Cross-multiplication

In mathematics, specifically in elementary arithmetic and elementary algebra, cross-multiplication is a useful tool for solving equations that involve fractions or rational expressions. This method helps simplify the equation or find the value of an unknown number.

The technique is sometimes called the "cross your heart" method because people often draw lines that look like a heart to remember which numbers to multiply together. For example, if you have an equation like a/b = c/d, where neither b nor d is zero, cross-multiplication lets you multiply a by d and b by c. This gives you ad = bc, making it easier to solve for the unknown.

In Euclidean geometry, this same idea can be understood by looking at the ratios of similar triangles. This shows how cross-multiplication connects algebra and geometry in a clear and practical way.

Procedure

Cross-multiplication is a helpful way to solve equations with fractions. Imagine you have an equation like $\frac{a}{b} = \frac{c}{d}$, where $b$ and $d$ are not zero. To solve it, you can "cross" the numbers: multiply the top of each fraction by the bottom of the other.

This gives you $ad = bc$. It’s like a quick trick to make the equation easier! The method works because multiplying both sides of an equation by the same number keeps them equal. This step-by-step shortcut helps students solve problems without dealing with complex fractions.

Use

Cross-multiplication is a helpful tool in math for solving problems with fractions. Imagine you have an equation like (\frac{x}{b} = \frac{c}{d}), where (x) is something you want to find out. By cross-multiplying, you can rearrange the numbers to find that (x = \frac{bc}{d}).

For example, if you want to know how far a car will travel in 7 hours, and you know it went 90 miles in 3 hours, you can set up the problem like this: (\frac{x}{7 \text{hours}} = \frac{90 \text{miles}}{3 \text{hours}}). By cross-multiplying, you multiply 7 hours by 90 miles and divide by 3 hours, which gives you (x = 210) miles. This method works for many problems involving fractions and helps make solving them easier.

Main article: clearing fractions

Rule of three

The rule of three is an old way to solve a special kind of math problem using cross-multiplication. It was a key part of math learning in the past and is still taught in some schools today.

For example, if you know that 4 yards of cloth cost 12 shillings, you can find out how much 6 yards will cost. You set up the problem like this: 4 yards divided by 12 shillings equals 6 yards divided by an unknown amount, called x. By cross-multiplying, you can find that x is 18 shillings. This method helps solve problems where you need to find a missing number in a proportion.