Text equations

In the oldest documents - the computing book of the Egyptian Ahmed (beginning of 2. millennium A.C) and the records of the Chinese (2000 A.C.) - you find a quite advanced theory of equations. However, the methods are not general and involve special tricks. During its further development, procedures were rather complicated by the lack of good mathematical symbols. One of the most important books had the title Aldschebr walmukâbala(~820 p.C.); dschebr means shifting negative terms to the other side, in order have only positive terms, mukâbala means combination of terms of the same kind on both sides. As it was widely available, its title named the theory of equations, later on in its Latin form Algebra et Almucabala. The second word disappeared gradually and Algebra became the theory of equations.

Determining equations are important because they let you solve many problems in many fields. You must now learn, how to see and recognize the mathematical aspects in every day life problems. This is one of the most difficult chapters of mathematics. We will show first by means of easily posed problems, how to extract from a problem's description its mathematical content and how to translate it into mathematical symbolism. Once you have understood these problems, create yourself similar ones and invent new ones.

Computation with x like with a known quantity

Task 1. 40 apples are placed on two dishes so that the second dish holds three times as many apples as the first dish. How many apples lie in the first dish?

If x denotes the number of apples in the first dish, the second dish contains 3x. You have therefore the equation 

Task 2: Translate the given text into arithmetical language:

Find your self other such statements and translate them.

Task 3. If you multiply a certain number by 3 and subtract 3 from it, you obtain the same result as, if you double it and add 3.

Where can you find here equality of two quantities? In the statement: "you obtain the same result".

Denoting the unknown number by x, the first step yields 3x - 3. You are to obtain the same result by adding 3 to twice the number so that

In order to check the result, you must start from the text of the task, because your equation might already contain a mistake!

Check: Three times the answer is 18, whence after subtraction of 3 you find 15. You are to obtain the same result by taking twice 6 and adding 3. You find again 15!

Task 4. A ten years old boy wants to know the age of his aunt. She answers that after 20 years their combined age will be 100.

The phrase: "Our combined age will be 100" yields equality of two expressions. Let the aunt be x years old, then 

Equality of two expressions follows from a basic principle.

Task 5. In a right-angled triangle (= 90º), the angle is twice as large as the angle . How large is ?

Geometry told you that ++=180º. This formula allows you to set up an equation. Let  = xº, so that  = 2xº and 

x + 2x + 90 = 180º

3x = 90º

x = = 30º

A relationship yields an equality of two expressions.

Task 6. A rectangle is 10 cm longer than wide and its circumference is 80 cm. How long are its sides?

The circumference can be expressed in two ways. It is 80 cm and it is the sum of the sides. Let the shorter side have the length x cm. You must use the geometrical fact that opposite sides are equally long. 

The rectangle is 15 cm x 25cm.

The equality of two expressions follows from a geometrical fact.

In a similar manner, laws of physics, economic relationships, etc. can yield equalities of two expressions.

Selection of unknown

Task 7. A mother and two daughters inherit 28,000 Baht. The mother is to receive 3 times as much as the daughters. How much does everyone receive?

In this problem, you must find several answers; you can denote the inheritance of any of the three persons by x.

1. Possibility

The mother receives 20,000 Baht, each daughter 4,000 Baht.

2. Possibility 

Every daughter receives 4,000 Baht, the mother 20,000 Baht.

Since the first version contains fractions, the second could be called simpler. In more complicated tasks, the calculations often differ considerably, but each method must yield the same result. There is no way to optimize this choice. Practice sharpens your wit!

Frequently, you can choose for x one or the other quantity.

Task 8. 34 birds sit on top of a tree, 26 underneath the tree. Several of those below fly up above, so that now there are on top three times as many birds as below. How many birds flew up?

Let x denote the birds who flew above: 

11 birds flew into the top of the tree.

However, if you denote the number of birds, which remained below, by x, then up above are three times as many as below, that is 3x. The total number of birds must remain the same: 34 + 26 = 60, whence 

You must now return to the significance of x; it is the number of birds which remained below. There were previously 26, whence 11 flew above.

It is often better to denote by x not the required quantity,
but another quantity, to which it can be reduced.

Task 9. Two cyclists Joan and Jill ride towards each other from locations A and B, which lie 5.4 km apart. Joan rides 0.3 km per minute, Jill 0.24 km per minute. At what distance from A will they meet?

You look first for an equality of quantities. Joan and Jill ride to their meeting point during the same time, whence we must express this time in two ways. If we denote the required distance by x km, Joan, who rides 0.3 km per minute, uses (x/0.3) minutes. Jill rides away from B, 5.4 km away, in the same time (5.4 - x) km, but since she only rides 0.24 km per minute, her time is (5.4 - x)/0.24 minutes. These times are equal, whence 

Joan and Jill meet at a distance of 3 km from A.

You have denoted the required quantity by x. You find another equality in the set task by realizing that the entire distance between A and B must equal the sum of its parts AC + CB. How can you reduce these sections to one unknown? If you denote, as before, AC by x, then CB = 5.4 - x, and you find that x + 5.4 - x = 5.4; this is a useless identity. Hence you must find another quantity, in order to solve the task in this manner. The cyclists use the same time to their meeting point. If you denote this time by x, Joan covers in this time 0.3·x km, Jill 0.24·x km.The two distances represent the entire distance AB, whence 

The cyclists meet after 10 minutes. Since you are to find the distance from A, you must still reduce x to the required quantity. Joan rides in 10 minutes 0.3·10 km, whence you obtain the same result, that is, 3 km.

The second approach, in which you did not denote the required quantity by x, was simpler!

Exercises: 

Answers

Unity of unknowns

Task 10. A, one of two dress makers, has worked 7 times as long as B. They receive together 40 Baht. How much does each of them receive, taking into account that B must be repaid 0.24 Baht for expenses.

B receives without expenses x Baht, A receives 7x Baht. But then B must still receive her expenses, that is she receives x + 24 Baht, whence 

The solution is wrong; it is easily seen that there is a mistake. Once you have selected a quantity to be denoted by x, you must confirm its unit:

The unit of x must be defined and is to be taken into consideration during setting up of the equation as also during the interpretation of the answer.

Dress maker B receives an additional x Baht, expressed in the same currency, 24 Baht (altogether (x + 0.24) Baht. A then receives 7 Baht, whence the equation yields 

You must use the unit, in which x was reckoned. B receives without expenses 4.97 Baht, with expenses 5.21 Baht. A receives 7·4.97 = 34.79 Baht.

It is unimportant what unit you use. You choose it so that the numbers will become convenient.

Relative numbers

Task 11: Joan is three years and 6 months old, Peter is 18 months old. When will Joan be twice as old as Peter?

Let time be the unknown x. In this task, you choose best the present time. Then the time counts from this starting point in two opposite directions. We must still agree on the significance of the leading sign and, when interpreting the answer, recall its significance. Let the future correspond to +, the past to -. This task falls into the area of relative numbers. As always, the calculations are performed purely formally, that is, you write for +a simply a, whence you obtain 

The result is positive. Six months later, Joan will be twice as old as Peter.

Task 12. Joan is 14 , Peter is 10 years old. When was Joan twice as old as Peter?

Let the time start now, the future be positive, the past negative. If x is the required time, the equations are 

Six years ago, Joan was twice as old as Peter, who was then four years old.

If your task is in the realm of relative numbers, you must select the starting point of the numbering and the significance of the leading signs.

Discussion of a solution

Task 13. There are at a party three times as many girls as boys, altogether 70 children. How many boys and girls are there?

Let x be the number of boys, so that there are 3x girls. 

This is not possible. Since you are to determine the number of persons, the task is in the realm of the number of persons. The formal calculation does not depend on the type of numbers and leads to a result, which in the realm of numbers of a unit leads to a division, which cannot be executed. In other words, there is a remainder! The formal calculation is correct, but the task demands an impossible answer.

If you would retain the numbers, but formulate the task in units which allow division, the answer could have been interpreted. For example, you have one glass with as much wine as water, altogether 70 cm³. How many cm³ of each are in the glass?

You arrive at the same formal calculation as above with x = 17½ cm³ water. Now the answer is possible and there are 52½ cm³ wine in the glass.

Similar problems arise elsewhere. Let a formal calculation yield 7 - 11 = - 4. If the task is in the realm of units, for example, 11 nuts are taken away from 7 nuts, the result demands an impossible subtraction. If the task is in relative numbers: "What is the result of 7 Bath intake and 11 Bath outlay?", the answer - 4 is possible.

The formal calculation does not depend on the type of number.
When you interpret a result, take due notice of the realm of the numbers.

Discussion of a task

You will now ask: "When is a task possible?". In order to answer this question, it is best to study a solution after using general numbers. As an example, consider the Task 13.

Three are a times as many girls as boys at a party, altogether b persons. How many boys and girls are present?

Let x be the number of boys, then ax is the number of girls, whence 

Since the task is in the realm of countable units, x must be an integer, that is, b must be a multiple of (1 + a).

If you set a = 1, b must be an even integer. If you set a = 2, you must choose b so that it is divisible by (1 + a) or 3; b can be any multiple of 3, for example 60, etc. If, as in Task 13, b = 70, then, for example, 5 is a factor in 70. Then you must have (1 + a) = 5 or a = 4. Thus, this task would be possible, for example, in the form: There are at a party four times as many girls as boys and altogether 70. etc.

Solution: There are 14 boys and 56 girls.

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