Finding the Square of the Largest Number from a Given Ratio and Sum Condition

The problem asks us to find the square of the largest of three numbers that are in a specific ratio, given a condition about their sum.

Let the three numbers be represented based on their ratio 2 : 3 : 5. We can use a common multiplier, say \(x\), to represent the actual numbers.

Representing the Numbers Based on Ratio

Since the numbers are in the ratio 2 : 3 : 5, we can write them as:

• First number = \(2x\)
• Second number = \(3x\)
• Third number = \(5x\)

Here, \(x\) is a non-zero constant.

Setting up the Equation Using the Sum Condition

We are given that twice their sum is 100. First, let's find the sum of the three numbers:

Sum = \(2x + 3x + 5x = (2 + 3 + 5)x = 10x\)

According to the problem, twice this sum is 100:

\(2 \times (\text{Sum}) = 100\)

\(2 \times (10x) = 100\)

\(20x = 100\)

Solving for the Value of x

Now, we need to solve the equation \(20x = 100\) to find the value of \(x\).

Divide both sides by 20:

\(x = \frac{100}{20}\)

\(x = 5\)

So, the common multiplier \(x\) is 5.

Finding the Actual Numbers

Now that we have the value of \(x\), we can find the three numbers:

• First number = \(2x = 2 \times 5 = 10\)
• Second number = \(3x = 3 \times 5 = 15\)
• Third number = \(5x = 5 \times 5 = 25\)

The three numbers are 10, 15, and 25.

Identifying the Largest Number

Comparing the three numbers (10, 15, and 25), the largest number is 25.

Calculating the Square of the Largest Number

The question asks for the square of the largest number, which is 25.

Square of the largest number = \(25^2\)

\(25^2 = 25 \times 25\)

\(25 \times 25 = 625\)

The square of the largest number is 625.

Let's verify the sum condition with these numbers: Sum = \(10 + 15 + 25 = 50\). Twice the sum = \(2 \times 50 = 100\), which matches the given condition.

Revision Table: Key Steps in Solving the Ratio Problem

Additional Information on Ratio and Proportion Concepts

What is a Ratio?

A ratio is a comparison of two or more quantities of the same kind, expressed as \(a : b\) or \(a : b : c\), etc. It shows how much of one quantity is in another quantity. In this problem, the ratio 2 : 3 : 5 means the numbers are proportional to 2, 3, and 5.

Using a Common Multiplier:

When numbers are given in a ratio like \(a : b : c\), we can represent the actual numbers as \(ak, bk, ck\), where \(k\) (or \(x\) as used above) is a non-zero constant. This constant scales the ratio to the actual values of the numbers. This is a standard technique to solve problems involving ratios and given conditions about the numbers.

Understanding "Twice their Sum":

This phrase means two times the result of adding the numbers together. If the sum of the numbers is \(S\), twice their sum is \(2S\).

Solving ratio problems often involves setting up an algebraic equation based on the given information (like sum, difference, product, etc.) and then solving for the common multiplier.