Are the ratios 2:4 and 5:10 equivalent?
The Correct Answer and Explanation is:
Yes, the ratios 2:4 and 5:10 are equivalent.
To determine if two ratios are equivalent, each ratio can be simplified to its lowest terms or written as fractions and compared. Begin with the first ratio:
2:4
This simplifies by dividing both terms by the greatest common divisor (GCD) of 2 and 4, which is 2:24=2÷24÷2=12\frac{2}{4} = \frac{2 \div 2}{4 \div 2} = \frac{1}{2}42=4÷22÷2=21
Now simplify the second ratio:
5:10
The GCD of 5 and 10 is 5:510=5÷510÷5=12\frac{5}{10} = \frac{5 \div 5}{10 \div 5} = \frac{1}{2}105=10÷55÷5=21
Both simplified ratios equal 1:2, confirming that 2:4 and 5:10 are indeed equivalent.
Another method to confirm equivalence is through cross multiplication. If the products of the outer and inner terms are equal, the ratios are equivalent:2×10=20,4×5=202 \times 10 = 20,\quad 4 \times 5 = 202×10=20,4×5=20
Since both products are equal, the ratios are equivalent.
Ratios represent relationships between two quantities. Two ratios are considered equivalent when they express the same proportion, even if the numbers are different. For instance, 2 apples out of 4 fruits and 5 apples out of 10 fruits both represent the same proportion—half.
This concept is crucial in solving problems involving proportions, scaling, maps, and models. By recognizing equivalent ratios, proportional relationships in various contexts can be identified and used to make accurate predictions or calculations.
In summary, 2:4 and 5:10 are equivalent ratios because they both reduce to 1:2, indicating the same proportional relationship.
