Cross-multiplication

In mathematics, especially in elementary arithmetic and elementary algebra, cross-multiplication is a helpful way to solve problems with fractions or rational expressions. This method makes it easier to solve equations or find missing numbers.

People sometimes call this the "cross your heart" method because they draw lines that look like a heart to remember what to do. For example, if you see an equation like a/b = c/d, where b and d are not zero, cross-multiplication means you multiply a by d and b by c. This gives you ad = bc, which simplifies the problem.

In Euclidean geometry, this idea relates to the ratios of similar triangles. It shows how cross-multiplication links algebra and geometry in a useful way.

Procedure

Cross-multiplication is a useful 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 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 shortcut helps students solve problems without dealing with complex fractions.

Use

Cross-multiplication is a useful way to solve problems with fractions in math. Imagine you have an equation like (\frac{x}{b} = \frac{c}{d}), where (x) is what you want to find. 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 fraction problems and makes 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.