Algebra

Decimal fractions, the periods of which start immediately after the decimal point, are called purely periodic, those which start later on mixed periodic.

Tasks: Convert a) 0.

..., b) 0.

..., c) 2.1

...into common fractions. Denote the required number by x:

10x = 3.	100x = 72.	1000x = 2142.
x = 0.	x = 0.	10x = 21.

9x = 3	99x = 0.72	990x = 2121
x = 1/3	x = 8/11	x = 2121/990 = 207/330 = 2⁴⁷/₃₃₀
0....= 1/3	0.... = 8/11	2.1... = 2⁷/₃₃₀

Since you can convert every finite decimal fraction into a common fraction, for example,

Every finite and every periodic decimal fraction
can be converted into a common fraction.

The inverse of this rule is also true. Every common fraction can be converted into a decimal fraction by extension or division. If the division is not complete, the digits will repeat and the fraction becomes a periodic fraction.

Per 100 Baht	p Baht	let the percentage equal z Baht, then
1	p/100	z = kp/100
k	kp/100

You do not multiply Bath by years, but only numbers. You must be clear in your mind regarding the significance of the numbers in a formula, that is, in what currency a payment is made. k and z have the same unit. " % " is not a unit, but an arithmetic symbol like + and -.

Task 1. 8 out of 40 students received the grade "good". What was their percentage?

Task 2. In a town, 420 persons fell sick; they represented 7 % of the town's population. How many people live in the town?

If you read the formula in the form z = (k/100)·p, you compute first 1%, then p%; 6% of 420 Baht are: 4.20 Baht·6 = 25.20 Baht; if you read the formula in the form z = k·(p/100), it means that you multiply k by p/100: 10% of 70.49 Baht are 70.49/10 = 7.05 Baht. If you have to evaluate a percentage, you employ mostly the formula p/100 = z/k, which follows from z = kp/100. You would compute the answer to Task 1. as follows: 8/40 = 0.20, that is, 20 %, etc.

Since the equation z = k·p/100 contains the three variables z, k and p and always two must be given, in order to determine the third, there arise the three groups of tasks:

If you want to check your result, you must change the sequence of your computations (Checking results). This is here also possible by changing from one task to another, whence you find in Task 1. that 20% of 40 students are 8 students.

1. How big is a garden, if the lawn covers 45% or 324 m²?

2. An elephant baby weighs 90 kg or about 3% of its mother. How heavy is the mother?

3. 144 students or 24% of all students walk to school. How many students are in the school?

It is incorrect to say that one will compute the interest of 400 Baht at 5%. A time statement must always accompany an interest statement. It is also required for the rate of interest. In fact, this is often omitted, because, as a rule, it is always assumed that it is for the same time, namely 1 year. However, if it is for half a year, this fact must be specially mentioned.

The percentage formula z = kp/100 states the interest which k Baht at p% yield in one year. Hence, in n years, you receive n times as much, that is z = kpn/100 Baht. If you now denote the interest by z, you obtain the

Since it contains the four variables z, k, p and n, of which three must always be given, in order to determine the fourth, you have now the four tasks:

III

given

sought

given

sought

given

sought

given

sought

k, p, n

z, k, n

z, p, n

z, k, p

Task: A capital yields in 1½ years at 4% interest the sum of 60 Baht. How big is the capital?

In practice, computation of interest beyond one year duration is unimportant. One computes only for 1 year, ½ year, etc, most of all daily interest. One day is 1/360 year, whence t days are t/360 of a year, that is, in the interest formula n = t/360:

Hence you must first of all divide k by 100, then multiply the result by t -the number of days, in order to obtain the interest rate. The partitioner is 360/p.

If there are frequent inputs and withdrawals, so that k and t vary, you obtain the formula

z=(k₁/100)·t₁/(360/p)+(k₂/100)·t₂/(360/p) . . .=
=(k₁/100)·t₁+(k₂/100)·t₂. . .)/(360/p)=
= Sum of interest rates/interest partitioner

A bank's extract of an account contains the following. The customer's credit and input from anywhere is denoted by C (credit), every withdrawal by D (debt), and a month has 30 days.

You must only divide 334 by the interest partitioner, that is by 360/3 = 120 to obtain 334:120 = 2.78. On the 31st of December, the account will be credited with 2.78. They will be added and also bring interest in the next year. This statement does not take account of postage and service charges.

Many arithmetical tasks, for which you have learned special methods, can be solved with the aid of equations. Two examples of calculation of gains and losses follow.

Task: At what cost was an article purchased, if it is sold with a profit of 24% at 256.68 Baht?

The equation tells that the purchase price + profit = sale price. Let x Baht be the purchase price, so that the profit = x·24/100 Baht, the sales price is 256.68 Baht, whence

x + x·24/100 = 256.68		100x + 24x = 25668		124x = 25668
x = 25668/142		x = 207		the purchase price is 207 Baht

Task: A merchant is prepared to give 15% discount on an article, but wants to receive after the discount has been subtracted 340 Baht. What must be his sales price?

The equation is: sales price - discount = 340 Baht. Let x be the sales price, so that the discount is x·15/100 Baht:

x - x·15/100 = 340		x - 3x/20 = 340		20x - 3x = 6800
17x = 6800		x = 400		The sales price must be 400 Baht

1. One person has about 5 l blood. John when tested had 0.6% alcohol in his blood. How much alcohol was in his body?

2. Vichai pays annually 2400 Baht fire insurance, which are 1.2 % of the sum for which his house is insured. For how many Baht has he insured his house?

3. In an election, A received 172 votes, B 114 votes and C the remainder of 350 votes. How many percent, rounded to integers, did everyone receive?

	I			II			III
given		sought	given		sought	given		sought
k,p		z	z,k		p	z,p		k