Rational And Non-Rational Numbers And Compound Interest JSS3 Mathematics Lesson Note

TOPIC: RATIONAL AND NON-RATIONAL NUMBERS AND COMPOUND INTEREST

RATIONAL AND NON-RATIONAL NUMBERS

Numbers that can be written as exact fractions or ratios in the form of PQ are called rational numbers. For example, we can write these numbers  8, 312,15,0.31,169, 0.1  as     81, 72, 15, 31100, 43, 110.

 In addition, rational numbers are also numbers that can be written as recurring decimals, for instance: 1799, 411, and 103 are equivalent respectively to the following:0.17171717, 0.36363636, 3.333333, etc.

 We can also write recurring decimals as 0.17, 0.36, 0.3. Numbers that cannot be written as exact fractions or recurring decimals are called non-rational numbers. Examples of non-rational numbers are 7=2.645751, 17=4.1231056256, 29=5.38516480713.

 SQUARE ROOTS

Since rational numbers are not perfect squares, their square roots cannot be obtained easily except by trial and error method or by the use of the Table of Square Roots in the four-figure table. 

Example 1:

Find 17  to three significant figures by the use of tables.

Solution:

17 gives 4.123 from the table. Hence, the answer is 4.12 to 3 s.f.

 Example2:

Find 293  to the nearest tenth by the use of tables.

Solution:

293 is equivalent to 2.93 X 100. This is equal to 2.93 X 100. We can now look up 2.93 from the table to give 1.712. So 293=2.93 X100=2.93 X 100=1.712X10=17.12.

Hence, the answer is 17.1 to the nearest tenth.

 EVALUATION

1.         Which of the following is an irrational number? a 0.243⃛b23c11
2.         Which of the following is a rational number? a 0.24578343 b5-15c11
3.         Find the square root of 7562, leaving your answer in one decimal place.

DIRECT AND INVERSE VARIATION

DIRECT VARIATION

This is used to describe quantities that vary in proportion 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 P∝R. The expression can now be written in equation form as 

P=KR

Where  P has been replaced by K “=and K”.K is a constant of variation. It can also be expressed as

K=PR

The equation P=KR is the equation of variation.

Example 1:

If p varies directly as the square of q, find the law of variation between p and q given that p=27 when q=3. Find the value of p when q is 2 and the value of q when p is 48.

Solution:

p∝q2     ∴p=kq2

k=pq2=2732=279=3 and the law of variation becomes p=3q2

For q=2,substitution gives p=3q2=322=3 X 4=12.

Then   p=12.

For p=48, substitution gives 48=3q2

such that   q2=483=16

thenq=16=4

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 a relationship.

Solution:

EVALUATION

1.         D varies directly as P and D=0.2 when P=10. Find D when P=18.
2.         If P increases by 30% from question 1, find the percentage change in D.

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 to S, symbolically this is written as T∝1S. The expression can now be written in equation form as T=KS.

Where T  has been replaced by “=and K”.K is a constant of variation. It can also be expressed as

K=TS

The equation T=KS is the equation of variation.

Example 3:

Given that T is inversely proportional to S, and that T=2 when S = 60, find the (a) relationship between T and S. (b) value of T when S = 90.

Solution:

T∝1 Such that  T=KS  and K=TS=260=120.

(a) T=120S is the required relationship between T and S. (b) T=12090=43=113

EVALUATION

1.         The current I 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?
2.         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.

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

1.         Factorize the expression p2-6p+16
2.         Factorize a2-8a-a+8
3.         What is the value of the digit 5 in the 734.95?
4.         What is the highest common factor of 8,9 and 12?
5.         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, numbers 1-5

WEEKEND ASSIGNMENT

1.         If x∝y and x=2 when y=4, find the value of x when y=8.  A. 2 B. 4C. 6  D.  8
2.         X∝1y and x=3 when y=4, find the value of y when x=6. A. 2 B. 4 C. 5 D.6
3.         If p varies directly as q and p=5,q=10, what is the value of p when q=40? 
4.         20 B. 10 C. 5 D.6
5.         M∝1n and m=4 when n=120. Find the relationship between m and n. 
6.         m=120nB. m=480nC. n=480m D. m =  n480
7.         Find the value of m when n = 80.  A.  60 B. 120 C. 48  D.84

THEORY

1.         R∝h  and R=5 when h=12, find (a) h when R=45 and (b) the percentage change in R if h increases by 25%.
2.         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 payments of N1350. Find the formula that connects m 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 in P∝QG. Also, H is directly proportional to Y and inversely proportional to M as in  H∝YM.

Example 1:

If  H∝YM. When H=42,Y=7 and M=3.

1.         Find the relation between H, Y, and M.
2.         Find H when Y=5 and M=9.

Solution:

1.         H∝YM and     H=KYM

After substituting, we have   K=HMY=42 X 37=18

The relation between them is given by H=18YM

1.         H=18YM=18 X 59=10

Example 2:

The universal gas law states that the volume Vm3 of a given mass of an ideal gas varies directly with its absolute temperature TK and inversely with its pressure PN/m2. A certain mass of gas at an absolute temperature of 425K and pressure of 1000N/m2 has a volume of 0.255m3. Find:

1.         the formula that connects P, V, and T.
2.         the pressure of the gas when its absolute temperature is 720K and its volume is 0.018m3 

Solution:

1.         V∝TP and V=KTP,    such that   K=PVT

Substituting the values, K becomes   K=1000 X 0.255425=255425=5185=35

and the relationship is  V=3T5P

1.         0.018=3 X 7205 X P

P=3 X 7205 X 0.018=3 X 7200005 X 18=3 X 400005=3 X 8000=24,000N/m2

EVALUATION

1.         Suppose Zx2y. When x=3, y=2 and Z=36. Find Z when x=4 and y=2116.
2.         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 the 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 a partial equation as T=750+bT, where N750 and b are constants depending on the customer.

Thus, partial variation statements can come in the 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+bQ

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+by2, where a and b are constants.

When X=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 them simultaneously,

a+16b=3

 a+25b =1

Subtracting:        -9b=2 and b=-29

Substitute for  b=-29 into equation (i), so that     a+16(-29)=3

anda=31+329=27+329=599. The law of variation becomes x=599-29y2

When   y=2,x becomes      x=599-2922=599-89=519

Example 4:

Tvaries 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+bV3, 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)3 becomes

    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=-2 into equation (i), so that     30=2a+8(-2)   and   30=2a-16

So that a=462=23.   The law of variation becomes    T=23V-2V3

When   V=4,T becomes      T=234-243=234-264=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 112 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 the first statement:     2700=a+3b

From the 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=200 into  equation (ii), so that     3100=a+5(200)

and 3100=a+1000.

So that is a=2100.   The law of variation becomes     C=2100+200T

If it takes 112 hours to make the dress, the cost becomes C=2100+2001.5=2100+300=N2400

EVALUATION

1.         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
2.         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

1.         Express 954 Kg in tonnes.
2.         Express 0.35 in a fraction in its lowest term.
3.         What is the sum of Nx and y kobo expressed in kobo?
4.         Factorize 25×2-1

•             A trader gives an 8% discount on an article in his kiosk marked  N1250.00. How much would a customer pay for such an article?

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 7.2, numbers 1-12.

WEEKEND ASSIGNMENT

1.         XABandX=1 when A=B=3. Find X when A=6  and  B=2. A. 19 B. 43C. 34  D. 5
2.         In the question above. A. X increases by 23.5% B. X decreases by 23.5% C. X increases by 25.0%  D. X  decreases by 25%
3.         X is partly constant and partly varies with y. This statement can be represented as 
4.         P=a+byB. P=a+bC. Y=a+py D. P = a + y
5.         If AB and BC2, then A. A∝C  B. AC2C. A2C2  D.  B α C
6.         How many constants do we have in partial variation? A. 1 B. 2  C. 3  D.6

THEORY

1.         If ABC and when B=4,C=9, A=6,calculate

(a)A when B=3 and C=10;   (b) C when A=20 and B=15.

1.         The charge for a pair of shoes is partly constant and partly varies with the number of pairs of shoes. If 90 pairs cost N1120 and the cost of 120 pairs is N1216.Find the charges for 150 pairs.

GENERAL EVALUATION

1.         Make L the subject of the formula A=πrL+πr2
2.         Change 84 in base ten to a number in base two.

•             Represent -1<x≤3 on the number line.
•             Convert 90 in a decimal number to an octal number.
•             A car travelled 80 Km in 48 minutes. What is the speed of the car in /h?

READING ASSIGNMENT

Essential Mathematics for J.S.S. 3 by Oluwasanmi A.J.S. 2014 edition; Pages 38-45, 49-53.

Essential Mathematics Workbook for J.S.S. 3 by Oluwasanmi A.J.S.; Exer.7.1, numbers 1-5; Exercise 5.1, numbers 6-10.

WEEKEND ASSIGNMENT

1.         The simple interest on N7,000 for 4 years at 712% is _________.
2.         N1,960 B.  N2,100 C.  N2,000 D. #3,000
3.         The amount on the above question is  A. N8,960 B. N9,100 C.  N9,000 D.  #9,600
4.         0.4054054054…can also be written as A.   0.4B.  0.405̌ C.  0.405⃛  D. 0.405
5.         The square root of 6.74 is A. 2.596 B. 26.94 C. 2.695 D. 2695
6.         Which of the following is a rational number?A. 0.74578343  B.  1-3 C.   11  D.  31`

THEORY

1.         What is the compound interest on N8,000 borrowed for 2 years at a 20% rate?
2.         Find the square root of  5720, leaving your answer in one decimal place.