## 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 | a_{1} |
a_{2} |
a_{3} |
… | a_{n} |

Quantity B | b_{1} |
b_{2} |
b_{3} |
… | b_{n} |

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

a_{1 · }b_{1} = a_{2 · }b_{2 }= a_{3 · }b_{3} = …= 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:**

- First you will find some examples. They use the reciprocal of the second ratio instead of the first ratio as we do.
- 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.
- 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.

- 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

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