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RATIONAL AND NON-RATIONAL NUMBERS AND COMPOUND INTEREST
RATIONAL AND NON-RATIONAL NUMBERS
Numbers which can be written as exact fractions or ratios in the form are called rational numbers. For example, we can write these numbers as
In addition, rational numbers are also numbers that can be written as recurring decimals, for instance: is equivalent respectively to the following:
We Numbers which cannot be written as exact fractions or recurring decimals are called non-rational numbers. Examples of non-rational numbers are
SQUARE ROOTS
Since rational numbers are not perfect squares, so their square roots cannot be obtained easily except by trial and error method or by the use of Table of Square Roots in the four-figure table.
Example 1:
Find to three significant figures by the use of tables.
Solution:
gives from the table. Hence, answer is to .
Example2:
Find to the nearest tenth by the use of tables.
Solution:
is equivalent to . This is equal to . We can now look up from the table to give . So that .
Hence, answer is to the nearest tenth.
EVALUATION
- Which of the following is an irrational number?
- Which of the following is a rational number?
- Find the square root of , leaving your answer in one decimal place.
DIRECT AND INVERSE VARIATION
DIRECT VARIATION
This is used to describe quantities which vary in proportions to each other, such that as one increases the other increases, and as one decreases the other decreases. Thus, if P varies directly as R, then the expression symbolically becomes . The expression can now be written in equation form as
Where has been replaced by is a constant of variation. It can also be expressed as
The equation is the equation of variation.
Example 1:
If varies directly as the square of , find the law of variation between given that when Find the value of when and the value of
Solution:
and the law of variation becomes
For substitution gives .
Then .
For substitution gives
such that
then
GRAPHICAL REPRESENTATION OF DIRECT VARIATION
Data collected from quantities that vary directly can be represented graphically. This will give a straight line graph through the origin as shown below.
Example 2:
Given that distance varies directly with time, consider the table below and plot a graph for such relationship.
| Distance | 5 | 10 | 15 | 20 | 25 |
| Time | 1 | 2 | 3 | 4 | 5 |
Solution:
EVALUATION
- varies directly as and when Find when
- If increases by from question find the percentage change in .
INVERSE VARIATION
This variation means that related quantities vary inversely or as reciprocal to each other. Hence as one increases the other decreases; and as one decreases, the other increases. Thus if T varies inversely as S, symbolically this is written as T∝1/S.The expression can now be written in equation form asT=K/S.
Where ∝ has been replaced by “=and K”.K is a constant of variation. It can also be expressed as
K=TS
The equation T=K/S is the equation of variation.
Example 3:
Given that T is inversely proportional to S, and thatT=2 when S = 60, find the (a) relationship between T and S. (b) value of T when S=90.
Solution:
T∝1/S such that T=K/S and K=TS=2(60)=120.
(a) T=120/S is the required relationship between T and S. (b) T=120/90=4/3=1 1/3
EVALUATION
The current I in in an electric circuit varies inversely with the resistance R. If a current of 10A is produced by a resistance of 20Ω, what current will be produced by a resistance of 80Ω?
Find the percentage change in the current from question (1) if the resistance is decreased by 10%
GRAPHICAL REPRESENTATION OF INVERSE VARIATION
The graph here will not be a straight line from the origin instead it will give us a curve.
Speed 80 40 20 10 5
Time 0.5 1 2 4 8
Example 4: Given that speed (S) varies inversely to time (t), use the below table to plot a graph of an inverse relationship between S and t.
Solution:
GENERAL EVALUATION
Factorize the expression p^2-6p+16
Factorize a^2-8a-a+8
What is the value of the digit 5 in the734.95?
What is the highest common factor of 8,9 and 12?
Simplify 7a-(4b+3c)-3b
READING ASSIGNMENT
Essential Mathematics for J.S.S. 3 by Oluwasanmi A.J.S. 2014 edition; Pages 49-53
Essential Mathematics Workbook for J.S.S. 3 by Oluwasanmi A.J.S.; Exercise 7.1, numbers1-5
WEEKEND ASSIGNMENT
If x∝y and x=2 when y=4, find the value ofxwheny=8. A. 2 B. 4C. 6 D. 8
x∝1/y and x=3 when y=4, find the value of y when x=6. A. 2 B. 4 C. 5 D.6
If p varies directly as q and p=5,q=10, what is value of p when q=40?
A. 20 B. 10 C. 5 D.6
m∝1/n and m=4 when n=120. Find the relationship between m and n.
A. m=120/n B. m=480/n C. n=480/m D. m = n/480
Find the value of m whenn = 80. A. 60 B. 120 C. 48 D.84
THEORY
R∝h and R=5 when h=12, find (a) h when R=45 (b) the percentage change in R if h increases by 25%.
When repaying a loan, the number of monthly payments, m, varies inversely with the amount of each payment, Na. The loan can be repaid by 10 monthly payment of N1350. Find the formula which connectsm and a. Hence find how long it takes to repay the loan with monthly payments of N750.
JOINT AND PARTIAL VARIATION
JOINT VARIATION
Joint variation is obtained when a quantity varies with more than one other quantity either directly and/or inversely. For instance, P is jointly proportional to both Q and G as inP∝QG. Also, H is directly proportional to Y and inversely proportional to M as inH∝Y/M.
Example 1:
If H∝Y/M .When H=42,Y=7 and M=3.
Find the relation between H,Y and M.
Find H when Y=5 and M=9.
Solution:
H∝Y/M and H=K Y/M
After substituting, we have K=HM/Y=(42 X 3)/7=18
The relation between them is given by H=18 Y/M
H=18 Y/M=18 X 5/9=10
Example 2:
The universal gas law states that the volume V(m^3 ) of a given mass of an ideal gas varies directly with its absolute temperature T(K) and inversely with its pressure P(N/m^2 ).A certain mass of gas at an absolute temperature 425K and pressure 1000N/m^2 has a volume0.255m^3. Find:
the formula that connects P,V and T.
the pressure of the gas when its absolute temperature is 720K and its volume is 0.018m3
Solution:
V∝T/P and V=K T/P , such that K=PV/T
Substituting the values, K becomes K=(1000 X 0.255)/425=255/425=51/85=3/5
and the relationship is V=3T/5P
0.018=(3 X 720)/(5 X P)
P=(3 X 720)/(5 X 0.018)=(3 X 720000)/(5 X 18)=(3 X 40000)/5=3 X 8000=24,000N/m^2
EVALUATION
Suppose Z∝x^2 y. When x=3,y=2 and Z=36. Find Z when x=4 and y=2 1/16.
Find the percentage change in Z when x increases by 20% and y decreases by 10%.
PARTIAL VARIATION
Partial variation problems occur everywhere around us. Some examples are described below:
When a hairdresser makes hair, the money he/she charges M, is dependent on both the cost of the wool (thread or weavon in some cases) C which is constant, and on the time T, taken to make the hair. The less the weaves, the less the time it will take to complete and the less the charges. We can write a partial equation for this as: M=C+bT, where C and b are constants.
Domestic electricity prepaid meter bills are prepared on two components which are N750 rental charge (independent of the amount of power consumed) and consumption charges (dependent on the quantity of power consumed). We can also write the total bill T in partial equation as: T=750+bT, where N750 and b are constants depending on the customer.
Thus, partial variation statements can come in these formats described below:
W is partly constant and partly varies as G is interpreted as W=a+bG
V varies partly as P and partly inversely as √Q can also be interpreted as V=aP+b/√Q
In these cases, a and b are constants that can be obtained simultaneously.
Example 3:
xis partly constant and partly varies as the square of y. Write an equation connecting x and y. Given that when x=3,y=4 and when x=1,y=5. Write down the law of variation. Find x when y=2.
Solution:
The equation connecting x and y is x=a+by^2, where a and b are constants.
Whenx=3,y=4, 3=a+b〖(4)〗^2 becomes
3=a+16b—–equation (i)
When x=1,y=5, we have 1=a+b〖(5)〗^2 becomes
1=a+25b—–equation (ii)
Combining the two equations and solving simultaneously,
a+16b=3
a+25b =1
Subtracting: -9b=2 and b=(-2)/9
Substitute for b=(-2)/9 into equation (i), so that a+16((-2)/9)=3
And a=3/1+32/9=(27+32)/9=59/9. The law of variation becomesx=59/9-2/9 y^2
When y=2,x becomes x=59/9-2/9 (2)^2=59/9-8/9=51/9
Example 4:
T varies as partly as V and partly as the cube of V. When T=30,V=2 and when T=15,V=3. Write the law connecting T and V. Find T when V=4.
Solution:
The equation connecting T and V is T=aV+bV^3, where a and b are constants.
when T=30,V=2, 30=2a+b〖(2)〗^3 becomes
30=2a+8b —–equation (i)
when T=15,V=3, 15=3a+b〖(3)〗^3becomes
15=3a+27b—–equation (ii)
Combining the two equations and solving simultaneously to eliminate a,
30=2a+8b—–equation (i)X 3
15=3a+27b—–equation (ii) X 2
6a+24b=90
6a+54b=30
Subtracting: -30b=60 and b=60/(-30)=-2
Alternatively, dividing through equation (i) by 2, gives 15=a+4b and dividing through equation (ii) by 3, gives 5=a+9b.
Then,
a+4b=15
a+9b=5
Subtracting: -5b=10
And b=-2 as obtained above.
Substitute for b=-2into equation (i), so that 30=2a+8(-2) and 30=2a-16
So that a=46/2=23. The law of variation becomes T=23V-2V^3
When V=4,T becomes T=23(4)-2(4)^3=23(4)-2(64)=92-128=-36
Example 5:
The cost in naira of making a dress is partly constant and partly varies with the amount of time in hours it takes to make the dress. If the dress takes 3 hours to make, it costs N2700, and if it takes 5 hours to make the dress, it costs N3100. Find the cost if it takes 1 1/2 hours to make the dress.
Solution:
Using C and T to represent the cost and time respectively, we can proceed by writing C=a+bT
From first statement: 2700=a+3b
From second statement: 3100=a+5b
Solving the simultaneously,
a+3b=2700
a+5b=3100
Subtracting: -2b=-400 and b=(-400)/(-2)=200
Substitute for b=200into equation (ii), so that 3100=a+5(200)
and 3100=a+1000.
So that a=2100. The law of variation becomes C=2100+200T
If it takes 1 1/2 hours to make the dress, the cost becomes C=2100+200(1.5)=2100+300=N2400
EVALUATION
Z varies partly directly with x and partly varies inversely with y. When Z=4,x=3,y=1 and whenZ=3,x=0.5,y=5.. Find Z when x=29,y=10
An examination fee is partly constant and partly varies with the number of subjects entered. When the examination fee is N800, three subjects are entered. When the fee is N1200, five subjects are entered. Find the number of subjects entered if the fee is N1400.
GENERAL EVALUATION
Express 954Kg in tonnes.
Express 0.35 in fraction in its lowest term.
What is the sum of Nx and y kobo expressed in kobo?
Factorize 〖25x〗^2-1
A trader gives 8% discount on an article in his kiosk marked N1250.00. How much would a customer pay on such article?
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