Proportional expressions are a mathematical concept that is often found in our surroundings.
For example, proportional expressions are used in many situations, such as shopping at the supermarket and in recipes.
In this article, we will explain the basic concept of proportional expressions and how to solve complex proportional expressions involving fractions and parentheses in an easy-to-understand way.
Let’s explore the world of proportional expressions in more depth, with calculation examples using Python.
Explaining how to solve proportional formulae
Basic concepts and how to solve proportional formulae.
A proportional equation is an equation that expresses the relationship of proportionality between two quantities.
For example, if a change in one quantity results in a proportional change in the other, such as ‘if one apple costs 100 yen, then two apples cost 200 yen’, this can be expressed by a proportional formula.
Proportional expressions have the property that ‘when A : B = C : D, the product of the inner terms (B and C) is equal to the product of the outer terms (A and D)’.
In other words, ‘when A : B = C : D, A x D = B X C’.
As an example, if 2 : 4 = 3 : 6, then 2 X 6 = 4 X 3 holds at 12 = 12.
That’s easy.
Now what about the case ‘3 : x = 9 : 12’?
This is basically the same way,
9 X x = 3 X 12
9x = 36
x= 4
This can be calculated.
Incidentally, if you use Python to calculate this, you get the following.
# Value of the given proportional expression A = 3 B = 9 C = 12 # Calculate x using the proportional property x = (A * C) / B print(‘x =’, x)
x = 4.0
Fractions and parentheses: A comprehensive overview of how to solve proportional expressions
How to solve proportional expressions involving fractions explained
Proportional expressions involving fractions are solved in the same basic way.
You can use the relationship ‘inner product = outer product’ to form an equation and solve it to find the unknown value.
The following is a more detailed explanation of how to solve proportional expressions involving fractions.
Example: 3/4 : x = 5/6 : 2/3
1. Using the basic properties of proportional expressions
In proportional expressions, the relationship ‘inner product = outer product’ holds.
In other words, in this example,
- Inner product: 3/4 × 2/3
- The outer product: x × 5/6 and we can formulate the equation that these are equal.
2. Based on the above ideas, the following equation can be formulated
(3/4) × (2/3) = x × (5/6)
3, Calculating fractions: Calculate the fractions on both sides.
(1/2) = (5/6)x
4. Eliminating the coefficient of x:To eliminate the coefficient of x, 5/6, multiply both sides by 6/5.
(1/2) × (6/5) = (5/6)x × (6/5)
5. Continue the calculation
3/5 = x
6. The answer: Therefore, x = 3/5.
The calculation using Python.
Let’s try the calculation using Python.
# Given the proportional equation a = 3/4 b = 5/6 c = 2/3 # Solve for x using the proportional property x = (a * c) / b print(‘x =’, x)
x = 0.6
Proportional expressions containing brackets: Explanation of how to solve proportional expressions containing brackets.
The basic solution to proportional expressions containing parentheses is the same.
The unknown value can be found by using the relationship ‘inner product = outer product’ to form an equation and then solving it.
Key points:
- Expanding parentheses: First, simplify the equation by expanding the parentheses.
- Inner and outer products: calculate the inner and outer products in the expanded equation.
- Solving equations: connect the calculated results with an equal sign and solve for the unknown characters.
The following example explains how to solve proportional equations involving parentheses in more detail.
Example problem: (2x + 1) : 3 = 5 : (x – 2)
1. using the basic properties of proportional expressions
The relationship ‘inner product = outer product’ holds in proportional expressions.
In other words, in this example,
- Inner product: (2x + 1) × (x – 2)
- The outer product: 3 × 5 and we can formulate the equation that these are equal.
2. Based on the above ideas, the following equation can be formulated
(2x + 1) × (x - 2) = 3 × 5
3. Expand the parentheses: Expand the parentheses on the left-hand side.
2x² - 3x - 2 = 15
4. Rearrange the equation: Transfer all terms to the left-hand side and rearrange.
2x² - 3x - 17 = 0
5. Solving a quadratic equation: As this is a quadratic equation, it can be solved using methods such as factoring and solution formulae. In this case, factorisation is not possible, so the solution formula is used. The solution formula is,
x = (-b ± √(b² - 4ac)) / (2a)
In the case of this problem, a = 2, b = -3 and c = -17, so the formula is substituted.
6. Answer: The calculation results in two values of x.
Using Python to do calculations
Let’s do some calculations using Python.
from sympy import symbols, Eq, solve # Define variables x = symbols(‘x’) # Set up an equation equation = Eq((2*x + 1) * (x - 2), 5 * 3) # Solve the equation solutions = solve(equation, x) # Print the solutions print(‘x =’, solutions)
x = [3/4 - sqrt(145)/4, 3/4 + sqrt(145)/4]
Two solutions were thus obtained.
Proportional expressions may seem complicated at first glance, but once you have the basic concepts down, you can solve any problem.
In this text, we have explained various aspects of proportional expressions, from their definition to how to solve problems involving fractions and parentheses.
By utilising Python, we have shown that even complex calculations can be easily performed.
Proportional expressions are an important concept underlying mathematics.
We encourage you to use this knowledge to try your hand at a variety of problems.
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