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.

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.