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Cross multiplication is a technique often used to solve equations involving proportions. It allows us to find the value of a missing variable quickly and efficiently. In a proportion, two ratios are set equal to each other, appearing in the form \( \frac{a}{b} = \frac{c}{d} \). To cross multiply, we multiply the numerator (top number) of one ratio by the denominator (bottom number) of the other ratio. This means:

• Multiply \(a\) by \(d\) (the denominator of the second fraction)
• Multiply \(b\) by \(c\) (the numerator of the second fraction)

The resulting equation is \(a \cdot d = b \cdot c\). This technique simplifies finding the unknown variable, such as when we encountered the equation \(280x = 530 \cdot 112\). We cross multiplied to set up an equation that we could then solve for \(x\). It's a powerful method because it reduces complex fractional equations to more manageable linear equations.

Fractions represent parts of a whole and are written as a number over another number, separated by a line. The top number is called the numerator, and it shows how many parts we have. The bottom number is the denominator and indicates the total number of equal parts the whole is divided into. For example, in \(\frac{3}{5}\), 3 represents the parts we have, while 5 is the total number of parts that make up a whole.

Fractions are used extensively in arithmetic for problems involving ratios or proportions. In our original problem, the fractions \(\frac{280}{530}\) and \(\frac{112}{x}\) represented two ratios. When solving for an unknown value, it's important to ensure that our final answer is expressed in its simplest form, also known as 'lowest terms'. This involves reducing the fraction by dividing both numerator and denominator by their greatest common divisor. This process makes the fraction easier to understand and work with.

The greatest common divisor (GCD) is the largest number that divides two numbers without leaving a remainder. It's crucial for simplifying fractions to their lowest terms, ensuring the result is easily interpretable. To find the GCD of two numbers, say 280 and 59360, follow these simple steps:

• List out the factors of each number.
• Identify the largest factor that appears in both lists. This is the GCD.

In practice, using the GCD to simplify a fraction like \(\frac{59360}{280}\) involves dividing the numerator and the denominator by 280, their GCD. This process helped us reduce \(\frac{59360}{280}\) to 1 over 1, simplifying our answer. Finding the GCD is an essential skill in math, as it helps clarify results and makes calculations more straightforward and accurate.