Choose All Ratios Equivalent To $48:60$.A. $3:4$B. \$4:5$[/tex\]C. $13:15$D. $24:30$
Understanding Equivalent Ratios
In mathematics, equivalent ratios are fractions or ratios that have the same value, even though they may look different. To determine if two ratios are equivalent, we need to simplify them to their simplest form. In this article, we will explore how to simplify ratios and find equivalent ratios.
What are Equivalent Ratios?
Equivalent ratios are fractions or ratios that have the same value. For example, the ratios 2:3 and 4:6 are equivalent because they both represent the same proportion. To find equivalent ratios, we need to simplify the given ratio to its simplest form.
Simplifying Ratios
To simplify a ratio, we need to find the greatest common divisor (GCD) of the two numbers. The GCD is the largest number that divides both numbers without leaving a remainder. Once we have the GCD, we can divide both numbers by the GCD to simplify the ratio.
Step 1: Find the Greatest Common Divisor (GCD)
To find the GCD of two numbers, we can use the Euclidean algorithm or list the factors of each number. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.
Step 2: Divide Both Numbers by the GCD
Once we have the GCD, we can divide both numbers by the GCD to simplify the ratio. In this case, the GCD of 48 and 60 is 12. So, we can divide both numbers by 12 to simplify the ratio.
48 ÷ 12 = 4 60 ÷ 12 = 5
Simplified Ratio
The simplified ratio is 4:5.
Finding Equivalent Ratios
Now that we have simplified the ratio, we can find equivalent ratios by multiplying or dividing both numbers by the same value. For example, if we multiply both numbers by 2, we get:
4 × 2 = 8 5 × 2 = 10
The ratio 8:10 is equivalent to the simplified ratio 4:5.
Applying the Concept to the Given Options
Now that we have understood how to simplify ratios and find equivalent ratios, let's apply this concept to the given options.
Option A: 3:4
To determine if the ratio 3:4 is equivalent to the simplified ratio 4:5, we need to find the GCD of 3 and 4. The GCD of 3 and 4 is 1. So, we can divide both numbers by 1 to simplify the ratio.
3 ÷ 1 = 3 4 ÷ 1 = 4
The simplified ratio is 3:4. Since the simplified ratio 3:4 is not equal to the simplified ratio 4:5, option A is not correct.
Option B: 4:5
The simplified ratio 4:5 is already in its simplest form. Since the simplified ratio 4:5 is equal to the simplified ratio 4:5, option B is correct.
Option C: 13:15
To determine if the ratio 13:15 is equivalent to the simplified ratio 4:5, we need to find the GCD of 13 and 15. The GCD of 13 and 15 is 1. So, we can divide both numbers by 1 to simplify the ratio.
13 ÷ 1 = 13 15 ÷ 1 = 15
The simplified ratio is 13:15. Since the simplified ratio 13:15 is not equal to the simplified ratio 4:5, option C is not correct.
Option D: 24:30
To determine if the ratio 24:30 is equivalent to the simplified ratio 4:5, we need to find the GCD of 24 and 30. The GCD of 24 and 30 is 6. So, we can divide both numbers by 6 to simplify the ratio.
24 ÷ 6 = 4 30 ÷ 6 = 5
The simplified ratio is 4:5. Since the simplified ratio 4:5 is equal to the simplified ratio 4:5, option D is correct.
Conclusion
In conclusion, the equivalent ratios to the given ratio 48:60 are 4:5 and 24:30. The simplified ratio 4:5 is already in its simplest form, and the simplified ratio 24:30 is equivalent to the simplified ratio 4:5. Therefore, the correct options are B and D.