Lesson 4: Numeric Proportionality. History of the Rule of Three (III)

The Rule of Three

The rule of three was a name given in earlier days to an algorithm for solving proportions. The method required setting up the problem so that the unkown quantity is always last “extreme” in the proportionality. In  http://www.pballew.net/arithm18.html you can see an image that shows the rules as given by a 1827 arithmetic.

This rule is covered in almost all the arithmetics up to the beginning of the 20th century.

The rule of three was such a common part of arithmetic education that it found its way into common expressions. In his autobiography, Lincoln writes that he that he learned to “read, write, and cipher to the rule of 3.” A poem often used in student copy books was:

Multiplication is vexation;
Division is as bad;
The Rule of Three doth puzzle me,
And Practice drives me mad

 

Abraham Lincoln 16th President of the United States, 1809 – 1865 In 1865 Lincoln was assasinated as he watched a play

Could you translate it into Spahish?

This extra work will give you an extra point in this lesson!

 

From  http://www.pballew.net/arithm18.html

Lesson 4: Numeric Proportionality. Inverse Proportionality (II)

4. Quantities in inverse proportion.

Two quantities (or variables) are in inverse (or reciprocal) proportion when they increase or decrease in the multiplicative inverse (or reciprocal) ratio. This means that if we multiply (or divide) by a number one quantity the other quantity is divided (or multiplied) by the same number.

Consider the quantities A and B and a table of their corresponding values:

Quantity A	a1	a2	a3	…	an
Quantity B	b1	b2	b3	…	bn

If we multiply the corresponding values of both quantities, the product is  constant:

            a_{1 ·} b₁ = a_{2 ·} b₂ = a_{3 ·} b₃ = …= a_{n ·} b_n = m

We will say that the quantity A and the quantity B are in inverse proportion, or that they are inversely proportional.

We name constant of inverse proportionality.

5. Solving problems involving inverse proportional quantities

Remember the first thing you have to check is if the quantities are in inverse proportion. They have to verify that if  we multiply  (or divide) by a number one quantity the other quantity is divided (or multiplied) by the same number.

There are two methods

5.1. Rule of three inverse (Regla de tres inversa simple)

 The “regla de tres simple inversa” is a procedure to find out an unknown quantity that is in inverse proportion to other known quantities which are in inverse proportion.

 In general, to solve a regla de tres inversa simple we apply the following rule:

a ————- b

c ————- x

then    a/c = x/b  (we write the reciprocal of the second ratio, but can be the other but only one!) 

therefore     a · b = c · x     and       x = (a · b)/ c

You can practise in an interesting site but first read these instructions.

 INSTRUCTIONS:

1.  First you will find some examples. They use the reciprocal of the second ratio instead of the first ratio as we do.
2.  At the bottom of the screen you will find an example below ‘Your turn’. Try it! Write the solution and click on check. You will go to another screen.
3.  If you want to practise more, click on Try it again

 Practise here.

5.2. Unitary method

Recall: If A and B are two proportional quantities the unitary method, in Spanish “reducción a la unidad”, consist in working out the quantity of B that corresponds to one unit of A.

 I couldn’t find any practise, sorry!

6. Percentages

6.1.Meaning of a percentage

When you say “Percent” you are really saying “per 100”, you divide a quantity in one hundred equal parts and you take the number that indicates the percentage

 Percent means “for every 100” or “out of 100.” The (%) symbol as a quick way to write a fraction with a denominator of 100. As an example, instead of saying “it rained 14 days out of every 100,” we say “it rained 14% of the time.”

A percentage, whose symbol is %, is a ratio with 100 as divisor (“consecuente” in Spanish)

 A Percentage can also be expressed as a Decimal or a Fraction.

 You have to interpret a percentage as a fraction of denominator 100. After this we can transform the fraction into a decimal as we know, dividing numerator by denominator

To convert a fraction or decimal to a percentage, multiply by 100

If you consider a fraction as the parts I have of the unit (divided into equal parts), the procedure is obvious. A percentage indicates the parts we take for every 100, therefore we have to multiply the fraction by 100.

The same happens if you have a decimal number. It indicates the parts we have out of one unit (it can be greater than the unit, of course). A percentage indicates the parts we take for every 100, therefore we have to multiply the decimal number by 100.

To determine the percent of a number do the following steps:

• Multiply the number by the percent
• Divide the answer by 100

 We can write this through the following formula:

a% of C = a · C/100

 Practise on

• Convert between percent, fractions, and decimals
• Compare percent to fractions and decimals
• Find what percent one number is of another
• Estimate percent of numbers
• Percent of numbers and money amounts
• Solve percent equations
• Percent of change

7. Percent word problems

 We can understand a percentage as a proportion because three quantities related in a direct proportion appear: the percentage, t%, the whole quantity, C, and the part, A. We have

t%  of  C  = A

 So we can set up a rule of three:

C ——— A

100———t

 We studied different types of word problems on percentages,

• Finding the part if we know the whole and the percentage.
• Finding the percentage, if we know the whole and the part.
• Finding the whole, if we know the percentage and the part.

but all of them can be solved by setting a rule of three direct.

• Find what percent one number is of another: word problems
• Percent of numbers: word problems
• Percent of change: word problems

In order to solve problems where a quantity is increased or decreased by a percentage we have to remember that

Increasing a quantity by a percentage, t%, it is the same as working out the (100 + t )% of the quantity.

 Decreasing a quantity by a percentage, t%, it is the same as working out the (100 – t )% of the quantity.

Practise below

• Percent of a number: tax, discount, and more
• Find the percent: tax, discount, and more
• Sale prices: find the original price
• Multi-step problems with percent
• Simple interest

Finally, here you are a couple of videos on solving percent word problems

 

 

Lesson 4: Numeric Proportionality (I)

Vocabulary on Numeric Proportionality

Lesson 4: Numeric Proportionality

Practice Problems on Proportional Distributions and Simple Interest

Practice Problems on Recognizing Proportionality in Tables and on Percentages

Rule of Three Word Problems

I Have to Know by the End of this Lesson 

1. Definitions

Ratio is a way of comparing amounts of something. It shows how much bigger one thing is than another.

Ratio between two numbers  and , this is the quotient a/b .

Is fraction the same as ratio?

Recall fraction and proportion are not the same.

• A fraction is always a proportion, this is a quotient.
• A proportion can be a fraction or not when one of its terms is a decimal number.

Proportions

A proportion is a name we give to a statement that two ratios are equal. It can be written in two ways:

–  two equal fractions,   a/b=c/d     or,

–  using a colon,  a:b = c:d

The outer terms  and , are called the extremes, and the middle terms  and , called the means.
                   

We name constant of proportionality of a proportion to the constant value of any of the ratios (this is the quotient).

It is also named factor of proportionality.

Two ratios a/b and c/d form a proportion if and only if the cross products of the ratios are equal this is a · c = b · d

In problems involving proportions, we can use cross products to test whether two ratios are equal and form a proportion.

To find the cross products of a proportion, we multiply the outer terms, called the extremes, and the middle terms, called the means.

Finding the missing term in a proportion.

We can also use cross products to find a missing term in a proportion.

In general to work out a term of a proportion (no matter if it is an extreme or a mean):

• Find the crossing products
• Solve for  dividing by the number which multiplies it.

In general:

a/b = c/x  then a · x = b · c  then x = (b · c)/a (x is an extreme)

a/b = x/d  then a · d = b · x  then x = (a · d)/b (x is a mean)

Study this technique closely, because we will use it often in algebra.

If you want to revise these definitions and practise ,click on below

Take into account that First Glance is easy and In Depth requires more structures in English. Examples are clear and simple. Workout is highly advisable.

Be careful because they name to the ratios fractions (there aren’t decimal numbers in their examples).

2. Quantities in direct proportion

Two quantities are in direct proportion when they increase or decrease in the same ratio. This means that if we multiply (or divide) by a number one quantity the other quantity is multiplied (or divided) by the same number.

Consider the quantities A and B and a table of their corresponding values:

Quantity A	a1	a2	a3	…	an
Quantity B	b1	b2	b3	…	bn

If we build up proportions with the corresponding values of both quantities, the constant of proportionality is the same:    

     a₁/b₁=  a₂/b₂ =  a₃/b₃= …. = a_n/b_n    = k

We will say that the quantity A and the quantity B are in direct proportion or that they are directly proportional.

We name k constant of direct proportionality.

Two quantities which are in direct proportion will always produce a graph where all the points can be joined to form a straight line.

NOTE:

In Spanish we say “Magnitudes directamente proporcionales”. Remember that a “magnitud” is any quality that can be measured. For instance length, temperature, weight …

3. Solving problems involving directly proportional quantities

Understanding proportion can help in making all kinds of calculations. It helps you work out the value or amount of quantities either bigger or smaller than the one about which you have information but it isn’t enough. They have to verify that if  we multiply  (or divide) by a number one quantity the other quantity is multiplied (or divided) by the same number.

There are two methods

3.1. Regla de tres simple directa (Rule of three direct)

 The “regla de tres simple directa” is a procedure to find out an unknown quantity that is in direct proportion to other known quantities which are in direct proportion.

In general, to solve a regla tres directa simple (rule of three direct) we apply the following rule:

a —— b

c —— x

then  a/c=c/b     and therefore        a · x = c · b

this is           x = (c · b)/a 

3.2.Unitary method

If A and B are two proportional quantities the unitary method, in Spanish “reducción a la unidad”, consist on working out the quantity of B that corresponds to one unit of A.

Example

Lizz makes three equal frames. She needs 2,79 m of strip. How many metres will she need to make four frames?

We know these two quantities are in direct proportion. To solve this problem we need to know the length of strip we need to make one frame.

We build a table:

Number of frames	3	1	4
Strip (metres)	2,79	0,93	3,72

This is because if you make the third part of frames, this is one frame, you will need the third part of metres of strip:

2,79 : 3 = 0, 93 metres.

So, if we need to make four frames, we will need four times metres of strip, this is:

0,93 · 4 = 3,72 metres of strip

Lizz will need 3,72 metres of strip.

Here you are some problems on direct proportionality

Find the constant of variation: word problems

Proportional relationships: word problems

and a video on solving proportionality word problems

Lesson 13: Simultaneous Linear Equations. A Joke (III)

Algebra is a language, you have to know its words, rules, how to translate,….

What do you think a maths teacher would do in an English class?

Here you are a true history

About application of mathematics in linguistics

A teacher of English was ill and a teacher of mathematics replaced him. He began to compose a table of irregular verbs:

Then he said:
– Okay, I mark this form as  x . Then it’s possible to compose the proportion:

Lesson 13: Simultaneous Linear Equations: Solving Word Problems (II)

 

3. Solving word problems with simultaneous linear equations

Systems of linear equations are a powerful tool to solve problems. Many problems lead themselves to being solved with systems of linear equations. In “real life”, these problems can be incredibly complex. This is one reason why linear algebra (the study of linear systems and related concepts) is a very important branch of mathematics. However, you will face with much simpler problems in Secondary Education. Here you are some typical examples.

Steps for Solving Word Problems

 1. Read the problem. Relax. Underline important words and values.

 2. (Read again). Introduce variables:

 Let x = describe in words

Let y = describe in words

3.(Read again). Make a Table:

 

	Description of x	Description of y	Total
Description for combined quantities	ax     bya’x     b’y		cc’

 

4. Write down the system of equations (from table):

ax + by  = c a’x + b’y = c’

 5.Solve for  x and  y using substitution, equalizing or elimination.

6.Write the answer in words including proper units.

7.Check the answers not only in the system but also in the context of the problem.

Here you are a good site to practise.

First you have got solved examples. At the bottom of the page you will find a generator of problems. But you can jump right to the exercises if you click on the link that says Jump right to the exercises! at the beginning.

Instructions:

1. Click on “new problem” to get started.
2. On these exercises, you will not key in your answer.
3. However, you can check to see if your answer is correct on answer. The solution is explained in detail.

Finally here you are a video where is explained how to solve a numeric word problem with simultaneous linear equations.