Explainer: Equivalent Ratios

In this explainer, we will learn how to simplify ratios, whether they be part-to-part or three-part ratios, and how to find equivalents to these ratios.

We are interested here, in particular, in ratios that compare two parts of a whole, usually written as 𝑎𝑏 (sometimes also as the fraction 𝑎𝑏). Ratios, like fractions, can be simplified and all have a simplest form.

Definition: Ratios

Ratios are used to compare two numbers.

For instance, to bake cookies, you need to mix 2 cups of flour with 1 cup of sugar. We can say that the ratio of flour to sugar is 2 to 1. We can write this ratio as 21 (read “two to one”).

It means that for every two cups of flour, there is one cup of sugar.

If I want to make many cookies, I may decide to multiply the quantities given in the recipe by, say, three.

I will mix 6 cups of flour with 3 cups of sugar. The ratio of flour to sugar is now 63. However, I still have one cup of sugar for every two cups of flour. We see that the ratios 21 and 63 are equivalent: the numbers 2 and 1 and the numbers 6 and 3 compare in the same way. In both cases, the rstnumber is double the secondnumber.

From this, we see that ratios are equivalent if it is possible to get one from the other by multiplying or dividing by a whole number (except zero) both sides of the ratio.

It follows that it is possible to simplify every ratio to a unique form, called the simplest form, defined as the form where it is not possible to further divide both sides of the ratio and still get integers. This situation happens when the two sides of the ratio have no common factor other than 1.

Let us see how this applies with the following examples.

Example 1: Expressing a Simple Ratio Given by a Diagram in Its Simplest Form

Using the figure below, express the ratio of the number of squares to the number of triangles in its simplest form.

Answer

We need first to count the number of squares, 8, and the number of triangles, 10, and then form the corresponding ratio: 810.

This ratio means that we have 8 squares for 10 triangles. To write it in its simplest form, we need to figure out whether there is a common factor (other than 1) between 8 and 10. The answer is yes, as 2 is a common factor. Therefore, we can divide both sides of the ratio by 2, and we get (8÷2)(10÷2)=45.

The ratio of the number of squares to the number of triangles in its simplest form is 45. It could be represented on the diagram by taking only half of the squares and half of the triangles.

Example 2: Expressing a Simple Ratio in Its Simplest Form

Express the ratio 14056 in its simplest form.

Answer

To simplify the ratio 14056, we need to find all the common factors of 140 and 56. Both numbers have 2 as a factor. By dividing both sides of the ratio by 2, we get (140÷2)(56÷2)=7028.

We can further divide by 2; we get a ratio of 3514.

The numbers 35 and 14 are both multiples of 7; we can divide both sides of the ratio by 7 and get (35÷7)(14÷7)=52.

The numbers 5 and 2 have no common factor other than 1. The answer is thus 52.

Note that we could have noticed that both 140 and 56 are multiples of 28 (i.e., they both have 28 as a factor) and divided both 140 and 56 by 28 directly.

Example 3: Writing a Ratio in Its Simplest Form

Given that there are 50 boys and 20 girls in a class, calculate the ratio of the number of girls to the number of boys in its simplest form.

Answer

There are 20 girls and 50 boys, so the ratio of the number of girls to that of boys is 2050. To simplify this ratio, we look for common factors of 20 and 50. Both are multiples of 10, so we have (20÷10)(50÷10)=25.

The ratio of the number of girls to the number of boys expressed in its simplest form is 25.

We are going now to look at a more complex question where the ratio is not directly given.

Example 4: Expressing a Ratio in Its Simplest Form in a Two-Step Problem

The side length of a square is 4 cm. A rectangle has length 4 cm and width 3 cm. Find the ratio between the area of the square and the area of the rectangle in its simplest form.

Answer

First, we find the area of the square: it is 4=16cm.

Then, we find the area of the rectangle: 4×3=12cm. Hence, the ratio between the area of the square and the area of the rectangle is 1612.

Finally, we express it in its simplest form. For this, we look for all common factors between 16 and 12. We see that their greatest common factor is 4. Dividing both sides of the ratio by 4 gives (16÷4)(12÷4)=43. The final answer is: 43.

Finally, we are going to deal with a ratio that involves fractions and see how to express it in its simplest form.

Example 5: Expressing a Ratio Involving Fractions in Its Simplest Form

Express the ratio 10753 in its simplest form.

Answer

The given ratio is not expressed in a proper form since the numbers are not integers but fractions. To express this ratio with integers, we need to multiply both sides of the ratio by an appropriate number so that both fractions become integers. The easiest way is then to multiply both sides by 7×3, that is, 21, since 7×3×107=3×10 (as 7×107=10) and 7×3×53=7×5. Hence, we have 10753=3035.

This ratio can now be simplified since 30 and 35 are both multiples of 5. Dividing now both sides of the ratio by 5 gives (30÷5)(35÷5)=67. The ratio 10753 is expressed as 67 in its simplest form.

Let us summarize this explainer.

Key Points

  1. The ratio of 𝑎 to 𝑏 compares the two numbers 𝑎 and 𝑏. Ratios usually compare two parts of the same whole and are written as 𝑎𝑏.
  2. The numbers forming the ratio must be integers. If they are not, the fractions or decimal numbers forming the ratio must be multiplied by an appropriate number so that all numbers are then integers.
  3. To express a given ratio in its simplest form, we search for a common factor of the numbers forming the ratio, and divide all the sides of the ratio by this common factor. After that, we search again for a common factor of the two or more obtained numbers and continue until the obtained numbers have no common factor. Then, the ratio is in its simplest form.
  4. If you already know how to find the greatest common factor of two numbers, then you just need to divide the two numbers forming the ratio by their greatest common factor, if there is any, to write the ratio in its simplest form.
  5. If the numbers forming the ratio have no common factor other than 1, then the ratio is written in its simplest form.

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