Math Calculators
Ratio Calculator

# Ratio Calculator

The ratio calculator simplifies ratios by bringing ratios to the lowest terms. Finds missing values in proportions and compares two given ratios finding if they are equal.

3 : 4 = 600 : 800

250:280 enlarge 2.5 times = 625:700

There was an error with your calculation. ## Ratio calculator

The ratio calculator allows you to simplify ratios, find missing values in proportions, and identify whether the two given ratios are equivalent. The calculator accepts integers, decimal numbers, and numbers in a scientific e-notation as inputs. An example of a number in a scientific e-notation is 2e5, which equals 2 × 10⁵. There is a 15-character input limit, meaning that each input (A, B, C, or D) cannot exceed 15 characters.

## Directions for use

1. To use the calculator as a ratio converter, or, in other words, to simplify a ratio, enter the numerator and the denominator for one side of the ratio. Enter A and B or C and D. Then press "Calculate." The ratios calculator will then simplify the given ratio and return the answer in the lowest terms.

Suppose the known values were inserted as integers or in the scientific e-notation. In that case, the calculator will also demonstrate the steps of the solution.

Suppose the inserted value is already in the lowest terms. In that case, the calculator will find an equivalent ratio by multiplying the numerator and the fraction denominator by 2.

1. To use the calculator for finding a missing value in a proportion, enter the three known values and leave the unknown value field blank. You can use any fields for the unknown value – A, B, C, or D. After entering the three known values, press "Calculate." The calculator will return the solved proportion with all four values. If the entered values were integers, the calculator would also demonstrate the 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 of the numbers by another number.

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 is in the fraction's denominator. Ratios are widely used in real life to compare any two quantities.

For example, if there are 2 girls and 6 boys in a class, the ratio of girls to boys would be 2:6, or, in a simplified form, 1:3, meaning that for every girl, there are three boys.

A proportion is an expression equating two ratios. In our previous example, the proportion could be written as follows:

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

or

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

or

$$2:6=1:3$$

In a proportion a:b=c:d, the second and the third terms, b and c, are called the "means" of the proportion. And the first and the last terms, a and d are called the "extremes." Proportions have a significant property, called the Means-Extremes Property, or the Proportion Formula.

## The proportion formula

In any proportion a:b=c:d, the product of means b × c equals the product of extremes a × d. Or, mathematically:

If

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

Then

$$a × d = b × c$$

This formula allows us to find a missing term of a proportion. For example, if we would need to solve the given proportion for a, we would regroup the proportion formula as follows:

$$a=\frac{b × c}{d}$$

Let's look at the calculation examples of all three scenarios described above.

## Example 1

Jane is a landscape designer creating designs of outdoor spaces for a client. The space has an area of 216 square meters, and she created a plan where a swimming pool takes 64 square meters. Right before Jane submits her design, the client comes up with the requirement that at least a third of the space has to be occupied by the pool. Does she have to make a new design, or can she submit the existing one?

To determine whether or not she has to create a new design, she has to figure out the ratio of the pool area to the total outdoor area and then compare that value to 1/3.

It is given that the pool occupies 64 square meters, while the total outside area is 216 square meters. Therefore, the needed ratio is: 64/216.

The ratio is not in the lowest terms. Therefore, we can simplify it. We can simplify the ratio by dividing the numerator and the denominator by the greatest common factor (the GCF).

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

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

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

Therefore,

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

The pool occupies 8/27 of the total outside area. The client, however, wants it to take up at least 1/3, or 9/27 of the total area. 8/27 < 9/27, and, unfortunately, Jane has to create a new design.

## Simplifying the ratio

To quickly find the solution to the problem, enter 64 and 216 in fields A and B (or C and D), respectively, and press "Calculate."

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

## Finding a missing value

Find a missing value in the following proportion:

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

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

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

99 and 4 are the means in this proportion, and 3 and the unknown value x are the extremes. Therefore:

$$3 × X = 4 × 99$$

and

$$x = \frac{4 × 99}{3}$$

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

$$x = 132$$

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

Helen wants to order a translator to translate several articles from English into Japanese. The translator's website shows an average rate of $20 for a 600-word translation. Helen's articles are around 20,000 words altogether. How will she calculate the order cost if the translator refuses to give her a discount? Enter some equivalent units in fields A and C. Enter other equivalent units in fields B and D. In this example, we use A and С for the number of words and B and D for money. The fields A and B are for the first case (the translator's current rate), and fields C and D are for the second case (the possible rate for Helen's order). • In field A, enter the number of words at the translator's rate - 600. • In field B, you enter the price for 600 words, i.e., 20. • In field C, enter the number of words in your order, i.e., 20,000. • And in field D, you get the result 666.66666666667. Then you can round up the result to$667. Don't forget that Helen can ask for a discount for bulk orders, but $667 can be a starting point in negotiations. ## Example 3 Jack is on vacation in Indonesia and wants to exchange his cash dollars for the Indonesian rupiah's local currency. He needs the money to pay cash to rent a Yamaha X-Max maxi-scooter, which costs 3,500,000 rupiahs per month. He knows that today the exchange rate at the nearest exchanger to his hotel is 14,750 rupiahs for one U.S. dollar. How many dollars does he need to exchange to get 3,500,000 rupees? And again, we use some equivalent units in fields A and C and other equivalent units in fields B and D. In this example, we use A and С for Indonesian rupiah and B and D for U.S. dollars. • In box A, enter the number of rupees per$1, i.e., 14,750.
• In field B, you enter the equivalent of that amount in dollars, i.e., 1.
• In field C, you enter the number of rupees you want to get, i.e., 3,500,000.
• In field D, you will get the amount you want in dollars, i.e., 237.28813559322.

It turns out that if the money changer does not take commission, he needs to exchange at least $237 to pay for scooter rent for a month. He will likely exchange a rounder sum -$250 or \$300.

## Example 4

You are buying groceries in the supermarket and want to get some apples. In the fruits and vegetable department, only pre-packaged apples are available. One of the packages has 4 apples in it and costs 16 dollars. The other package has 3 apples, but it costs 12 dollars. You don't want to spend too much money, so you want to figure out, in which case the price per one apple is smaller, and buy that package. Which package should you buy? Is there a price difference per one apple between the two packages?

Let's say that package 1 has 4 apples and costs 16 dollars. That means the price per apple will be 4/16. Package 2 contains 3 apples and costs 12 dollars. Therefore, the price per one apple, in this case, will be 3/12. To figure out if there is a price difference per apple between these two packages, you need to answer the following question: Are the following two ratios equivalent: 4/16 and 3/12?

To determine whether the given ratios are equivalent, we need to write them down in the lowest terms. To write a ratio in the lowest terms, we need to simplify it by dividing the numerator and the denominator by their GCF.

The greatest common factor of the numerator and the denominator of the first ratio, 4 and 16, is 4. Therefore, we can simplify it as follows:

$$\frac{4}{16}=\frac{4/4}{16/4}=\frac{1}{4}$$

The greatest common factor of the numerator and the denominator of the second ratio, 3 and 12, is 3. Therefore, we can simplify it as follows:

$$\frac{3}{12}=\frac{3/3}{12/3}=\frac{1}{4}$$

The two ratios are equal in the lowest terms, meaning that the original ratios are equivalent. The original ratios are equivalent. Therefore, the price per apple is the same for both available packages. You can choose any of them!

## Using the calculator to find the solution

To use the calculator for comparing the two ratios, 4/16 and 3/12, enter 4 in field A and 16 in field B, to complete one side of the proportion. Enter 3 in field C and 12 in field D to complete the other side of the proportion. Then press "Calculate."

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

is TRUE

## Proportion properties

The most important property of proportions (and the most useful) is the Means-Extremes property. Proportions do, however, have some other interesting properties.

The means and the extremes permutation:

If

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

Then, with the means permutation, the following is true:

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

And, with the extreme's permutation, the following is true:

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

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

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, the two values are in a golden ratio if the ratio of the larger value to the smaller one is the same as the ratio of the sum of these values to the larger value. Or, in mathematical terms: for a>b>0, the golden ratio can be written as follows:

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

The human brain considers the golden ratio the perfect ratio of the parts to a whole. And the golden ratio is often observed in nature, science, and art.