Example of Ratios and Proportions

The ratios and proportions, we call reason to the quotient that is indicated by two numbers and that represents the relationship between two quantities and a proportion to the equality that exists between two or more reasons.

1. Reason

A ratio indicates in division form the relationship between two quantities. It tells us how many units there are in relation to the others, and it is usually indicated by simplifying the fractions.

For example, if in a classroom we have 24 girls and 18 boys, then we will represent it in one of the following ways:

24/18
24:18

And since we can simplify the fraction by dividing it by 6, then we will have:

4/3
4:3

And it reads that there is a ratio of 4 to 3, or 4 for every 3.

Each of the values ​​of a ratio has a name. The value that is on the left side of the relationship is called antecedent, and the value on the right hand side is called consequent.

In this case, the ratio of girls to boys is a ratio of 4 to 3, or 4 girls for every 3 boys.

2. Proportion

The proportion indicates by means of an equality the comparison of two ratios. To write a proportion, we must take into account that the antecedent values ​​are always on the same side, as are the consequent ones.

In our classroom example, we can compare the ratio we have, of 4 girls for every 3 boys, and we can calculate how many boys are in a room in relation to the number of girls or vice versa. For this, first of all we will write the proportion that we already know:

4:3

Then an equals sign

4:3=

And then the total amount, for example that of the same room, remembering that we must respect the order of the antecedent and the consequent. In our example, the antecedent will be the number of girls, and the consequent the number of boys.

4:3=24:18

To check the equality of the proportion, two multiplications are carried out. In a proportion, we will take the equal sign as a reference. The numbers that are closest are called the centers, and the furthest numbers are the extremes. In our example, the numbers 3 and 24 are closest to the equals sign, so they are the centers. The 4 and the 18, are the extremes. To check that the proportion is correct, the product of the multiplication of the centers must be equal to the product of the multiplication of the extremes:

3 X 24 = 72
4 X 18 = 72

2.1 Direct proportion and inverse proportion

Proportions can express relationships in which increasing the quantity of the antecedent increases the quantity of the consequent. This variation is called direct proportion. The example above is a direct ratio.

In an inverse proportion, the increase of the quantity in the antecedent, means the decrease of the quantity in the consequent.

For example, in a furniture store, 6 workers make 8 chairs in 4 days. If we want to know how many workers are needed to build the 8 chairs in 1, 2 and 3 days, we will use an inverse proportion.

To determine it, we will use the number of workers as the antecedent figure, and the number of days as the consequent figure:

6:4=

Following the same order, on the other side of equality we will have as a precedent again the number of workers, and as a consequence the days that will take. We will have something like the following:

6:4 = ?:3
6:4 = ?:2
6:4 = ?:1

To determine the inverse proportion, we will multiply the factors of the known ratio, in our example, 6 and 4, and we will divide the result by the known data of the second ratio. Thus, in our example, we will have:

6 X 4 = 24
24 / 3 = 8
24 / 2 = 12
24 / 1 = 24

Thus we will have the following proportions:

6:4 = 8:3
6:4 = 12:2
6:4 = 24:1

With what we can calculate that to produce the 8 armchairs in three days, we need 8 workers; to make them in two days, we need 12 workers, and to make them in 1 day, we need 24 workers.

Examples of reasons

1. In a box we have 45 blue marbles and 105 red marbles. We express it as 45: 105 and dividing by 15, we have that the ratio is 3: 7 (three for every seven), that is, three blue marbles for every seven red marbles.
2. In a school class, each ball is used by each team of five children, that is, we have five students for each soccer ball. We have then in this example of reason that the relation between students - balls is 5 to 1. This ratio is written 5: 1 and we conclude that there is a ratio of five students to each soccer ball.
3. In a parking lot there are cars from Asian factories and American factories. In total there are 3060 cars, of which, 1740 are of Asian manufacture and the rest, 1320, are of American manufacture. This will give us that the ratio is 1740/1320. To simplify it, we first divide it by 10, which leaves us 174/132. If we now divide it by 6, we will have the ratio 29:22, that is, in the parking lot there are 29 Asian cars for every 22 American cars.

Examples of proportions:

Direct proportion:

1. In a store, national and imported sweets are sold at a ratio of 3: 2 If we know that 255 national sweets are sold per day, how many imported sweets are sold per day?

3:2=255:?
2 X 255 = 510
510/3 = 170 imported sweets.
3: 2 = 255: 170 (three is to two as 255 is to 170).

1. Boys and girls were invited to a party. If we know that 6 girls attended for every 4 boys, and there are 32 boys at the party, how many girls were there?

6:4 = ?:32
32 X 6 = 192
192/4 = 48 girls went to the party.
6: 4 = 48:32 (6 is 4 as 48 is 32)

1. To assemble a table, 14 screws are needed. How many screws do we need to assemble 9 tables?

14:1 = ?:9
14 X 9 = 126
126/1 = 126 screws are required.
14: 1 = 126: 9 (14 is to 1 as 126 is to 9)

Inverse ratio:

1. Two cranes move 50 containers in an hour and a half. How many cranes are needed to move the 50 containers in half an hour?

2:1.5 =?:.5
2 X 1.5 = 3
3 / .5 = 6 cranes are needed.
2: 1.5 = 6: .5 (two cranes is an hour and a half, like six cranes is half an hour)

1. If 4 students do a teamwork in 45 minutes, how long will it take if the team is made up of 6, 8, 10 and 12 students?

We will have the following proportions:

a) 4:45 = 6 :?
b) 4:45 = 8 :?
c) 4:45 = 10 :?
d) 4:45 = 12 :?

4 X 45 = 180

a) 180/6 = 30 minutes
b) 180/8 = 22.5 minutes
c) 180/10 = 18 minutes
d) 180/12 = 15 minutes

So the proportions will be:

a) 4:45 = 6:30
b) 4:45 = 8: 22.5
c) 4:45 = 10:18
d) 4:45 = 12:15

• Keep reading: Simple rule of three.