# Cross Multiplication Calculator

By Maciej Kowalski, PhD candidate
Last updated: Apr 29, 2021

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:

2

x
=
5

7

-1

3.4
=
9x

10

2

3
x =
21

8

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 of fraction multiplication:

2

3
x =
2

3
*
x

1
=
2 * x

3 * 1
=
2x

3

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 proves 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 take the product of the values in a cross pattern:

`left numerator * right denominator = right numerator * left denominator`.

And if we were to use the symbols appearing in Omni's cross multiply calculator, i.e., take the equation:

A

B
=
C

D

we'd get:

`A * D = B * C`.

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`:

`(A * D) / D = (B * C) / D`.

A
=
B * C

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:

`left numerator * right denominator = right numerator * left denominator`,

into:

`right numerator * left denominator = left numerator * right denominator`.

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`:

3.5

x
=
1

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:

A

B
=
C

D

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:

3.5

x
=
1

100

`x * 1 = 3.5 * 100`

`x = 350`

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?

## FAQ

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

To solve for x with fractions, you need to:

1. Transform both sides into quotients.
2. Cross multiply the fractions.
3. Simplify the two expressions.
4. Divide by what's in front of `x`.
5. Enjoy having solved for `x` with fractions.

### How do I cross multiply fractions?

To cross multiply fractions, you need to:

1. Make sure you have only a fraction on each side.
2. Multiply the numerator of the first by the denominator of the second.
3. Multiply the numerator of the second by the denominator of the first.
4. Combine steps 2-3 into an equation.
5. If needed, solve the resulting equation with basic methods.
6. 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:

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

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

To compare fractions using cross multiplication, you need to:

1. Make sure you have only a fraction on each side.
2. Multiply the numerator of the first by the denominator of the second.
3. Multiply the numerator of the second by the denominator of the first.
4. Compare values from steps 2 and 3.
5. If the one in step 2 was:
• Smaller, then the first fraction is smaller; or
• Larger, then the first fraction is larger.
6. If one of the multipliers was negative, change the relation to its opposite.
7. Enjoy having compared fractions using cross multiplication.

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

To solve proportions using cross multiplication, you need to:

1. Make sure you only have a fraction on each side.
2. Multiply the numerator of the first by the denominator of the second.
3. Multiply the numerator of the second by the denominator of the first.
4. Combine steps 2-3 into an equation.
5. Solve the resulting equation with basic methods.
6. Enjoy having solved a proportion using cross multiplication.
Maciej Kowalski, PhD candidate Enter any three numbers:
A
B
C
D
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