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Lesson 4: Numeric Proportionality (I)

March 20, 2011

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= …. = an/bn    = 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

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