Solving Ratio Problems with Sum of Squares

Let's break down this problem involving ratios and the sum of squares of numbers. We are given that three numbers, A, B, and C, are in the ratio 2 ∶ 3 ∶ 5. We are also given that the sum of the squares of these numbers is 3800. Our goal is to find the value of the number C.

Representing Numbers Based on Ratio

When numbers are in a specific ratio, we can represent them using a common multiplier. Let this multiplier be 'x'.

• Number A can be represented as $A = 2x$.
• Number B can be represented as $B = 3x$.
• Number C can be represented as $C = 5x$.

Here, 'x' is a common factor for all three numbers. Since A, B, and C are likely positive numbers in such problems (especially when dealing with sums of squares), x will be a positive value.

Setting Up the Equation from Sum of Squares

The problem states that the sum of the squares of these numbers is 3800. We can write this as an equation:

$A^2 + B^2 + C^2 = 3800$

Now, substitute the representations of A, B, and C in terms of 'x' into this equation:

$(2x)^2 + (3x)^2 + (5x)^2 = 3800$

Solving the Equation for the Multiplier 'x'

Let's simplify and solve the equation:

• Square each term: $(2x)^2 = 4x^2$, $(3x)^2 = 9x^2$, and $(5x)^2 = 25x^2$.
• Substitute these back into the equation: $4x^2 + 9x^2 + 25x^2 = 3800$.
• Combine the terms with $x^2$: $(4 + 9 + 25)x^2 = 3800$.
• This simplifies to: $38x^2 = 3800$.
• Now, isolate $x^2$ by dividing both sides by 38: $x^2 = \frac{3800}{38}$.
• Calculate the value: $x^2 = 100$.
• To find 'x', take the square root of both sides: $x = \sqrt{100}$.
• Since we assume x is positive in this context: $x = 10$.

Finding the Value of C

Now that we have the value of 'x', we can find the values of A, B, and C.

• $A = 2x = 2 \times 10 = 20$
• $B = 3x = 3 \times 10 = 30$
• $C = 5x = 5 \times 10 = 50$

The question asks specifically for the value of C.

The value of C is 50.

Verification (Optional but Recommended)

Let's check if the sum of the squares of these calculated numbers equals 3800:

$A^2 + B^2 + C^2 = 20^2 + 30^2 + 50^2$

$= 400 + 900 + 2500$

$= 1300 + 2500$

$= 3800$

This matches the given information, confirming our values for A, B, and C are correct.

Revision Table: Key Concepts for Ratio Problems

Additional Information: Understanding Ratios and Squares

Ratios are fundamental in mathematics and are used to compare quantities. When dealing with problems involving ratios and operations like summing squares or cubes, using a common multiplier 'x' is a standard and effective method. This transforms the ratio problem into an algebraic equation that can be solved.

The term 'sum of squares' means you take each number, square it (multiply it by itself), and then add those results together. Squaring numbers ($n^2$) always results in a non-negative value. In this problem, solving for $x^2$ led to $x^2 = 100$. The square root of 100 is $\pm 10$. However, in typical ratio problems involving real-world quantities or simple numbers defined this way, the multiplier 'x' is considered positive to yield positive numbers unless specified otherwise. If x were -10, the numbers would be -20, -30, -50, but their squares would still be the same: $(-20)^2 = 400$, $(-30)^2 = 900$, $(-50)^2 = 2500$, leading to the same sum of squares. The question asks for "the value of C", and without context suggesting negative numbers, we assume the positive value derived from the positive 'x'.