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Lesson 4: Numeric Proportionality. Inverse Proportionality (II)

March 25, 2011

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:

            a1 · b1 = a2 · b2 = a3 · b3 = …= an · bn = 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

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.

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

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

 

 

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