If three positive numbers are in the ratio 2 : 3 : 5 and the sum of their squares is 1368, then what is sum of all the numbers ?
60
Solving Ratio and Sum of Squares Problems
This problem asks us to find the sum of three positive numbers given their ratio and the sum of their squares. Understanding ratios and how to represent numbers based on a ratio is key here.
Understanding the Problem Statement
We are given:
- Three positive numbers are in the ratio 2 : 3 : 5.
- The sum of the squares of these numbers is 1368.
- We need to find the sum of these three numbers.
Representing the Numbers Using the Ratio
When numbers are in a ratio, we can represent them using a common variable, say 'x'. Since the ratio is 2 : 3 : 5, the three positive numbers can be represented as:
- First number: \(2x\)
- Second number: \(3x\)
- Third number: \(5x\)
Here, \(x\) must be a positive value since the numbers are positive.
Setting Up the Equation from the Sum of Squares
The problem states that the sum of the squares of these numbers is 1368. We can write this as an equation:
\((2x)^2 + (3x)^2 + (5x)^2 = 1368\)
Solving the Equation for 'x'
Now, let's simplify and solve the equation to find the value of \(x\).
- Square each term:
\(4x^2 + 9x^2 + 25x^2 = 1368\)
- Combine the like terms (\(x^2\)):
\((4 + 9 + 25)x^2 = 1368\)
\(38x^2 = 1368\)
- Isolate \(x^2\) by dividing both sides by 38:
\(x^2 = \frac{1368}{38}\)
\(x^2 = 36\)
- Take the square root of both sides to find \(x\). Since the numbers are positive, \(x\) must be positive:
\(x = \sqrt{36}\)
\(x = 6\)
Finding the Three Numbers
Now that we have the value of \(x\), we can find the three positive numbers:
- First number = \(2x = 2 \times 6 = 12\)
- Second number = \(3x = 3 \times 6 = 18\)
- Third number = \(5x = 5 \times 6 = 30\)
Let's quickly check if the sum of their squares is indeed 1368:
\(12^2 + 18^2 + 30^2 = 144 + 324 + 900 = 1368\). This matches the information given in the problem.
Calculating the Sum of All the Numbers
The final step is to find the sum of these three positive numbers:
Sum = First number + Second number + Third number
Sum = \(12 + 18 + 30\)
Sum = \(60\)
Thus, the sum of all three numbers is 60.
Revision Table: Key Steps in Ratio Problems
| Step | Description | Application in this Problem |
|---|---|---|
| 1 | Represent numbers using ratio and a variable. | Numbers are \(2x\), \(3x\), \(5x\). |
| 2 | Formulate an equation based on the given condition (e.g., sum, sum of squares). | \((2x)^2 + (3x)^2 + (5x)^2 = 1368\). |
| 3 | Solve the equation for the variable. | Found \(x = 6\). |
| 4 | Calculate the actual numbers using the variable's value. | Numbers are 12, 18, 30. |
| 5 | Answer the specific question asked (e.g., find the sum, difference, etc.). | Calculated the sum: \(12 + 18 + 30 = 60\). |
Additional Information: Ratios and Proportions
A ratio is a way to compare two or more quantities. It shows how much of one quantity there is compared to another. Ratios can be written as \(a:b\), \(a/b\), or "a to b". In this problem, the ratio 2:3:5 means that for every 2 units of the first number, there are 3 units of the second number and 5 units of the third number.
A proportion is an equation that states that two ratios are equal. While this problem doesn't directly involve setting two ratios equal, the concept of using a common multiple (\(x\)) relates to proportionality – each number is proportional to its part in the ratio.
Problems involving ratios often require setting up algebraic equations based on given information, such as the sum, difference, product, or sum of squares, and then solving for the unknown multiple represented by a variable like \(x\).
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Important Questions from Simple Ratios
If x : y = 5 : 2, then the value of (8x + 9y) : (8x + 2y) is:
If the value of \(\frac{{p - q}}{q} = \frac{4}{3}\) .find p : q
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The ratio of two numbers A and B is 5: 8. If 5 is added to each of A and B, then the ratio becomes 2 : 3. The difference between A and B is:
In population of a city the ratio of men and women is 12 : 11. If total population of that city is 4,60,000, then the population of women is:
