Why is 60: 45 and 4 : 3 equivalent?
The Correct Answer and Explanation is :
To understand why the ratios (60:45) and (4:3) are equivalent, we need to simplify the first ratio and compare it to the second.
Step 1: Simplifying the Ratio (60:45)
A ratio is expressed as a fraction, so we can write (60:45) as:
[
\frac{60}{45}
]
Next, we simplify this fraction. Both 60 and 45 can be divided by their greatest common divisor (GCD). To find the GCD of 60 and 45, we can list the factors or use the Euclidean algorithm.
- The factors of 60 are (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60).
- The factors of 45 are (1, 3, 5, 9, 15, 45).
The largest common factor is (15). Now, we divide both terms of the ratio (60:45) by (15):
[
\frac{60 \div 15}{45 \div 15} = \frac{4}{3}
]
Step 2: Understanding the Ratio (4:3)
The ratio (4:3) is already in its simplest form, indicating that for every 4 parts of one quantity, there are 3 parts of another quantity.
Conclusion: Equivalence of Ratios
Since we simplified (60:45) to (4:3), we can conclude that the two ratios are equivalent. They represent the same relationship between two quantities, even though the actual numbers are different.
Real-World Example
This concept is commonly applied in various scenarios, such as cooking, mixing solutions, or scaling recipes. If a recipe requires (60) grams of one ingredient and (45) grams of another, it can also be expressed as needing (4) parts of the first ingredient for every (3) parts of the second ingredient, providing flexibility in measurements without changing the ratio of the ingredients.
Thus, understanding the equivalence of ratios is crucial in maintaining proportions and ensuring consistency in various applications.