The duplicate ratio of 4 : 5 is:
16 : 25
Understanding Duplicate Ratios in Mathematics
The question asks us to find the duplicate ratio of 4 : 5. Let's first understand what a duplicate ratio is in the context of mathematical ratios.
What is a Duplicate Ratio?
In mathematics, the duplicate ratio of two numbers \(a\) and \(b\) is defined as the ratio of their squares. If we have a ratio \(a : b\), its duplicate ratio is \(a^2 : b^2\).
This concept is useful when comparing areas or other quantities that scale with the square of a linear dimension.
Calculating the Duplicate Ratio of 4 : 5
We are given the ratio 4 : 5. To find its duplicate ratio, we need to square both the antecedent (the first term, 4) and the consequent (the second term, 5).
Let the given ratio be \(a : b\), where \(a = 4\) and \(b = 5\).
The duplicate ratio is \(a^2 : b^2\).
- Square the first term (antecedent): \(4^2 = 4 \times 4 = 16\)
- Square the second term (consequent): \(5^2 = 5 \times 5 = 25\)
So, the duplicate ratio of 4 : 5 is \(16 : 25\).
Analyzing the Options
Let's look at the given options based on our calculation:
- 16 : 25
- 64 : 125
- 4 : 5
- 5 : 4
Our calculated duplicate ratio, 16 : 25, matches the first option.
The other options represent different concepts:
- 64 : 125 is the triplicate ratio (\(4^3 : 5^3\)).
- 4 : 5 is the original ratio.
- 5 : 4 is the inverse ratio of 4 : 5.
Therefore, the duplicate ratio of 4 : 5 is 16 : 25.
Step-by-Step Solution
Here is a summary of the steps to find the duplicate ratio:
- Identify the given ratio, \(a : b\). In this case, it is 4 : 5, so \(a=4\) and \(b=5\).
- Calculate the square of the first term, \(a^2\). Here, \(4^2 = 16\).
- Calculate the square of the second term, \(b^2\). Here, \(5^2 = 25\).
- The duplicate ratio is \(a^2 : b^2\). So, it is 16 : 25.
| Ratio Type | Formula for \(a : b\) | Example with 4 : 5 |
|---|---|---|
| Original Ratio | \(a : b\) | 4 : 5 |
| Duplicate Ratio | \(a^2 : b^2\) | \(4^2 : 5^2 = 16 : 25\) |
| Sub-duplicate Ratio | \(\sqrt{a} : \sqrt{b}\) | \(\sqrt{4} : \sqrt{5} = 2 : \sqrt{5}\) |
| Triplicate Ratio | \(a^3 : b^3\) | \(4^3 : 5^3 = 64 : 125\) |
| Sub-triplicate Ratio | \(\sqrt[3]{a} : \sqrt[3]{b}\) | \(\sqrt[3]{4} : \sqrt[3]{5}\) |
| Inverse Ratio | \(b : a\) | 5 : 4 |
Based on the definition and calculation, the duplicate ratio of 4 : 5 is 16 : 25.
Revision Table: Ratio Concepts
| Term | Definition | Formula (for \(a:b\)) |
|---|---|---|
| Ratio | Comparison of two quantities by division | \(a : b\) or \(\frac{a}{b}\) |
| Antecedent | The first term in a ratio | \(a\) in \(a:b\) |
| Consequent | The second term in a ratio | \(b\) in \(a:b\) |
| Duplicate Ratio | Ratio of the squares of the terms | \(a^2 : b^2\) |
| Sub-duplicate Ratio | Ratio of the square roots of the terms | \(\sqrt{a} : \sqrt{b}\) |
Additional Information on Ratios
Ratios are fundamental in mathematics for comparing quantities. They can be simplified like fractions by dividing both terms by their greatest common divisor. For example, the ratio 10 : 15 can be simplified to 2 : 3 by dividing both terms by 5.
Understanding different types of ratios like duplicate, sub-duplicate, triplicate, and sub-triplicate ratios is important for various mathematical problems, especially those involving geometry (areas, volumes) and scaling.
A ratio can also be expressed as a fraction or a decimal, though the colon notation (\(a:b\)) is common when discussing ratio properties like the duplicate ratio.
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Important Questions from Ratio and Proportion
In a mixture of 156 litres, the ratio of milk and water is 7 : 6. How much water should be added to make the ratio 14 : 13?
If the ratio of the first to second number is 3 : 4 and that of the second to the third number is 8 : 5, and sum of three numbers is 190 then the third number is:
The third proportional to 9 and 15 is:
The average age of 3 persons is 30 years.If their ages are in the ratio of 3 : 5 : 7 respectively, then the age of the eldest person is:
The income of A and B are in the ratio 5 : 3. The expenses of A, B and C are in the ratio of 8 : 5 : 2. If C spends 2000 and B saves ₹ 700, then A saves:
