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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.
Example
1. 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.
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
Rates can also written in a ratios form.
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
Rate of Exchange
People 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.


COUNTRY
CURRENCY
EQUIVALENT SHILLINGS
United states
Europe
Japan
Britain
Switzerland
Canada
Australia
Kenya
Uganda
South Africa
Soud Arabia
India
Sweden
Zambia
Mozambique
Botswana
1 Dollar
1 Euro
1 Yen
1 Pound stg
1 Franc
1 Dollar
1 Dollar
1 Shilling
1 Shilling
1 Rand
1 Rial
1 Rupee
1 Kronor
1 Kwacha
1 Meticais
1 Pula
1272.50
1720.33
10.02
2513.68
1038.76
1152.48
1049.54
18.525
0.745
181.60
338.695
31.105
186.42
0.317
0.0535
209.85
Examples
1. 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. 186.42
1,000Kronor= ?
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
=T shs. 186420
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
The tourist will receive Tsh. 186420
2. 2. How much 20,600 Tanzania shillings worth in Indian Rupees?
Soln.
1 Rupee = Tsh. 31.105
? = Tsh. 20,600
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
= 662.273 Rupees
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
Variations
Direct Variation
The two variables x and y are said to vary directly of the ratio is constant.
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
The real number K is called the constant of variation.
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
And relationship may be written as EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONSwhich reads as “y is proportional to x
If y varies directly as the square of x, then EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS=Constant.
And can be written as EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONSand the algebraic relation is y=kx2
When having pairs of different corresponding values of x and y, this equation hold true.
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
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 thatEcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS, 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 like
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONSEcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
Examples
If x varies directly as the square of y, and x=4 where y=2, find the value of x when y=8.
Solution
Let x1= 4 , y1 = 2, y2 = 8, x2 is required
But
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS


Inverse variation
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
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=3
Find the value of y when x=18.
Solution.
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
Example 2
3 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.
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
– The number of tailors in directly proportional to the number of clothes.
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS

When t = 5, c= 20, d can be found as
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
It takes 4days for to tailors to sew 20 clothes
JOINT 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 = kEcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS yEcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONSz where k is a fixed real number then x varies jointly as y and z.
Similarly if x1 EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONSy1 EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONSz1 and x2 EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONSy2 EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONSz2 are corresponding values of the variables x, y and z, then x1 = k × (y1 EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONSz1) and x2 = k × (y2 × z2)
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
From these we get
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
Examples 1
1. 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 =4
Solution
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS
Example 2
9 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.
Solution
Let w= workers
h=hours
d=days
It is a joint variation problem and can be written as
EcoleBooks | MATHEMATICS O LEVEL(FORM THREE) NOTES - RATES AND VARIATIONS




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