Proportional expressions are one of the fundamental concepts in high school mathematics.

Proportional expressions are formulae that describe the relationship between two quantities changing in a certain ratio and can be used to solve a wide range of everyday problems and mathematical tasks.

In particular, it is also closely related to understanding functions and equations, and a solid understanding of the concept of proportional expressions can help in solving more complex mathematical problems.

This article explains the basic concepts of proportional expressions and the procedure for solving specific problems, with application examples.

It also includes real calculations using Python code to show how to work with proportional expressions in a practical approach.

- Useful in high school mathematics! Solving and applying problems with proportional expressions
- Master the proportional equation and don’t be afraid of graphs! How to express y in terms of the x formula
- How do you make a proportional formula? Includes examples of its use in everyday life
- How to make a proportional formula Explains the procedure for making a proportional formula.
- Proportionality in everyday life Examples of proportional formulae used in everyday life
- Speed problems Solving speed problems with proportional expressions
- Percentage problems Solving percentage problems with proportional formulae

- SUMMARY.

## Useful in high school mathematics! Solving and applying problems with proportional expressions

### Relationship to functions: a deeper look at the relationship between proportional expressions and functions

A proportional formula is an expression that shows that two quantities are in a proportional relationship.

Conversely, a function expresses a relationship in which the value of one variable determines the value of another variable that corresponds to just one.

In other words, a proportional expression can be considered a type of function.

A function represented by a proportional formula is a straight line passing through the origin when graphed. The slope of this line corresponds to the proportionality constant, and the greater the proportionality constant, the steeper the slope of the graph.

For example, the graph of y = 100x is a straight line passing through the origin. The slope of this line corresponds to the proportionality constant of 100.

The greater the slope, the steeper the line on the graph will be.

Thus, proportional expressions can be seen as part of the wider concept of a function.

Understanding proportional expressions is the first step towards understanding functions as a whole.

More complex functions can also be learned by deepening the concept of proportional expressions.

### Linking with equations: combining proportional expressions with equations

The topic ‘Linking with equations: combining proportional expressions and equations to solve’ is explained in this section.

Combining proportional expressions and equations can help you solve more complex problems.

Such problems usually involve two variables, where it is known that there is a proportional relationship between them.

#### 1. what is a proportional equation?

A proportional equation is an equation that describes the relationship between two quantities that change in a certain ratio.

For example, if 𝑦 is proportional to 𝑦, the relationship is expressed as

$$y=kx$$

where is called the constant of proportionality.

This equation shows a linear relationship such that doubling the value of 𝑦 also doubles the value of 𝑦.

#### 2. linking equations and proportional expressions

Equations and proportional expressions can be combined to solve a variety of real-life and mathematical problems. For example, consider the following problem

##### Example problem

The price of a certain commodity is proportional to the number of pieces.

This means that the price of a commodity is proportional to the number of pieces and is expressed by the following equation

$$P=kn$$

As a further condition, let us assume that if we buy five of the goods, the total price will be 2000 yen. Use this condition to find the price per unit.

##### Solution.

1. find the proportional constant

$k$

to find 𝑘

The formula for the total price if five units are purchased is as follows

$$P=kn$$

where,

$P = 2000$ $n=5$

Assuming,

$$2000=k\times 5$$

Solving this equation, the proportionality constant

𝑘

for the proportionality constant 𝑘

$$k=\frac{2000}{5}=400$$

This means that the price per unit is 400 yen.

The following is the Python code.

# Number of goods and total value n = 5 # number of goods total_price = 2000 # Total price of 5 pieces # Calculate the proportional constant k k = total_price / n # Output the price per piece print(f ‘The price per piece is {k} yen.’)

`Price per piece is 400.0 JPY`

2. how to find the total amount for any number of pieces

Proportionality constant

$k=400$

Now that we know that 𝑘, we can find the total amount for any number of

𝑛

the total amount for an arbitrary number of 𝑃

𝑃

can be calculated as follows

$$P=400n$$

For example, if 10 units are purchased, the total amount is,

$$P=400\times 10=4000\text{\hspace{0.17em}yen}\text{}$$

The amount is 4000 yen.

## Master the proportional equation and don’t be afraid of graphs! How to express y in terms of the x formula

### Finding proportional expressions from graphs How to find proportional expressions from graphs

This section explains how to find proportional expressions from graphs.

The graph of a proportional relationship is a straight line passing through the origin. Proportional expressions are generally

$y=kx$

and is of the form,

𝑘

is the proportionality constant.

To find the proportionality equation from the graph, follow these steps.

1. identify the point through which the line passes: check that the line on the graph passes through the origin. For proportional relationships, it always passes through the origin.

2. choose any point: any point on the line other than the origin.

$({x}_{1},{y}_{1})$

3. the proportionality constant

𝑘

4. calculate k: using the coordinates of the chosen point,

$k=\frac{{y}_{1}}{{x}_{1}}$

is calculated. This gives the proportionality constant

𝑘

can be found.

5. create the proportional equation: the proportional constant 𝑘

𝑘

is found, the proportionality equation

$y=kx$

is written.

For example, a point on the graph

$(2,6)$

If there is a line passing through the line, the proportionality constant

𝑘

is given by

$k=\frac{6}{2}=3$

and the proportionality equation is

$y=3x$

import matplotlib.pyplot as plt import numpy as np # Range of values for x x = np.linspace(0, 4, 100) # Proportionality for y = 3x y = 3 * x # Create graph plt.figure(figsize=(6, 6)) plt.plot(x, y, label=‘y = 3x’, colour=‘blue’) # Line y = 3x plt.scatter(2, 6, colour=‘red’) # Point (2, 6) plt.text(2, 6, ‘(2, 6)’, fontsize=12, verticalalignment=‘bottom’, horizontalalignment=‘right’) # Label the points # Graph title and label plt.title(‘Graph of y = 3x’) plt.xlabel(‘x’) plt.ylabel(‘y’) # Display grid plt.grid(True) # Proportion plt.axhline(0, colour=‘black’,linewidth=0.5) plt.axvline(0, colour=‘black’,linewidth=0.5) # Show legend plt.legend() # Show graph plt.show()

## How do you make a proportional formula? Includes examples of its use in everyday life

### How to make a proportional formula Explains the procedure for making a proportional formula.

A proportional equation is an expression that expresses that two ratios are equal, and it is important to understand the concept of ‘ratio’ in order to create a proportional equation.

A ratio represents the relationship between two quantities and a ratio equation is expressed as ‘a=c’. This section explains the steps to create a proportional equation using this formula.

- Finding the two quantities involved

First, identify two quantities that are in a proportional relationship. For example, speed and time, or price and quantity. - Establish a ratio

Next, set up a ratio between those quantities. For example, if the ratio of speed to time is constant, the proportional relationship is expressed in the form ‘speed 1:time 1 = speed 2:time 2’. - Converting to a proportional equation

Finally, use the ratios you have found to convert them into proportional expressions. It is common to convert the ratio symbol ‘:’ into the form of a fraction in the form ‘a/b = c/d’.

Proportional expressions can thus be used in a variety of everyday situations and are a useful tool for clarifying relationships between numbers.

### Proportionality in everyday life Examples of proportional formulae used in everyday life

Proportionality is used naturally in many situations in everyday life. An everyday example is a cooking recipe.

If a recipe calls for ‘ingredients for four people’ and you want to double it to ‘eight people’, you double the quantities of all the ingredients. This is because the amount of ingredients is proportional to the number of people.

The concept of proportionality is also often used in shopping. For example, when buying apples at 100 yen each, the price is proportional to the number of apples purchased: 200 yen for two apples, 500 yen for five apples, and so on.

Similarly, the cost of petrol is proportional to the distance traveled. The longer you drive, the more petrol you need and therefore the more you pay.

Proportionality is thus frequently used in situations where quantities increase or decrease in conjunction with each other and is a natural part of our lives.

### Speed problems Solving speed problems with proportional expressions

Speed problems can be solved using the proportional relationship established between distance traveled, time, and speed.

The basic relationship is

**speed = distance ÷ time. **

Using this, for example, even if speed and time are unknown, the proportional formula can be used to solve the problem.

For example, if a car travels 160 km in two hours at a speed of 80 km/h, you can use the proportional equation to solve for how far it would travel in three hours without changing speed.

As the speeds are the same, there is a proportional relationship between the running time and distance. This relationship can be expressed as an equation as follows.

$\frac{160}{2} = \frac{x}{3}$

where x is the distance travelled in 3 hours. To solve this equation, write the code in Python.

# Solve the relationship between distance and time using the proportional formula distance_2_hours = 160 # 160 km in 2 hours time_2_hours = 2 time_3_hours = 3 # Use the proportional formula to calculate the distance travelled in 3 hours distance_3_hours = (distance_2_hours / time_2_hours) * time_3_hours print(distance_3_hours) # Result is 240 km

Running this code, the distance traveled in three hours is calculated to be 240 km.

Thus, speed problems can be solved efficiently using proportional formulae.

### Percentage problems Solving percentage problems with proportional formulae

Percentage problems consider the ratio of a part to the whole and can be solved using proportional formulae.

The basic idea is to formulate the equation as ‘what proportion of the part to the whole’.

For example, in the question ‘What percentage of the whole does the price of a product correspond to?’, the price and the percentage of the whole are proportional.

If a product worth $5,000 is discounted by 20%, the price after the discount should be calculated.

This problem can be solved using the following proportionality equation.

$\frac{x}{100} = \frac{割引後の価格}{5000}$

100x = 5000 Price after discount

Here, x is the formula to find the price after the discount.

Solve this problem in Python code.

# Original price and discount rate of the product original_price = 5000 discount_rate = 20 # 20%. # Calculate the discount amount discount_amount = original_price * (discount_rate / 100) # Calculate the discounted price discount_price = original_price - discount_amount print(discounted_price) # Result is 4000 yen

In this code, the price of a 5000 yen item is calculated to be 4000 yen after a 20% discount.

Percentage problems can also be solved easily using proportional expressions like this.

## SUMMARY.

Proportional formulae are a useful tool for clarifying the relationship of change between two quantities and are used not only in high school mathematics but also in many aspects of everyday life.

For example, in speed and proportion problems, proportional formulae can be used to derive answers efficiently.

Proportional expressions are also closely related to functions and equations and are an important element in helping to understand mathematics as a whole.

Through calculation examples using Python, students also learned how to calculate proportional expressions in practice.

Thus, by understanding and utilizing proportional expressions, you will be able to deal with more advanced mathematical problems, so please make sure you have the basics down and use them in your applications.

## コメント