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Cross-multiplication is a powerful method used in solving equations that involve proportions. Whenever you have an equation in the form of \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication allows you to eliminate the fractions by multiplying across the equals sign in a diagonal manner. Here's how it works:

• Start with the proportion \( \frac{a}{b} = \frac{c}{d} \).
• Multiply the numerator of the first ratio by the denominator of the second ratio, resulting in \( a \times d \).
• Do the reverse for the other side: multiply the numerator of the second ratio by the denominator of the first \( c \times b \).
• The result of this process gives the equation \( a \times d = c \times b \).

Cross-multiplication simplifies solving equations by removing the fractions early in the problem. It becomes particularly handy as you get more complex equations. In the original exercise, this technique transformed \( \frac{0.5}{0.15} = \frac{0.3}{x} \) into the simpler equation \( 0.5x = 0.3 \times 0.15 \).
Knowing how and when to use cross-multiplication will greatly help in understanding and solving mathematical proportions.

Solving equations is a fundamental skill in mathematics that involves finding the value of an unknown variable that makes an equation true. In the original exercise, once cross-multiplication was applied, we were left with the equation \( 0.5x = 0.045 \).
Here's a breakdown of how to solve such equations:

• Isolate the variable: The first step is to rearrange the equation to get the variable on one side. This often involves moving other terms to the opposite side. For the equation \( 0.5x = 0.045 \), dividing both sides by \( 0.5 \) isolates \( x \).
• Perform arithmetic: Carry out any necessary multiplication, division, addition, or subtraction. In the example, you divide \( 0.045 \) by \( 0.5 \) to solve for \( x \).
• Check your solution: Whenever possible, substitute your solution back into the original equation to ensure it satisfies the equation.

Solving equations is all about finding the steps that simplify the problem enough to isolate the unknown variable, then performing the necessary operations to solve for it. This turns what appears to be a complex problem into manageable steps.
With practice, this process becomes more intuitive, enabling a seamless transition from the problem to the solution.

Mathematical ratios are a way of comparing quantities that have the same unit. They express how much of one thing there is compared to another and are usually written in the form \( a:b \), which is equivalent to \( \frac{a}{b} \). Ratios are critical in understanding proportions, and they play a large role in solving problems like the one from the original exercise.

• Understanding ratios: Ratios tell us how two quantities relate. For example, a ratio of \( 0.5:0.15 \) could tell us how two particular measurements compare numerically.
• Converting ratios: Converting a ratio into a fraction \( \frac{a}{b} \) makes it easier to use in equations, especially in setting up proportions.
• Working with ratios in proportions: When two ratios are set equal, they form a proportion. Cross-multiplication then helps in finding the unknown quantity within this setup.

By grasping the concept of ratios, students develop a deeper understanding of not just mathematical comparisons but the foundational basis for many problems involving fractions and proportional relationships.
This gives them the tools needed to work through exercises, such as finding the missing term in a proportion by applying these comparisons effectively.