Theory

PROPOTINALITY

Ratio

A ratio is like a fraction. If we want to compare two quantities we can divide both numbers, then we can express it

    1. As a fraction
    2. As a decimal number
    3. As a ratio

For example: Comparing the numbers 7 and 2, we write 7/2 which means that 7 contains the number 2, 7/2 times (as a fraction, 7/2 ; as a decimal number, 3.5 ; as a ratio, 7/2).

The three mean the same, but in a ratio we can use decimal numbers and in fractions we only use integers ones.

We can multiply/divide the terms in a ratio by the same number.

Direct Proportionality

We say that there is a direct proportionality between two magnitudes if an increase on one magnitude causes a proportional increase on the other and a decrease on the first quantity causes a proportional decrease on the second.

For example: A man walks 5200m in 2 hours and a half. How much will he walk in 7 hours at the same speed?


Indirect Proportionality

We say that there is an inverse proportionality between two magnitudes if an increase in one magnitude causes a proportional decrease in the other and a decrease in the first magnitude causes a proportional increase in the other. That is if one magnitude is multiplied by 2, 3, ... this causes in the second a division by 2, 3, ... etc.

For example: 18 men can do a job in 10 days, in how many days will 45 men do the same job?


Mixed Proportionality

We say that there is a mixed proportionality between three or more magnitudes the relation between them is direct and indirect together. The way to solve this type of activity is to reduce the three magnitudes into two and work as a direct/indirect proportion.

For example: Four farmers harvest 10.000kg of apples in 9 days work. How many kilograms will six farmers harvest if they work for 15 days?

Lets suppose that a we only have a farmer. So the farmer will need 4·9 days in the first case to harvest 10.000kg and 6·15 days in the second one. All this argument transform a three column proportionality problem into a two column one:

PERCENTAGES

A percent is a ratio to 100.

A percentage can be considered as a fraction with denominator 100.

Conversion

To convert from a decimal to a percent, just move the decimal 2 places to the right.

To convert a fraction into a percentage we multiply the numerator by 100 and we make the division.

There are some very easy cases of percentages that we can calculate mentally:

      • 50% = 1/2
      • 25% = 1/4
      • 20% = 1/5
      • 10% = 1/10

How to calculate a percentage of a quantity

If we want to calculate the percetal value of a quantity we only have to divide by 100 the percent and multiply it by the quantity we have.

For example: The 40% of 25 is 0'40·25 = 10

How to calculate a total from the percent

We do it using the direct proportionality.

For example: In the class 13 students didn’t do their homework; this was 52% of the class. How many students are in this class?

Students Percent

13 52

x 100

As the percent grows the number of students should grow, too. So, 52/100 = 13/x what is identical to 100/52 = x/13. As the 13 is dividing we pass to the other side of the equation multiplying.

And we obtain: x = 100·13/52 = 25 students.

Increasing and Decresing with percents

We can do it in two ways:

Adding and subtracting the percentual value:

          • Calculate the percentual value.
          • Add (increasing) or subtract (decresing) to the value.

Variation Index:

          • Divide the percentage by 100
          • Add (increasing) or subtract this value to 1
          • Multiply it to the quantity we want to optain

For example: The population of a town is 63500 and last year it increased by 8%, what is the population now?

      1. 63500 · 0'08 = 5080 and we add the value, 63500+5080 = 68580
      2. 63500 · (1 + 8/100) = 63500 · 1'08 = 68580

For example: The price of a jaket is 68€ and there is a discount of 7%, what is the final price?

      1. 68 · 0'07 = 4'76€ so as it's a discount we subtract to the original price: 68 - 4'76 = 63'24€
      2. 68 · (1-7/100) = 68 · 0'93 = 63'24€

How to find the original value

If we know the % increase or decrease and the value we can find the total using direct proportionality.

For example: The net salary of an employee is 1230€ after paying 18% of IRPF, what is the gross of his/her salary?

As 18% is what the emproyee pays as IRPF, hi will have the 82% net in his/her salary.

€ %

1230 82

x 100

In this case both the percentage and the euros must grow up so, we are speaking about a direct proportionality; what means that: 100/82 = x/1230

Isolating the x, we obtain that: x=100·1230/82 = 1500€ is the employee's salary without retention.

Fro example: I have bought a house for 187.000€, at a 10% of IVA. What was the price before IVA?

In this case, the 10% is added to the final price, so we have paid 110% as a total.

€ %

187000 110

x 100

In this case as the percentage goes down the euros must also go down, so we have a direct proportionality. So we have: 100/110 = x/187000

Isolating the x, we obtain that: x = 100·187000/110 = 170000€ cost my house.