The ratio of two numbers is 1 ∶ 5 and their product is 320. What is the difference between the squares of these two numbers?
1536
Solving the Ratio and Product of Two Numbers Problem
This problem involves finding two numbers based on their ratio and product, and then calculating the difference between their squares. Let's break it down step-by-step.
Identifying the Numbers Based on Ratio and Product
Let the two numbers be represented by \(a\) and \(b\). We are given two key pieces of information:
- The ratio of the two numbers is \(1 \ratio 5\). This means we can write the numbers as \(a = k\) and \(b = 5k\) for some constant \(k\).
- The product of the two numbers is \(320\). This means \(a \times b = 320\).
Now, we can use the second piece of information to find the value of \(k\). Substitute the expressions for \(a\) and \(b\) in terms of \(k\) into the product equation:
\( (k) \times (5k) = 320 \)
\( 5k^2 = 320 \)
To find \(k^2\), divide both sides by 5:
\( k^2 = \frac{320}{5} \)
\( k^2 = 64 \)
Now, take the square root of both sides to find \(k\). Assuming positive numbers based on typical ratio problems:
\( k = \sqrt{64} \)
\( k = 8 \)
With the value of \(k\), we can find the two numbers:
- First number, \(a = k = 8\)
- Second number, \(b = 5k = 5 \times 8 = 40\)
Let's quickly verify: Ratio \(8/40 = 1/5\) (correct). Product \(8 \times 40 = 320\) (correct).
Calculating the Squares and Their Difference
The problem asks for the difference between the squares of these two numbers. The squares are:
- Square of the first number (\(a\)): \(a^2 = 8^2 = 64\)
- Square of the second number (\(b\)): \(b^2 = 40^2 = 1600\)
The difference between the squares is the larger square minus the smaller square:
Difference = \(b^2 - a^2\)
Difference = \(1600 - 64\)
Difference = \(1536\)
Therefore, the difference between the squares of the two numbers is 1536.
| Description | Value |
|---|---|
| Ratio of numbers | \(1 \ratio 5\) |
| Product of numbers | \(320\) |
| First number (\(k\)) | \(8\) |
| Second number (\(5k\)) | \(40\) |
| Square of first number | \(8^2 = 64\) |
| Square of second number | \(40^2 = 1600\) |
| Difference of squares | \(1600 - 64 = 1536\) |
Revision Table: Ratio and Product Concepts
| Concept | Explanation | Formula/Example |
|---|---|---|
| Ratio | A comparison of two quantities. Can be written as \(a \ratio b\), \(a/b\), or "a to b". | Ratio of 8 to 40 is \(8 \ratio 40\) or \(1 \ratio 5\). |
| Product | The result of multiplying two or more numbers. | Product of 8 and 40 is \(8 \times 40 = 320\). |
| Representing numbers by ratio | If the ratio is \(m \ratio n\), the numbers can be \(mk\) and \(nk\) for some constant \(k\). | Ratio \(1 \ratio 5\) means numbers are \(k\) and \(5k\). |
| Difference of squares | Subtracting the square of one number from the square of another. | Difference between \(b^2\) and \(a^2\) is \(b^2 - a^2\) or \(a^2 - b^2\). |
Additional Information: Working with Ratios in Problems
When solving problems involving ratios, it's common to introduce a variable (like \(k\) in this case) to represent the common factor that scales the ratio to the actual numbers. This allows you to use the other given information (like product, sum, difference, etc.) to solve for this variable and find the actual numbers.
- Sum: If the ratio is \(1 \ratio 5\) and the sum is 48, then \(k + 5k = 48\), so \(6k = 48\), \(k = 8\). Numbers are 8 and 40.
- Difference: If the ratio is \(1 \ratio 5\) and the difference is 32, then \(5k - k = 32\), so \(4k = 32\), \(k = 8\). Numbers are 8 and 40.
- Product: As shown in this problem, if the ratio is \(1 \ratio 5\) and the product is 320, then \((k)(5k) = 320\), so \(5k^2 = 320\), \(k^2 = 64\), \(k=8\). Numbers are 8 and 40. Note that for products, you work with \(k^2\), not just \(k\).
Always read the question carefully to identify what calculation is needed after finding the numbers (sum, difference, product, difference of squares, etc.).
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