Understanding the Weight Division Problem

The question describes a total weight that is separated into two distinct parts. We are given a relationship between these two parts and the total weight based on a specific equation. Our goal is to determine the ratio of the first part to the second part. This involves setting up an algebraic equation based on the problem statement and solving for the relationship between the two unknown parts.

Setting Up the Equation for Weight Parts

Let's denote the first part of the weight as \(p_1\) (in kg) and the second part as \(p_2\) (in kg). The total weight is the sum of these two parts.

Total Weight \(W = p_1 + p_2\)

The problem states a condition: "5 times the first part added to 11 times the second part makes 7 times the whole weight." We can translate this statement directly into a mathematical equation:

\(5 \times p_1 + 11 \times p_2 = 7 \times W\)

Substituting the expression for the total weight \(W\) into the equation, we get:

\(5p_1 + 11p_2 = 7(p_1 + p_2)\)

Solving the Weight Ratio Equation

Now, we need to solve this equation to find the relationship between \(p_1\) and \(p_2\), which will give us the ratio \(p_1 : p_2\).

First, distribute the 7 on the right side of the equation:

\(5p_1 + 11p_2 = 7p_1 + 7p_2\)

Next, we need to gather the terms involving \(p_1\) on one side and the terms involving \(p_2\) on the other side of the equation. Let's move the \(p_1\) terms to the right and \(p_2\) terms to the left.

Subtract \(5p_1\) from both sides:

\(11p_2 = 7p_1 - 5p_1 + 7p_2\)

\(11p_2 = 2p_1 + 7p_2\)

Subtract \(7p_2\) from both sides:

\(11p_2 - 7p_2 = 2p_1\)

\(4p_2 = 2p_1\)

To find the ratio \(p_1 : p_2\), we need to express \(\frac{p_1}{p_2}\). We can do this by dividing both sides of the equation \(4p_2 = 2p_1\) by \(2p_2\) (assuming \(p_2 \neq 0\)).

\(\frac{4p_2}{2p_2} = \frac{2p_1}{2p_2}\)

Simplify both sides:

\(\frac{4}{2} = \frac{p_1}{p_2}\)

\(2 = \frac{p_1}{p_2}\)

This means the ratio of the first part to the second part, \(p_1 : p_2\), is 2 : 1.

Final Weight Ratio Result

Based on the given problem statement and the equation derived, the ratio of the first part to the second part is 2 : 1.

Revision Table: Weight Ratio Calculation Steps

Additional Information: Understanding Ratios and Proportions

A ratio is a comparison of two quantities. It shows how much of one quantity there is compared to another. Ratios can be written with a colon (e.g., 2:1), as a fraction (e.g., \(\frac{2}{1}\)), or using the word "to" (e.g., 2 to 1).

In this problem, finding the ratio \(p_1 : p_2\) tells us the relative sizes of the two parts of the weight. A ratio of 2:1 means the first part is twice as large as the second part.

Proportion is an equation stating that two ratios are equal. For example, if \(a:b = c:d\), this is a proportion, which can also be written as \(\frac{a}{b} = \frac{c}{d}\). Proportions are often used to solve problems involving scaling quantities.

In algebraic problems like this one, setting up and solving linear equations is a common method to find unknown values or relationships like ratios between quantities. The key is accurately translating the word problem into a correct mathematical model.