fraction numerator 3.5 over denominator x end fraction equals fraction numerator 4.5 over denominator y end fraction explain the problem with given data and calculate the fraction respectively
The correct answer and explanation is :
To solve the proportion ( \frac{3.5}{x} = \frac{4.5}{y} ) for the variables ( x ) and ( y ), we can use the method of cross-multiplication. This technique involves multiplying the numerator of each fraction by the denominator of the other fraction, setting the products equal to each other:
[ 3.5 \times y = 4.5 \times x ]
To isolate ( x ), we rearrange the equation:
[ x = \frac{3.5 \times y}{4.5} ]
Similarly, to isolate ( y ):
[ y = \frac{4.5 \times x}{3.5} ]
These equations express ( x ) and ( y ) in terms of each other. However, without additional information or specific values for either ( x ) or ( y ), we cannot determine their exact numerical values. To find specific values for ( x ) and ( y ), at least one of them must be known.
Understanding proportions is fundamental in mathematics, as they describe the equality of two ratios and are widely applicable in various fields, including science, engineering, and everyday problem-solving. The cross-multiplication method is a reliable and straightforward technique for solving proportions, especially when variables are present in the denominators. By cross-multiplying, we eliminate the fractions, simplifying the equation to a more manageable form. This method is particularly useful when dealing with equations where the variable is in the denominator, as it allows for direct manipulation and isolation of the variable.
In practical applications, proportions are used to solve problems involving scaling, conversions, and comparisons. For example, if you know that 3.5 liters of paint cover a certain area, and you want to find out how much area 4.5 liters would cover, setting up a proportion like the one above would allow you to solve for the unknown variable. This approach ensures accuracy and consistency in calculations, making it an essential tool in both academic and real-world scenarios.
For a visual explanation and further examples of solving proportions with variables in the denominators, you can refer to educational resources such as Khan Academy, which offers detailed videos and practice exercises on this topic.
By mastering the concept of proportions and the cross-multiplication method, you can effectively tackle a wide range of mathematical problems that involve ratios and proportional relationships.