# Cross Multiplication Calculator

Welcome to Omni's **cross multiplication calculator**, where we'll be solving for $x$ with fractions. The expressions we tackle here are often called proportions, and there's an easy algorithmic way to deal with them: **cross multiply the fractions**. In fact, whichever of the four values is unknown, once we cross multiply and divide, **we're sure to get our result**, no strings attached.

But before we get ahead of ourselves, let's slow down a bit and learn how to do cross multiplication **nice and easy**.

## Solving for x with fractions

Typically, we use cross multiplication when we have **one-variable equations including fractions**. For instance, it involves expressions of the form:

Note how we can have negative numbers or decimals in the numerators or denominators. Also, in the last example, $x$ **is outside of the fraction**, but we can easily get it inside by following the basic rules that tell us how to calculate the fraction multiplication:

Such equations, even if they look fancy, **must follow the same rules as any other**. In particular, we can add or subtract any number, and we can multiply) or divide by any non-zero value as long as we **do it on both sides of the equality sign**. For our purposes, the latter pair of arithmetic operations prove crucial: **we'll cross multiply and divide** to find the value of $x$.

## How to cross multiply fractions

The clue is in the name "*cross multiplication*". We'll calculate the product of the values **in a cross pattern**:

Where $\text{num}$ and $\text{den}$ are, respectively, numerator and denominator. And if we were to use the symbols appearing in Omni's cross multiply calculator, i.e., take the equation:

we'd get:

In fact, **that's all there is** to solving for $x$ with fractions. After all, once we cross multiply, we don't have fractions anymore, so we can turn to **other well-known methods** of dealing with equations. For instance, if we wanted to find $A$ from the formula above, it'd be enough to divide both sides by $D$:

Note how whichever letter we need (i.e., whichever numerator or denominator), the procedure would still be the same: cross multiply and divide. The difference is only in what we divide by in the second step. To be precise, we divide by:

- $D$ when seeking $A$;
- $C$ when seeking $B$;
- $B$ when seeking $C$; and
- $A$ when seeking $D$.

Furthermore, recall that **we can always exchange the sides of an equation**. Therefore, we can always change the formula:

into:

However, observe that **the pairs stay the same**: we need to preserve the cross multiplication pattern.

Alright, the instructions on how to do cross multiplication seem easy enough, don't you think? It's high time we **move from symbols and theory to numbers and practice**. And, to kill two birds with one stone, we'll take the opportunity to **let our cross multiply calculator shine**.

## Example: using the cross multiplication calculator

Suppose that **you're constructing an aircraft model**. After a few hours of meticulous gluing, the plane is ready: it will look awesome displayed on the shelf for everyone to admire. But now that the DIY part is over, why don't we learn something about it? Let's calculate **how large the real-life equivalent is**.

The box says that the model **is done on a** $1:100$ scale. You grab a ruler and check that **your creation is** $3.5$ **inches long**. Believe it or not, that's all we need to find the answer. The trick is in **a good understanding of how scaling works**: the unit rate on the box is proportional to the ratio of the model and real-life lengths. In our case, this means that **the** $3.5$ **inches is to the actual aircraft's length what** $1$ **is to** $100$:

where $x$ denotes the value we seek. However, before we rush to solve it ourselves, let's see **how easy the task is with Omni's cross multiplication calculator at hand**.

At the top of our tool, we see the formula:

and four variable fields corresponding to the letters. As the cross multiplication calculator states, **it's enough to input three** of the values, so we look back at our problem and write:

- $A = 3.5$, $C = 1$, and $D = 100$.

The moment we input the third number, **the tool will spit out the answer**. Note how it also provides **a step-by-step explanation** underneath: the same that we'll give right now.

We begin by doing what the above section taught us: **we cross multiply the fractions**:

Hence:

Which gives:

Normally, we'd still need to divide the result by the number standing in front of $x$. However, in our case, that number turned out to be $1$, **so there's no need**.

**We got our answer!** The real-life aircraft is $350$ inches long. Well, we should probably move on from the cross multiplication calculator to a length converter to get **a more reasonable answer**, don't you think?

Models are often built in different scales: $1:72$, $1:48$, and so on. Learn how to quickly pass from one to the other with our scale calculator.

## FAQ

### How do I solve for x with fractions?

To **solve for x with fractions**, you need to:

**Transform**both sides into quotients.**Cross multiply**the fractions.**Simplify**the two expressions.**Divide**by what's in front of`x`

.**Enjoy**having solved for`x`

with fractions.

### How do I cross multiply fractions?

To **cross multiply fractions**, you need to:

**Make sure**you have only a fraction on each side.**Multiply**the numerator of the first by the denominator of the second.**Multiply**the numerator of the second by the denominator of the first.**Combine**steps 2-3 into an equation.- If needed,
**solve**the resulting equation with basic methods. **Enjoy**having cross multiplied the fractions.

### Why does cross multiplication work?

Cross multiplication is, in fact, **simple multiplication done twice**.

Firstly, we **multiply both sides by the left side's denominator**, which leaves only the numerator on the left (according to fraction simplification rules), and multiplies the right numerator (according to fraction multiplication rules). Next, we **multiply both sides by the right side's denominator**, which gives a product on the left and kills the denominator on the right.

All in all, since we can always multiply both sides of an equation by a non-zero number, **cross multiplication indeed works**.

### How does cross multiplication work?

To **use cross multiplication**, you need to:

**Make sure**you have only a fraction on each side.**Multiply**the numerator of the first by the denominator of the second.**Multiply**the numerator of the second by the denominator of the first.**Combine**steps 2-3 into an equation.- If needed,
**solve**the resulting equation with basic methods. **Enjoy**having used cross multiplication.

### How do I compare fractions using cross multiplication?

To **compare fractions using cross multiplication**, you need to:

**Make sure**you have only a fraction on each side.**Multiply**the numerator of the first by the denominator of the second.**Multiply**the numerator of the second by the denominator of the first.**Compare**values from steps 2 and 3.- If the one in step 2 was:
**Smaller**, then the first fraction is smaller; or**Larger**, then the first fraction is larger.

- If one of the multipliers was negative,
**change**the relation to its opposite. **Enjoy**having compared fractions using cross multiplication.

### How do I solve proportions using cross multiplication?

To **solve proportions using cross multiplication**, you need to:

**Make sure**you only have a fraction on each side.**Multiply**the numerator of the first by the denominator of the second.**Multiply**the numerator of the second by the denominator of the first.**Combine**steps 2-3 into an equation.**Solve**the resulting equation with basic methods.**Enjoy**having solved a proportion using cross multiplication.