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.

1. RATES AND VARIATIONS
RATES:-
When sets or quantitiesof different kinds arerelated, we use the word rate.
i.e 1. A rateof pay of 10,000/=Tsh per hour (money- time)
2. The priceof juiceis 700/=Tsh per litre(money -weight of juice)
3. The average speed of 80 kilometresper hour (distance- time)
Thereforethe rateis the constant relationbetweentwo sizes of two
quantitiesconcerned.
NOTE:
Rates deals with the comparisonoftwo quantitiesof different kinds.
Example
1. Hiring a car at a charged rateof Tsh 2,000/=per kilometer.
(a) A journey of 40 kilometerswill 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 statethe ratewe always give two quantitiesconcerned and theunit
measurement.
E.g: Averagespeed is writtenas100 kilometresper 2 hours or 50
kilometresper one hour.
Rates canalso writtenin a ratiosform.

2. Rate of Exchange
People in any countryexpect to pay and be paid in currencyof their own
country. It is necessary to exchangethecurrencyof the first countryfor that
of thesecond, when money is moved from one country to another.
i.e: The rateof exchangelinked together variouscurrenciesofthe world,
which enable transfer of money and payment for goods to take place
betweencountries.
Consider tablebelow shows the exchangeratesas supplied by the CRDB
bankeffective on May 17, 2007.
COUNTRY CURRENCY EQUIVALENT
SHILLINGS
United states
Europe
Japan
Britain
Switzerland
Canada
Australia
Kenya
Uganda
South Africa
Soud Arabia
India
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
1272.50
1720.33
10.02
2513.68
1038.76
1152.48
1049.54
18.525
0.745
181.60
338.695
31.105

3. Sweden
Zambia
Mozambique
Botswana
1 Kronor
1 Kwacha
1 Meticais
1 Pula
186.42
0.317
0.0535
209.85
Examples
1. 1. A tourist from Sweden wishes to exchange1,000 Kronorsinto
Tanzanianshillings. How much does she receive?
Soln.
From the tableabove
1kron =Tsh. 186.42
1,000Kronor=?
=T shs. 186420
The tourist will receiveTsh. 186420
2. 2. How much 20,600 Tanzaniashillingsworth in IndianRupees?
Soln.
1 Rupee = Tsh. 31.105
? = Tsh. 20,600
= 662.273Rupees

4. Variations
Direct Variation
The two variablesx and y aresaid to vary directlyof the ratioisconstant.
The real number K is called the constant of variation.
And relationship maybewrittenas which readsas “y is proportional
to x”
If y variesdirectlyas the squareof x, then =Constant.
And canbe writtenas and the algebraic relationisy=kx2
When having pairsof different corresponding values of x and y, this
equationhold true.
Therefore, we say that x and y vary directlyif the ratiosof the values of y to
the values of x areproportional.
NOTE:
If x and y represent variablessuch that , theny=kx,
The form of this equationy=kx is similar toy=mx. The graph of y=mx is a
straight linepassing through theorigin, M being thegradient sameto the
equationy=kx,
The graph is a straightlinepassing through the originand gradient isk.

5. A sketch is like
Examples
If x variesdirectlyas the squareof 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
Inverse variation

7. NOTE: The graph does not touch the axisbecausedivisionby 0 (zero) is
impossible.
Example 1
If x variesinversely as y, and x=2, when y=3
Find the value of y when x=18.
Solution.

8. Example 2
3 tailorsare sewing 15 clothesin 5 days. How long would it take for 5
tailorsto sew 20 clothes?
Solution
- Let t = tailors, d = days c= clothes.
A number of tailorsis inversely proportionaltothe number of days.
- The number of tailorsin directlyproportionaltothe number of clothes.

9. When t = 5, c= 20, d can be found as
It takes4days for to tailorsto sew 20 clothes
JOINT VARIATION
If a quantityisequalto a constant timestheproduct of the twoother
quantities, thenwe say that the first quantifyvariesjointlyas the other two
quantities.
If x = k y z where k is a fixed real number then x variesjointly as y and z.
Similarlyifx1 y1 z1 and x2 y2 z2 arecorresponding valuesof the
variablesx, y and z, then x1 =k × (y1 z1 ) and x2 = k × (y2 × z2)
From these we get

10. Examples 1
1. If x varies directlyas y and inversely proportionalasz and x = 8, when y=
12 and z = 6. Find the value of x when y = 16 and z =4
Solution
Example 2

11. 9 workers working 8 hours a day to completea pieceof work in 52 days.
How long will it takes13 workers to completethe samejob by working 6
hours a day.
Solution
Let w= workers
h=hours
d=days
It is a joint variationproblem and canbe writtenas