 A math course on proportionality in the fourth grade. In this lesson, the student should be able to demonstrate whether a table is proportional and know how to calculate the proportionality coefficient. In addition, they will be able to develop skills in using the cross product rule to determine a fourth proportional. Proportionality in math is very important for student progress. You will learn physical science through compound quantities like average speed. You should also be able to recognize a proportional situation graphically.

We will end this lesson by solving problems from everyday life in the eighth grade.

## I. The fourth proportional

Example 1:

Six jars of honey cost 21€. It is assumed that the price paid is proportional to the number of pots purchased. How much are five pots?

• The data are grouped in a proportionality table: • We determine x by calculation using the cross product rule. €.

• We deduce that five pots of honey cost 17.5 €.

Example 2:

How long does it take to download a 850 MB file under the same conditions?

• It is assumed that the connection speed is constant, i.e. the download time is proportional to the file size.
• The data are grouped in a proportionality table. • We determine the value of x using the cross product rule seen in seventh grade: .
• We deduce that the download time for a file with a capacity of 850 MB is 204 seconds.

Remark:

Example 1 can also be solved by determining the price of a pot: € so a pot costs 3.50 euros.

Then, we determine the price of 5 pots: €.

On the other hand, this transition to unity is more delicate in example 2.

## II. Apply proportionality in math

### 1. percentages

Example:

In a middle school with 475 students, there are 323 half-boarders.

What is the percentage of half boarders at this college?

We are looking for the number of half boarders per 100 students in which the proportion of half boarders would be the same.

• The data are grouped in a proportionality table. • We determine x by the calculation: .
• This means that 68% of the students are half-boarders at this college.

### 2. average speed

Definition:

The average speed v is defined as a function of time t and speed d. .

Example:

During a hike in the mountains, we covered 12.6 km in 4h 30 min.

What was our average speed?

• Here, d = 12.6 km and t = 4h 30 min=4.5 h.
• So we have .

Remark:

• Care must be taken to ensure consistency of units in applications.
• One passes from a speed in m/s in km/h by multiplying by 3,6.

### 3. Compound quantities

Example:

Gold is one of the densest metals. Its density is .

The Bank of France keeps this metal in the form of blocks, called ingots, 2.65 dm high and whose base has an area of 0.244 dm².

How much does such an ingot weigh?

• To say that the density of gold is means that of gold weighs 19.3 kg.
• The mass of an ingot of volume is sought. • • A gold bar weighs 12.48 kg.
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