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Cross-multiplication is a powerful mathematical technique used to solve equations involving proportions, where two fractions are set equal. This method is straightforward and particularly handy when dealing with ratios, which are often expressed as fractions.
In the original exercise, you have a proportion like \( \frac{3}{5} = \frac{a}{105} \). The key step is to multiply the numerator of each fraction by the denominator of the other fraction. This moves the variables out of the fractional notation and into a simple equation. For example, in our case, multiply 3 by 105 and 5 by \(a\) to get:

• \(3 \times 105 = 5 \times a\)

This simplifies to \(315 = 5a\). By doing this cross-multiplication, you transform the proportion into an algebraic equation that can be easily solved. This method can only be used when two ratios are truly equal, providing an efficient pathway to calculate the unknown variable.

Ratios are mathematical expressions that describe the relationship between two numbers, showing how many times one value contains or is contained within the other. In simplest terms, ratios compare two quantities. They are written in the form \( \frac{a}{b} \) or expressed with a colon as \(a:b\).
In the context of the original exercise, each problem presents a ratio that is set equal to another, forming a proportion. For example, \( \frac{3}{5} \) is a ratio showing that for every 3 units of one value, there are 5 units of another. The other ratio, \( \frac{a}{105} \), follows the same logic but involves a variable \(a\).
Working with ratios involves understanding this relationship. When you equate two ratios, it indicates that the two sets of quantities have the same relative size or rate. Cross-multiplication can then be used to solve for unknowns within these proportional relationships, making ratios highly applicable in various real-world problems and mathematical exercises.

Solving equations involves finding the value of the unknown variable that makes the whole equation true. When solving proportion problems like those in the original exercise, you often end up with a simple linear equation after using cross-multiplication.
Consider the equation \(315 = 5a\), which resulted from the cross-multiplication in our problem. To solve for \(a\), you need to isolate the variable. This is done by performing arithmetic operations that transform the equation step-by-step to achieve \(a\) on one side and a number on the other side. Here, you divide both sides by the coefficient of \(a\) (i.e., 5):

• \(a = \frac{315}{5}\)

This division gives the result \(a = 63\).
Each step in solving an equation focuses on simplifying and isolating the variable to find its value. Understanding this process is crucial for tackling more complex equations and developing strong problem-solving skills in algebra and beyond.