Form 3 Mathematics – RATES AND VARIATIONS msomimaktaba, November 13, 2018August 17, 2024 RATES AND VARIATIONS RATES:-When sets or quantities of different kinds are related, we use the word rate.i.e 1. A rate of pay of 10,000/= Tsh per hour (money- time)2. The price of juice is 700/= Tsh per litre (money -weight of juice)3. The average speed of 80 kilometres per hour (distance- time)Therefore the rate is the constant relation between two sizes of two quantities concerned.NOTE:Rates deals with the comparison of two quantities of different kinds.Example1. Hiring a car at a charged rate of Tsh 2,000/= per kilometer.(a) A journey of 40 kilometers will cost 40 x Tsh 2,000= Tsh 80,000/=(b) A journey of 100 kilometres, costs 100 x Tsh. 2,000= Tsh.200,000/=If we state the rate we always give two quantities concerned and the unit measurement.E.g: Average speed is written as 100 kilometres per 2 hours or 50 kilometres per one hour.Rates can also written in a ratios form.Rate of ExchangePeople in any country expect to pay and be paid in currency of their own country. It is necessary to exchange the currency of the first country for that of the second, when money is moved from one country to another. i.e: The rate of exchange linked together various currencies of the world, which enable transfer of money and payment for goods to take place between countries.Consider table below shows the exchange rates as supplied by the CRDB bank effective on May 17, 2007.COUNTRYCURRENCYEQUIVALENT SHILLINGSUnited statesEuropeJapanBritainSwitzerlandCanadaAustraliaKenyaUgandaSouth AfricaSoud ArabiaIndia Sweden Zambia Mozambique Botswana1 Dollar1 Euro1 Yen1 Pound stg1 Franc1 Dollar1 Dollar1 Shilling1 Shilling1 Rand1 Rial1 Rupee 1 Kronor 1 Kwacha 1 Meticais 1 Pula1272.501720.3310.022513.681038.761152.481049.5418.5250.745181.60338.69531.105 186.42 0.317 0.0535 209.85 Examples1. 1. A tourist from Sweden wishes to exchange 1,000 Kronors into Tanzanian shillings. How much does she receive?Soln.From the table above 1kron =Tsh. 186.421,000Kronor= ?=T shs. 186420The tourist will receive Tsh. 1864202. 2. How much 20,600 Tanzania shillings worth in Indian Rupees?Soln.1 Rupee = Tsh. 31.105? = Tsh. 20,600= 662.273 RupeesVariationsDirect VariationThe two variables x and y are said to vary directly of the ratio is constant.The real number K is called the constant of variation.And relationship may be written as which reads as “y is proportional to x”If y varies directly as the square of x, then =Constant.And can be written as and the algebraic relation is y=kx2When having pairs of different corresponding values of x and y, this equation hold true.Therefore, we say that x and y vary directly if the ratios of the values of y to the values of x are proportional.NOTE:If x and y represent variables such that, then y=kx,The form of this equation y=kx is similar to y=mx. The graph of y=mx is a straight line passing through the origin, M being the gradient same to the equation y=kx,The graph is a straight line passing through the origin and gradient is k.A sketch is likeExamplesIf x varies directly as the square of y, and x=4 where y=2, find the value of x when y=8.SolutionLet x1= 4 , y1 = 2, y2 = 8, x2 is required ButInverse variation NOTE: The graph does not touch the axis because division by 0 (zero) is impossible.Example 1 If x varies inversely as y, and x=2, when y=3Find the value of y when x=18.Solution.Example 23 tailors are sewing 15 clothes in 5 days. How long would it take for 5 tailors to sew 20 clothes?Solution– Let t = tailors, d = days c= clothes. A number of tailors is inversely proportional to the number of days. – The number of tailors in directly proportional to the number of clothes. When t = 5, c= 20, d can be found asIt takes 4days for to tailors to sew 20 clothesJOINT VARIATION If a quantity is equal to a constant times the product of the two other quantities, then we say that the first quantify varies jointly as the other two quantities. If x = k yz where k is a fixed real number then x varies jointly as y and z. Similarly if x1 y1 z1 and x2 y2 z2 are corresponding values of the variables x, y and z, then x1 = k× (y1 z1) and x2 = k× (y2× z2)From these we getExamples 11. If x varies directly as y and inversely proportional as z and x = 8, when y= 12 and z = 6. Find the value of x when y = 16 and z =4Solution Example 29 workers working 8 hours a day to complete a piece of work in 52 days. How long will it takes 13 workers to complete the same job by working 6 hours a day.SolutionLet w= workersh=hoursd=daysIt is a joint variation problem and can be written as ALL NOTES FOR ALL SUBJECTS QUICK LINKS:AGRICULTURE O LEVEL PURE MATHEMATICS A LEVELBAM NOTES A LEVELBASIC MATH O LEVELBIOLOGY O/A LEVELBOOK KEEPING O LEVELCHEMISTRY O/A LEVELCIVICS O LEVELCOMPUTER(ICT) O/A LEVELECONOMICS A LEVELENGLISH O/A LEVELCOMMERCE O/A LEVELACCOUNTING A LEVELGENERAL STUDIES NOTESGEOGRAPGY O/A LEVELHISTORY O/A LEVELKISWAHILI O/A LEVELPHYSICS O/A LEVELMOCK EXAMINATION PAPERSNECTA PAST PAPERS Basic Mathematics Study Notes Form 3 Basic Mathematics Study Notes FORM 3MATHEMATICSPost navigationPrevious postNext postRelated Posts Basic Mathematics Study Notes Form 4 Mathematics – THREE-DIMENSIONAL FIGURES November 13, 2018August 17, 2024THREE-DEMENSIONAL FIGURES INTRODUCTION A three dimensional figure is a solid figure having three measures. Some examples are pyramids, cylinders, prisms, cubes and cuboids. – Apart from three dimensional figures, there are also one and two dimensional figures. Examples A line is one – dimensional. There is one direction to move… Read More Basic Mathematics Study Notes Form 3 Mathematics – STATISTICS November 13, 2018August 17, 2024STATISTICS Is the study or the methods of collecting, summarizing and presenting data and interpreting the information. MEASURES OF CENTRAL TENDENCY 1. Mean 2. Median 3. Mode MEAN “X” Is obtained by adding up all the data values then divide by the number of characters. I.e. i.e. =mean x1+x2+x3………. Sum… Read More Basic Mathematics Study Notes Form 2 Mathematics – LOGARITHMS November 13, 2018August 17, 2024LOGARITHMS STANDARD NOTATIONS Standard notation form is written in form of A x 10n whereby 1≤ A< 10 and n is any integers Example Write the following in standard form (i) 2380 Solution: 2380 = 2.38 x 103 (ii) 97 Solution: 97 = 9.7 x 101 (iii) 100000 Solution: 100000… Read More Leave a Reply Cancel replyYour email address will not be published. Required fields are marked *Comment *Name * Email * Website Save my name, email, and website in this browser for the next time I comment. Δ
Basic Mathematics Study Notes Form 4 Mathematics – THREE-DIMENSIONAL FIGURES November 13, 2018August 17, 2024THREE-DEMENSIONAL FIGURES INTRODUCTION A three dimensional figure is a solid figure having three measures. Some examples are pyramids, cylinders, prisms, cubes and cuboids. – Apart from three dimensional figures, there are also one and two dimensional figures. Examples A line is one – dimensional. There is one direction to move… Read More
Basic Mathematics Study Notes Form 3 Mathematics – STATISTICS November 13, 2018August 17, 2024STATISTICS Is the study or the methods of collecting, summarizing and presenting data and interpreting the information. MEASURES OF CENTRAL TENDENCY 1. Mean 2. Median 3. Mode MEAN “X” Is obtained by adding up all the data values then divide by the number of characters. I.e. i.e. =mean x1+x2+x3………. Sum… Read More
Basic Mathematics Study Notes Form 2 Mathematics – LOGARITHMS November 13, 2018August 17, 2024LOGARITHMS STANDARD NOTATIONS Standard notation form is written in form of A x 10n whereby 1≤ A< 10 and n is any integers Example Write the following in standard form (i) 2380 Solution: 2380 = 2.38 x 103 (ii) 97 Solution: 97 = 9.7 x 101 (iii) 100000 Solution: 100000… Read More