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Equations Questions
1. Solve the simultaneous equation (3 mks)^{ }
2. A photograph is mounted on a frame that it leaves a uniform border at the bottom and at the top. At each side a uniform border which is half the border at the bottom is left.
If the side of the square photograph is 5cm and area of frame is 75cm^{2}.
(a) Write down an equation for the area of the frame (simplified) (2mks)
(b) What are the dimensions of the frame? (4mks)
(c) What is the percentage area of the frame that is not covered by the photograph? (4mks)
3. Solve the equation (3mks)
4. Factorize completely: 75x^{2} – 27y^{2 } (2mks)
5. Solve for x given that (3mks)
6. Solve for x and y in. (4mks)
7. The attendance at a party consisted of 35 men, a number of women and some children. The number of women was one and a half times that of the children present.
 If there were a total of 65 participants, how many women attended the party? (2mks)
 During the party, each child took one bottle of soda, the men took two bottles each while some women took two and others three. Given that five crates each containing 24 bottles of soda were consumed, how many women took two bottles of soda? (6mks)
 Each crate of soda was bought for sh.432 plus a deposit of sh.10 per bottle incase it broke. How much money did the party organizers pay at the soda depot? (2mks)
8. Simplify the expression (3 marks)
p^{2} – 4m^{2 }
2m^{2} – 7mp + 3p^{2}
9. During a certain ceremony goats and chicken were slaughtered. The number of heads (for both chicken and goats) was 45. The total number of legs was 100. Determine the exact number of goats and chicken slaughtered. (3 marks)
10. A tailor bought a number of suits at a cost of Sh. 57,600 from Kensuit wholesalers. Had he bought the same number of suits from Umoja wholesalers, it would have cost him Sh. 480 less per suit. This would have enabled him to buy 4 extra suits for the same amount of money.
(a) Find the number of suits the tailor bought (4 marks)
(b) The tailor later sold each suit for Sh. 720 more than he had paid for it. Determine the percentage profit he made? (4 marks)
11. Use factor method to evaluate the following expression leaving your answer as a product of prime factors in power form.
(3mks)
12. Solve for x in the given equation. (3mks)
64^{x} – 121 = 7 – 4^{3x}
13. Given the 4x = 3y, evaluate (4mks)
14. Find the value of y in the equation
(4mks)
15. Factorise completely the expression
45a^{2} – 20b^{2}
and hence or otherwise find its value when a = 5 and b = 3 (3mks)
16. Simply (3mks)
17. Find the value of M in the following equation. (3 mks)
1 M 1
27 X 81 = 243
18. Rotich bough 4 pencils and 6 pens for kshs. 184 and Kamau bought 3 pencils and 8 pens of the same type at kshs 222. Find the price of each item. (3 mks)
19. A 60m by80m parking lot is torn up to install a side walk of uniform width around its perimeter. The new area of the parking lot is two thirds of the old area. How wide is the side walk? (4 mks)
20. Mukku bought 4 text books and 3 novels at total cost of sh. 4250. if he had bought 2 text books more and one novel less, he would have spent sh. 250 less. On another occasion makku bought 3 text books and 3 novel at the same prices, find how much he spent in the second occasion. (4 marks)
21. A Kenyan businessman US$100 to a company in the United States of America. The Kenyan
can either pay through his account in Kenya or through his account in the United Kingdom.
Which method is cheaper and by how much? Give your answer in Kenya shillings given that;
1 US dollar = 76.84 Kenya shillings
1 Sterling Pound = 1.53 US dollars
1 Sterling pound = 115.70 Kenya shillings
22. Foreign exchange on 27/5/2010 was given as follows:.
Currency  Buying (Kshs)  Selling (Kshs) 
1 Euro  84.15  84.26 
1 Sterling pound  118.35  121.47 
A tourist came to Kenya from London with 6000 Euros which he converted to Kenya shillings
at a bank. While in Kenya he spent a total of Kshs.300,000 then converted the balance into sterling pounds at the Same bank. Calculate the amount in sterling pounds he received.
23. A Kenyan football fan visited South Africa from Kenya. He changed his currency from
Kenya shillings to South African rand. The exchange rates in Kenya were as per the table below:
Buying Selling
9.9399 10.0166
He has a total of shs.2, 8000,265 and must spend 13 days. During his stay, he spent 8900
Rands on food and accommodation, 97,000 Rands on a return air ticket and 53689
Rands on entertainment. On his return, he converted the remaining amount into Kshs.
How much did he receive to the nearest cents?
24. A French tourist changes 3000 Francs into Kenyan shillings at 1 Franc = Kshs.1.89.
He spends shs.4695, then exchanges the remaining shillings back into Francs at 1 Franc =1.95.
How many Francs does he receive?
25. Hamisi arrived in Nairobi from USA with 40 travelers cheques each with 75 US dollars. How
much does she receive in Kshs from the bank on a day when 1 US dollar was equivalent to Kshs
81.40 and the bank charges commission at the rate of Kshs.100 per travelers cheque
26. A Kenya bank buys and sells foreign currencies as shown below Buying in Kshs. Selling in Kshs.
1 Hong Kong dollar 9.74 9.77
1 South African rand 12.03 12.11
A tourist arrived in Kenya with 105,000 Hong Kong dollars and changed the amount to
Kenya shillings. While in Kenya, she spent Shs.403,879 and changed the balance to South
African rand before leaving for South Africa. Calculate the amount, in South African rand
that she received
27. A Japanese tourist entered Kenya with Kshs.500,000 Japanese Yen which he converted
to Kenya currency. While in Kenya, he spend Kshs.16200 in all. He then converted all the
remaining money into Euros before leaving for Italy. If he carried out all his transactions
at the Stanbic bank using rates shown below, calculate to the nearest Euro, how much money
he left Kenya with. (Do not use mathematical tables for this question)
Selling (Kshs  Buying Kshs  
100 Japanese Yen  66.35  66.05 
1 Euro  78.15  77.85 
28. Do not use mathematical tables in this question. Equity bank buys and sells foreign currencies
as shown:
Buying (Kshs.) Selling Kshs.
1 US dollar 77.43 78.10
I South African Rand 9.03 9.51
A tourist arrived in Kenya with 5,600 US dollars and changed the whole amount to Kenya
shillings while in Kenya he spend shs.201,367 and changed the balance to South African rand
before leaving to S. Africa . Calculate the amount in S. African rand that she received
29. A tourist arriving from Britain had UK ₤ 9000. He converted the pounds to Kenyan shillings at a
commission of 5%. While in Kenya he spent ¾ of his money. He exchanged the remaining to US
dollars with no commission. Calculate to the nearest US dollars the amount using the exchange
rate below.
Currency  Buying Kshs.  Selling Kshs. 
1 US Dollar 1 UK Dollar  63.00 125.30  63.20 125.95 
30. A company was to import goods worth Kshs.100,000 from U.K and changed the money to
Sterling pounds. The company later realized that it was cheaper to import the same goods
from U.S.A and changed the sterling pounds to dollar. Unfortunately the transaction failed
and the money was converted to Kenya shillings. How much money did the company endup
with, given that;
1 US dollar = Kshs.78
1 Sterling pound = Kshs.120
1 Sterling pound = 1.79 U.S dollar
31. A tourist arrives in Kenya from England with S.£ 50,000 and uses the money to buy Kenya
shillings. He quickly changes his mind and sells the Kenya shillings to get back his s£.
How much money in S.£ did he get? Use the table below
buying selling
1 Sterling pound 120.7131 120.9294
32. A Kenyan bank buys and sells foreign currencies at the exchange rates shown below:
Buying Selling
(Kshs.) (Kshs.)
1 Euro 147.86 148.00
1 US Dollar 74.22 74.50
An American arrived in Kenya with 20000 Euros. He converted all the Euros to Kenya
shillings at the bank. He spent Kshs.25100 200 while in Kenya and converted the remaining
Kenya shillings into US Dollars at the bank. Find the amount in dollars that he received
33. Simplify;
(a) 6a – 2b + 7a – 4b + 2
(b) 2x – 2 3x + 2
2x 4x
Equations Answers
1  2x – 3 = y
When 
M_{1}
M_{1}
A_{1}
B_{1} 
partial fact or equivalent both answers of x or y both answers 
4  
2.  Length of the frame (x + x + 5) = (2x + 5) Width of the frame ( ½ x + ½ x + 5) = (x + 5) Area = (2x + 5)(x + 5) (a) 75 = 2×2 + 10x + 5x + 25 2×2 + 15x – 50 = 0 (b) 2×2 + 15x – 50 = 0 2×2 + 5x + 20x – 50 = 0 x(2x – 5) + 10(2x – 5) = 0 (x + 10)(2x – 5) = 0 x = 10 or 2 ½ length (2 x 5/2 + 5) = 10cm width (5/2 + 5) = 7.5cm (c) Area not covered = area of frame – area of photo = 75 – 25 = 50cm2 % area = 50/75 x 100 = 66.67%/66 2/3% 
B1
M1 A1
M1
A1 B1 B1
M1
M1 A1 
Dimension with unknowns
Factorization
Difference in area 
10  
3.  M1
M1
A1  Multiplication by LC.  
4  3(25×2 – 9y2) 3(5x – 3y) (5x + 3y)  M1 A1 2  For 3 out

5.  2(x – 3) = 10(4) – 5(x – 2) 2x – 6 = 40 – 5x + 10 7x = 56 x = 8 
M1
M1
A1  
03 
8. (p + 2m) (p – 2m) B1 factors for numerator
2m – p m – 3p B1 factors for denominator
= – p + 2m B1
m – 3p
3
9. Let of goats be a
.” .” chicken be b
a + b = 45 B1 for both equations
4a + 2b = 100
a + b = 45
2a + b = 50 M1 method for solving
any of the unknown
a = 5
a = 5
b = 40
Goats were 5
Chicken were 40 A1
3
10. Ken – suit;
Let the number of suits bought be x and the cost
per suit be y
Then xy = 57600
y = 57600
x M1 Eq
Umoja
No. of suits bought is (x + 4)
Cost per suit is (y – 480)
= P(x + 4) (y – 480) = 57600 M1 Eq
y = (x + 4) (57600 – 480 M1 Sub of y
x
– 480x^{2} – 1920 + 230400 = 0
x^{2} + 4x – 480 = 0 M1 formation of quadratic eq
(x – 20) (x + 24) = M1 fact
No of suits = 20 A1
(b) Cost per suit = 57600 M1 exp of cost
20
Sh. 2880 A1
Profit per suit = Sh. 720
% profit = 720 x 100 M1 exp
2880
= 25% A1
10
11.  13824 = 2^{9} x 3^{3} 0.000125 = 1 = 1 8000 2^{6}x5^{3} 
M1
M1
A1 
Reciprocal
Prime products  
03  
12.  64^{x}+4^{3x} = 128 (4^{3})^{x} + 4^{3x }= 128 4^{3x }+ 4^{3x} = 128 2(4^{3x}) = 128 4^{3x}= 64 = 4^{3} 3x = 3 X = 1 
M1
M1
A1  
03  
13.  4x = 3y y = ^{4}/_{3} x 
M1
M1 M1 A1 
x = ¾ y
For subst
For num For den  
04  
14.  3^{5} x 3^{2y} = 3^{4} 3^{6} x 3^{y2y+1} 3^{5+2y} = 3^{4} 3^{7y} y = 2  M1
M1
A1  For factorization
For simplification  
03  
15. 
B1
M1
A1 
For simplied factorization
substitutes and simplification  
03  
16.  M1
M1
A1 
Condone 0.667  
03  
17.  27^{m} x 1 = 243 81
(3^{3})^{m} x 1 = 3^{5} 3^{4}
3 ^{3m – 4} = 3^{5}
– 3m – 4 = 5
– 3m = 9
m = 3 
M1
M1
A1 3 
Exp. in powers of 3
Equating powers of 3  
18. 
4p + 6q = 184….x3 3p + 8q = 222….x4
12p + 18q = 552 12p + 32q = 888
14q = 336 14 14 q = 24
4p + 144 = 184
4p = 40
P = 10 
M1
M1
A1 3 
Formation of Equation
Elimination of p or q or equivalent
In both 
19.  Old area = 80 x 60 = 4800m^{2} New area = (80 – 2x)(60 – 2x) = 2 4x^{2} – 280x + 4800 = 2/3 x 4800 4x^{2} – 280 + 1600 = 0 X = 6.28m  M1
M1 M1 A1  old area
exp solving CAO  
04  
20 
5t =350 n=20001050 = 950
 B1
M1
A1
B1  2 equations
solving
For t and n 
21. Through A/C in Kenya
1000000 X 76.84 = Shs.7684000
through A/C in UK
1000000 X 115.70 = Shs.7,562,091.15
1.53
Through UK less by
768400 – 7562091.85 = 121,908.85
22. 6000 turn ______ 6000 X 84.15
= Kshs.504900
Balance = 504900 – 300000
= 204900
჻ sterling pound = ^{204900}/_{121.47}
= 1686.8
23. In Rand = 2800265 = 279562.4264
10.0166
Expenses = (115,700 + 97000 + 53689)
= 266389 Rand
Remainder = 279562.4264
266,389.000
13, 174.4264
Amount in Kshs. = 13174.4264 x 9.9399
= 130,942.50
24. Kshs. (3000 x 1.89) = 5670
Remain = 5670 – 4695 = 75
Francs =(975) = 500
1.95
25. Amount in dollars = 75 x 40 = 3,000
Amount in Ksh = 3000 x 81.40 = 244,200/=
Less commission 4,000
Total received sh 240,200
26. Hong Kong 8105,000 x 9.74 = ksh.1022700
Amount spent in Kenya = 403879
Balance = 1,022,600 – 403,879 = 618,821
Amount in South Africa = 618821
12.11= 51100rands
27. 500000 J yen into Kshs.= (500000 x 66.5)
100
= Kshs. 330,250
Amount spend in Kenya= Kshs. 16200
Remained with Kshs. (330250 – 16200)
= Kshs. 314,040
Kshs. 314040 into Euros:
= (314040)
78.15
= 4,018.554063Euros
He left Kenya with = 4,019 Euros (nearest Euro
28. 1 $ Kshs. 77.43
5600$ = (5600 x 77.43)
= 433608
Spent 201,367
Remainder = (433608 – 201367)
= 232241
1SR shs.9.51
Shs.232241
1 x 232241
9.51
= shs.24420.715
29. 1UK ₤ = 125.30
9000 UK ₤ = 125.30 x 9000
= 1,127,700
Commission = 5/100 x 1,127,700 = 56,385
He got 1,071,315
Expenditure = ¾ of 1,071,315 = 803,486.25
Amt. left = 267,828.75
In US $ = 267,828.75
63.20
= 4237.7966 ≃ 4237 US $
30. 1 sterling pound = Kshs.120
? = Kshs.100000
^{100000}/_{120} = 833.3 sterling pounds
1 sterling pound = 1.79 U.S dollars
833.3 = ?
= 833.3 x 1.79 = 1491.7 dollars
1 U.S dollar = Kshs.78
1491.7 dollars = Kshs?
1491.7 x 78 = 116350 Kenya shillings
31. Amount received in Kenya shillings
= 50,000 x Shs.120.7131
= Kshs.6035655
Amount received in sterling pound
= 1 x Kshs.6035655
120.9294 = 49910.568
32. Sh(20000 x 147.86) = sh.2957200
To US Dollars = 44700
74.5 = 6000
He received 6000 US Dollars
33. a) 6a + 7a – 2b – 4b + 2
= 13a – 6b + 2
b) 2x – 2 – 3x + 2 = 2(2x – 2) – (3x + 2)
2x 4x 4x
= 4x – 3x – 4 – 2
4x
= x – 6
4x
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