   6674
       Tax on 9680 earned
       9680 X ¹⁰/₁₀₀ = 968
       Tax on 9120 earned
       9120 x ¹⁵/₁₀₀ = Shs. 1368
       Tax on next 9120 x ²⁰/₁₀₀ = Shs.1824
       Tax on next 9120 x ²⁵/₁₀₀ = 2280
       Total 968 + 1368 + 1824 + 2280 = 6440
       6674 – 6440 = 234
       Let x be charged at 30%
       ³⁰/₁₀₀ X x = 234
       X = ^{(234 X 100)}/₃₀ = Shs.780
       Total chargeable Income
       780 + (9120 x 3) + 9680 = 37820
       Salary 37820 – 15220 = Shs.2260 per month.
    2.  Net salary (37820 – 1270 – 6674) = Shs.29876
16.  
    
    1. 1st year after dep. Of 20%
       800 000 x ⁸⁰/₁₀₀
       = Khs. 640,000……………………………………………………
       2nd year after dep. of 5%
       = 640000 x ⁹⁵/₁₀₀
       = 608,000 .....................................
       The next 3 yrs
       A = P [1 – ^R/₁₀₀] = 608,000 ( 1 – ¹⁰/₁₀₀)³
       = 698 000 (0.9)
       = Sh. 443,232 ……………………………………………………
       800,000 – 443,232 = Sh. 356,768 ………………………………………
    2. S.I = 3000 x ¹⁵/₁₀₀ x 2
       = Sh 900 …………………………………………………
       A = 3000 [1 + ¹⁵/₁₀₀]²
       = 3000[1.15]²
       = sh. 3967.50 ……………………………………………….
       C.I = sh 967.50
       967.50 – 900 = sh 67.50 ……………………………………….
17.  
    
    1. 1. Taxable Income
          [¹¹⁵/₁₀₀ x 24 800[ + 12000 - 1220
          = 28520 + 12000 – 1220
          = Ksh.39,300
          = K₤ 1965 p.m.
       2. Tax due 325 x 2 = sh 650
          650 x 3 = sh 1950
          325 x 5 = sh 1725 ………………………..
          325 x 6 = sh 1950
          340 x 7.50 = sh 2250
          Total tax = sh. 8825 P.m………………………(without relief)
    2. 1. Total deduction
          = sh (7280 + 2400 + 1200 + ²/₁₀₀ of 24 800)……………….
          = (7280 + 2400 + 1200 + 496) + 1220
          = sh (11376 +_ 1220) = sh. 12,596 P.m …………………………………………
       2. Net income = sh (24800 + 1200 – 12596) = sh. 24,204 P.m……………………………
18.  
    
    1. Total instalments = (24 x 1250)= Shs.30000
       H.P = 7200 + 30000= 37200
    2. 124% = 37200
       100% =
       C.P = ¹⁰⁰/₁₂₄ x 37200
       = 30000
    3. A = 30000 (1 + 18/100)2
       = 30000 (1.18)2 = 41772
       Total interest = 41772 – 30000 = 11772
19.  
    1.  
       1. (10, 500 + 6,500) x 12 = K₤ 10,20 p.a
          20
       2. 1st 1980 x 2 =   Kshs. 3960
          2nd 1980 x 3 =  Kshs 5940
          3rd 2480 x 5 =  Kshs. 12 400
          4th 1480 x 7 =  Kshs. 10360
          5th 1980 x 9 =  Kshs.17 820
          Last 300 x 10 = Kshs 3 000
                                  Kshs. 53 480
          PAYE = ⁵³⁴⁸⁰/₁₂ - 3000 x ¹²/₁₀₀
          = Shs. 4156.70
       3. Net monthly pay
          17000 – 320 + ²/₁₀₀ x 17000
          = 17000 – 660
          = Kshs 16 340.0