Rates, Ratio and Percentages Questions and Answers - Form 1 Topical Mathematics

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Questions

If 5 men can erect 2 cottages in 21 days, how many more men, working at the same rate will be needed to erect 2 cottages in the same period?
The length and width of a rectangular paper were measured to the nearest centimeter and found to be 18cm and 12cm respectively. Find the percentage error in its perimeter.
1. Two pipes A and B can fill a tank in 3hrs and 4 hrs respectively. Pipe C can empty the full tank in 6 hrs.
  1. How long would it take pipes A and B to fill the tank if pipe C is closed?
  2. Starting with an empty tank, how long would it take to fill the tank with all pipes running?
2. The high quality Kencoffee is a mixture of pure Arabica coffee and pure Robusta coffee in the ratio 1 : 3 by mass. Pure Arabica coffee costs shs. 180 per kg and pure Robusta coffee costs sh 120 per kg. Calculate the percentage profit when the coffee is sold at sh 162 per kg.
A number of nurses working at Sotik Health Centre decided to raise shs.144,000 to buy a plot of land. Each person was to contribute the same amount. Before the contributions were collected five of the nurses retired. This meant that the remaining contributors had to
pay more to meet the target.
1. If there were n nurses originally, find the expression of the increase in contribution per person
2. If the increase in the contribution per person was shs.2,400, find the number of nurses originally at the health centre
3. How much would each person have contributed to nearest shilling if the 5 people had not retired.
4. Calculate the percentage increase in the contribution per person because of the retirement
3 taps X,Y and Z can fill a tank in 40 hours, 15 hours and 20 hours respectively.
The three taps are turned on at 8.00a.m when the tank is empty for five hours then Z is turned off. After two hours tap Y is turned off. Work out ;-
1. The proportion of water in the tank after seven hours
2. The proportion of water in the tank after seven hours
3. The time the tank will be completely full
Jane and Philip working together can do a piece of work in 6 days. Jane working alone takes 5 days longer than Philip. How many days does it take Philip to do the work alone?
Sixteen men working 9 hours a day can complete a piece of work in 14 days. How many more men working 7 hours a day would complete the same job in 12 days?
A group of people planned to contribute equally towards buying land at a price of shs.180000. However 3 members of the group withdrew from the project. As a result, each of the remaining members were to contribute kshs.3000 more.
1. find the original number of members in the group
2. How much would each person have contributed if the 3 people had not withdrew
3. Calculate the percentage increase in the contribution per person caused by the withdrawal
Kori and Mue decided to start a business. Korir contributed shs.40,000 and Mue shs.64000. The two men agreed that in any year, 15% of the profit shall be divided equally between them and 20% of the profit will be used to meet the cost of running the business the following year. They also agreed to share the rest of the profit in the ratio of their contributions. The profit made after the first year was shs.43200.
1. How much did they set aside towards the cost of running the business for the second year?
2. How much did Mue receive at the end of the first year?
3. Korir bought cows with his share of the profit. If each cow cost shs.1800, how many cows did he buy?
Given the ratio x : y = 2:3, find the ratio (7x – 3y) : (2x + 3y)
Abdul bought five bulls and thirty goats at an auction spending a total of Kshs.117000. His friend Ali bought four bulls and twenty five goats at the same auction and spent Kshs.22,250 less.
1. Find the cost of each animal at the auction
2. Abdul later sold all his animals at a profit of 40% per bull and 30% per goat. Ali sold all his animals at a profit of 50% per bull and 40% per goat. Determine who made more profit and by how much?
The cost of providing a commodity consists of transport, labour and raw material in the ratio 8:4:12 respectively. If the transport cost increases by 12% labour cost 18% and raw materials by 40%, find the percentage increase of producing the new commodity
A mother is now 2½ times as old as her daughter Mary; four years ago the ratio of their ages was 3:1. Find the present age of the mother
Sixteen men working at the rate of 9hrs a day can complete a piece of work in 14 days. How many more men working at the rate of 7 hours a day would complete the same job in 12 days.
Two business partners, Kago and Beatrice contributed 90, 000/= and 120,000/= in order to start a business. They agreed that 25% of the profit made after end of the year will be put back into the business. They also estimated that 40% of the profit will cover salaries and other expenses for the year. The remainder would be shared between the partners in the ratio of their contributions. At the end of the first year the business realized a gross profit of shs.181,300.
1. Calculate how much each received after end of the year.
2. At the end of 2nd year the business realized the same gross profit as the previous year and the partners decided to dissolve the business and share everything. Determine how much
  money each received.
A number is such that the product of its digits is 24. When the digits are reversed, the number so formed exceeds the original number by 27. Find the number.
The radius of a cylinder is increased by 30% while its height is decreased by 20%. Find the percentage change in the volume of the cylinder
Tap A fills a tank in 6 hours, tap B fills it in 8 hours and tap C empties it in 10 hours. Starting with an empty tank and all the three taps are opened at the same time, how long will it take to fill the tank?
Sixteen men working 9 hours a day can complete a piece of work in 14 days. How many more men working 7 hours a day would complete the same job in 12 days?
Three businessmen Langat, Korir and Koech contributed shs.160,000, Shs.200,000 and shs.240,000 respectively and started a business. They agreed that 30% of the profit each year will go to expenses, 15% of the reminder would go back to the business. The rest of the profit would be shared in the ratio of their contribution. At the end of the first year, the business realized a profit of kshs.60,000.
Calculate how much;
1. 1. Langat received
  2. Korir received
  3. Koech received
2. Express what Korir received as a percentage of the total profit
The price of a book is increased by 25%.
1. In what ratio has the price increased?
2. What is the new price if the book was shs.400 before the change?
1. A chemist added 120 liters of a solution A containing 25% alcohol to 180 liters of solution B containing 20% alcohol. What percentage of the resulting solution in alcohol?
2. He removed X liters of resulting mixture and added an equal amount of pure alcohol to the resulting mixture. If the new mixture contains 22% of the alcohol, find the value of X
The length and width of a rectangular paper were measured to the nearest centimeter and found to be 18cm and 12cm respectively. Find the percentage error in its perimeter.
Given that a:b = 1: 2 and b:c = 3:4. Find a:b:c

Answers

Max Perimeter = 2( 18.5 + 12.5)
= 62 cm
Working Perimeter = 2(18 +12)
= 60 cm
% error = ²/₆₀ X 100 = 3.33%
1. 1. In 1 hr; Tap A fills ¹/₃ B - ¹/₄Capacity filled in 1 hr = ¹/₃ + ¹/₄
    =⁷/₁₂
    ⁷/₁₂ = 1 hr
    1 = 1 x 1 x ¹²/₇
    = 1⁵/₇ hrs.
  2. ¹/₃ + ¹/₄ - ¹/₆ = ⁵/₁₂ ⇒ in one hr
    ⁵/₁₂ = 1hr
    1 = 1 x 1 x ¹²/₅
    = 2²/₅ hrs
1. In 1 hour ¹/₄₀ + ¹/₁₅ + ¹/₂₀ of the tank will be filled
  = ¹⁷/₁₂₀
  In 5 hours = ¹⁷/₁₂₀ x 5
  =¹⁷/₂₄
2. In two hours taps x and y
  (¹/₄₀ + ¹/₁₅) x 2 of the tank to be filled
  = ¹¹/₆₀
  In 7 hours = ¹¹/₆₀ + ¹⁷/₂₄
  =¹⁰⁷/₁₂₀
3. Remaining fraction = 1 – ¹⁰⁷/₁₂₀
  = ¹³/₁₂₀
  In ¹/₄₀ hour proportion, time taken
  = ¹³/₁₂₀ x 40h
  = 4 ¹/₃
  Time taken = 7 + 4¹/₃ = 11 hrs 20 min.
  Tank will be full at 8.00 + 11hrs 20 min
  1920 hrs or 7.20 p.m
Let Philip take x days to finish the job alone.
¹/_x + ¹/_(x+5) = ¹/₆
6 (x + 5) 6x = x(x + 5)
6x + 30 + 6x = x² + 5
x² - 7x – 30 = 0
(x – 10) (x + 3) = 0
x = 10 and x = -3
16 9 14
X 7 12
X = 16 x ⁹/₇ x ¹⁴/₁₂
= 24men
Extra men = 24 -6
= 8men
1. Let the original no. of people be x
  Originally each would contribute
  ¹⁸⁰⁰⁰⁰/_x
  New contribution per person
  ¹⁸⁰⁰⁰⁰/_{(X -3)}
  
  ¹⁸⁰⁰⁰⁰/_{(X – 3)} - ¹⁸⁰⁰⁰⁰/_x = 3000
  180000x – 180000x + 540000 = 30000 – 9000
  30x² – 90x – 5400 = 0
  3x² – 9x – 540 = 0
  X² – 3x – 180 = 0
  (x- 15) (x + 12) = 0
  X = 15 or -12
  Original number of people 15
2. ¹⁸⁰⁰⁰⁰/₁₅ = ¹⁸⁰⁰⁰⁰/₁₅
3. Original contribution per person
  Shs.12000
  New contribution per person
  = ¹⁸⁰⁰⁰⁰/₁₂ = 15000
  % increase
  ^{(15000 – 12000)}/₁₂₀₀₀ X 100%
  ³⁰⁰⁰/₁₂₀₀₀ X 100%
  = 25%
1. cost of running the business
  ²⁰/₁₀₀ X 43200
  = Shs.8640
2. 15% of profit
  ¹⁵/₁₀₀ X 43200 = Shs. 6480
  Rest of the profit
  = 43200 – (8640 + 6480) = 28080
  Ratio of contribution
  40000 : 64000
  5 : 8
  Mue received
  ¹/₂ X 6480 = Shs.3240
  ⁸/₁₃ X 28080 = Shs. 17280
  = Shs.20320
3. Konie received
  Shs.3240 + 10800 = 14040
  ¹⁴⁰⁴⁰/₁₈₀₀ = 7.8
  1800
  = 7 cows
(7x – 3y) : 2x + 3y
x= 2 : y= 1
14 – 9 : 4+ 9
5 : 13
1. B - bulls
  G - Goats
  5B + 30G = Kshs.117000 ………. Equation (i)
  4B + 25G = Kshs.(117000 – 22250)
  4B + 225G = Kshs.94750 ………… Equation (ii)
  From equation (i) 5B + 30G = Kshs.117000 (dividing through by 5)
  = (B + 6G = 23400) x 4
  = 4B + 24G = 93600 …………..(iii)
  Equation (ii) – q(iii) = 4B + 24G = 94750 -
  4B + 24G = 93600
  G = 1150
  1 goat costs Kshs.1150
  Substituting in (i)
  5B + 30 (1150) = 117000
  5B + 34500 = 117000
  5B = 825000
  B = Kshs.16500
2. Abduls selling price
  Bull ¹⁴⁰/₁₀₀ x 16500 = 23100 x 5 = Kshs.115,500
  Goat ¹³⁰/ ₁₀₀ x 1150 = 1495 x 30 = Kshs.44850
  Total 44850 + 115500 = Kshs.160350
  = Kshs.160350
  Ali’s selling price
  Bulls ¹⁵⁰/₁₀₀ x 16500= 24750 x 4 = Shs.99000
  Goats ¹⁴⁰/₁₀₀ x 1150 = 1610 x 25 = Shs.40250
  Total 99000 + 40250 = Kshs.139,250
  Profit made
  Abdul - Kshs. (160350 – 117000) = Kshs.43350
  Ali - Kshs. (139250 – 94750) = Kshs.44500
  Ali made more profit by Kshs.1150/=
Original costs
T = ⁸/₂₄ x = ^x/₃
L= ⁴/₂₄x = ^x/₆
R= ¹²/₂₄x = ^x/₂
New T = ^x/₃ x 1.12 = 0.3733x
L = ^x/₆ x 1.18 = 0.1967x
R = ^x/₂ x 1.4 = 0.7x
Therefore % change
^{(0.3733x + 0.967x + 0.7x)}/_x – x x 100
= 0.27 x 100
= 27%
Let Mary’s yrs be x
Mothers age = 2¹/₂x
4yrs ago Mary was x – 4
4yrs ago mother was 2¹/2x – 4
^{(21/₂x – 4)}/_{(x – 4)} = ³/₁
⁵/₂x – 3x = -12
- ¹/₂x = -12
x = 24yrs
mother’s age is =(⁵/₂ x 24)
= 60yrs
16 x 9 x 14
7 x 12
= 24
Extra men = 24 – 16
B1= 8 more men
Ratio K : B = 3 : 4
1. Kongo got ³/₇ x ³⁵/₁₀₀ x 181300 = 27195/=
  Beatrice got ⁴/₇ x ³⁵/₁₀₀ x 181300 = 36260/=
2. Kongo got ³/₇ x ⁶⁰/₁₀₀ x 181300 + 9000
  = 136,620/=
  Beatrice got ⁴/₇ x ⁶⁰/₁₀₀ x 181300 + 120000
  = 182,160/=
Let no. be mn
M + n = 9...(i)
10m + n, reversed 10n + m
10n + m – 10m + n = 27
1n – 9m
V₁ = Πr²
h
R = 130r = 1.3r
H = ^80h/₁₀₀ = 0.8h
V₂ = ΠR²h = (1.3r)² x 0.8h
= 1.352V₁
% change = ^{(V₂ – V₁)}/_V1 x 100
= ^{(1.352 – 1) V₁}/_V1 x 100
0.352 x 100 = 35.2%
In 1hr both fills = 1 + 1 -10 = 23
Tina to fill = 120 = 5 5/23
5hrs 13min
16 9 14
X 7 12
X = 16 x ⁹/₇ x ¹⁴/₁₂
= 24men
Extra men = 24 -6
= 8men
1. Expenses = ³⁰/₁₀₀ x 600,000
  = sh. 180,000
  Business = ¹⁵/₁₀₀ x 420,000
  = sh. 63,000
  Rest of profit = 357,000
  Ratio 160 : 200 : 240
  4 : 5 : 6
  1. Langat received = sh ⁴/₁₅ x 357,000
    = sh 95,200
  2. Korir received = sh ⁵/₁₅ x 357,000
    = sh119,000
  3. Koech received = sh ⁶/₁₅ x 357,000
    = 142,800
2. % = 119, 000 x 100
  600,000
  = 19.83
1. 125 :100 = 5:4
2. ⁵/₄ x 400 = 500
Alcohol A = ²⁵/₁₂₀
= 30cm³
Alcohol in B = ²⁰/₁₀₀ x 180
= 36cm³
Results = ^{(36 + 30)}/_{(120 + 180)}
= ⁶⁶/₃₀₀ x 100
=22%
Remaining = 300-x
Volume of alcohol = (300 –x) x ²²/₁₀₀ = 66-0.22x
Total volume of alcohol = 66-0.22x + x
= 66 + 0.78x
% alcohol = ^{66 + 0.78x x 100}/₃₀₀ = 35
= 66 + 0.78x = 105
0.78x=39
x=50
Max Perimeter = 2( 18.5 + 12.5)
= 62 cm
Working Perimeter = 2(18 +12)
= 60 cm
% error = ²/₆₀ X 100 = 3.33%
a :b = 1:2
b : c = 3:4
a:b = 3:6
b:c = 6:8
 a:b:c = 3:6:8