Ratio Calculator

Ratio calculator

Our versatile ratio calculator allows you to effortlessly simplify ratios, find missing values in proportions, and determine if two given ratios are equivalent. This tool accepts a variety of inputs, including integers, decimals, and numbers in scientific e-notation. For example, a number in scientific e-notation like 2e5 represents 2 × 10⁵. Please note that there is a 15-character limit for each input field, meaning fields A, B, C, or D cannot exceed this length.

Directions for use

1. To use the calculator as a ratio converter—or, in other words, to simplify a ratio—enter the numerator and denominator for one side of the ratio. Enter these into fields A and B, or C and D. Then, click "Calculate." The ratio simplifier will immediately process the given ratio and return the answer in its lowest terms.

If the known values are entered as integers or in scientific e-notation, the calculator will also display the step-by-step solution.

If the inserted value is already in its simplest form, the calculator will generate an equivalent ratio by multiplying both the numerator and the denominator by 2.

2. To use the proportion calculator to find a missing value, input the three known values and leave the field for the unknown value blank. You can leave any of the fields empty—A, B, C, or D. After entering the three known values, click "Calculate." The tool will solve the proportion and display all four values. If you input integers, the calculator will also provide the step-by-step solution to the problem.

Definitions and important formulas

In mathematics, a ratio is defined as an ordered pair of numbers, a and b. We use ratios to compare two values by dividing one number by the other.

A ratio of a to b can be written as \$\frac{a}{b}\$, a/b, or a:b. It is generally assumed that b ≠ 0, since b represents the denominator of the fraction. Ratios are widely used in everyday life to compare any two quantities.

For example, if a class consists of 2 girls and 6 boys, the ratio of girls to boys is 2:6. In simplified form, this is 1:3, meaning that for every one girl, there are three boys.

A proportion is a mathematical expression equating two ratios. Using our previous example, the proportion can be written as follows:

$$2:6::1:3$$

or

$$\frac{2}{6}=\frac{1}{3}$$

or

$$2:6=1:3$$

In the proportion a:b=c:d, the second and third terms (b and c) are known as the "means" of the proportion. The first and last terms (a and d) are referred to as the "extremes." Proportions possess a fundamental property known as the Means-Extremes Property, or the Proportion Formula.

The proportion formula

In any proportion a:b=c:d, the product of the means (b × c) equals the product of the extremes (a × d). Mathematically, this is expressed as:

If

$$\frac{a}{b} = \frac{c}{d}$$

Then

$$a × d = b × c$$

This formula allows us to easily find a missing term in any proportion. For example, if we need to solve a given proportion for a, we simply rearrange the proportion formula as follows:

$$a=\frac{b × c}{d}$$

Let's explore some practical calculation examples covering all three of the scenarios described above.

Example 1

Jane is a landscape designer planning an outdoor space for a client. The total area of the space is 216 square meters, and she has drafted a layout featuring a swimming pool that covers 64 square meters. Right before Jane submits her proposal, the client adds a new requirement: the pool must occupy at least one-third of the total space. Does Jane need to draft a new design, or can she submit her current one?

To determine this, Jane must calculate the ratio of the pool's area to the total outdoor area and compare that value to 1/3.

Given that the pool is 64 square meters and the total area is 216 square meters, the initial ratio is: 64/216.

Since this ratio is not in its lowest terms, we can simplify it. We simplify the ratio by dividing both the numerator and the denominator by their greatest common factor (GCF).

The greatest common factor for 64 (the numerator) and 216 (the denominator) is 8. Dividing both terms by the GCF of 8, we get:

$$\frac{64}{8} = 8$$

$$\frac{216}{8} = 27$$

Therefore,

$$\frac{64}{216} = \frac{8}{27}$$

The pool currently occupies 8/27 of the total outdoor area. However, the client requested it take up at least 1/3, which is equivalent to 9/27. Since 8/27 < 9/27, Jane unfortunately needs to create a new design.

Simplifying the ratio

To quickly find the solution using our ratio simplifier tool, simply enter 64 and 216 into fields A and B (or C and D), respectively, and click "Calculate."

Answer:

$$\frac{64}{216} = \frac{8}{27}$$

Finding a missing value

Let's find the missing value in the following proportion:

$$\frac{3}{99} = \frac{4}{x}$$

To solve for an unknown value in a proportion, we apply the proportion formula, which states that the product of the means always equals the product of the extremes. We can write the given proportion as follows:

$$\frac{3}{99} = \frac{4}{x}$$

Here, 99 and 4 are the means, while 3 and the unknown value x are the extremes. Therefore:

$$3 × X = 4 × 99$$

and

$$x = \frac{4 × 99}{3}$$

$$x = \frac{396}{3}$$

$$x = 132$$

Answer

$$\frac{3}{99} = \frac{4}{132}$$

Example 2

Helen needs to hire a freelance translator to convert several articles from English into Japanese. The translator's website lists an average rate of $20 per 600 words. Helen's articles total approximately 20,000 words. How can she calculate the total cost of her order if the translator does not offer a bulk discount?

You can easily solve this by entering equivalent units into the calculator. Use fields A and C for one set of equivalent units, and fields B and D for the other.

In this scenario, we'll use fields A and С for the word count, and fields B and D for the cost. Fields A and B represent the known rate (the translator's current pricing), while fields C and D represent Helen's specific order.

• In field A, enter the number of words at the translator's base rate, which is 600.
• In field B, enter the price for those 600 words, which is 20.
• In field C, enter the total number of words in your order, which is 20,000.
• In field D, the calculator will output the result: 666.66666666667.

Helen can round this result up to $667. While she can always negotiate a discount for a bulk order, $667 gives her a solid starting point for negotiations.

Example 3

Jack is vacationing in Indonesia and needs to exchange his U.S. dollars for the local currency, Indonesian rupiahs. He needs cash to rent a Yamaha X-Max maxi-scooter, which costs 3,500,000 rupiahs per month.

He knows that today's exchange rate at the currency exchange nearest to his hotel is 14,750 rupiahs per 1 U.S. dollar. How many dollars does he need to exchange to get exactly 3,500,000 rupiahs?

Once again, we place equivalent units in fields A and C, and the other equivalent units in fields B and D.

In this example, A and С will represent the Indonesian rupiah, while B and D will represent U.S. dollars.

• In field A, enter the number of rupiahs per $1, which is 14,750.
• In field B, enter the dollar equivalent of that amount, which is 1.
• In field C, enter the total number of rupiahs Jack needs, which is 3,500,000.
• In field D, you will get the required amount in dollars: 237.28813559322.

Assuming the money changer doesn't charge a commission, Jack needs to exchange at least $237 to cover his scooter rental for the month. Realistically, he will likely exchange a rounder sum, such as $250 or $300.

Using the calculator to find the solution

To use the equivalent ratio calculator to compare two ratios—such as 4/16 and 3/12—enter 4 into field A and 16 into field B to complete one side of the proportion. Then, enter 3 into field C and 12 into field D for the other side. Finally, click "Calculate."

Answer

$$\frac{4}{16} = \frac{3}{12}$$

is TRUE

Proportion properties

The most critical and useful property of proportions is the Means-Extremes Property. However, proportions also feature several other interesting mathematical properties.

The means and the extremes permutation:

If

$$\frac{a}{b}=\frac{c}{d}$$

Then, applying the means permutation, the following holds true:

$$\frac{a}{c}=\frac{b}{d}$$

And, applying the extremes permutation, the following is true:

$$\frac{d}{b}=\frac{c}{a}$$

Increasing and decreasing the proportion can be done according to the following rules:

If

$$\frac{a}{b}=\frac{c}{d}$$

Then the proportion can be increased as follows:

$$\frac{a+b}{b}=\frac{c+d}{d}$$

And decreased as follows:

$$\frac{a-b}{b}=\frac{c-d}{d}$$

Composing a proportion by addition and subtraction If

$$\frac{a}{b}=\frac{c}{d}$$

Then the following is true:

$$\frac{a+c}{b+d}=\frac{a}{b}=\frac{c}{d}$$

And

$$\frac{a-c}{b-d}=\frac{a}{b}=\frac{c}{d}$$

The golden ratio

In mathematics, two values are in the golden ratio if the ratio of the larger value to the smaller value equals the ratio of their sum to the larger value. In mathematical terms, for a>b>0, the golden ratio formula is written as follows:

$$\frac{a}{b}=\frac{a+b}{a}$$

The human brain inherently perceives the golden ratio as the most aesthetically pleasing proportion between parts and a whole. Unsurprisingly, the golden ratio is frequently observed throughout nature, science, and art.