# Proportions and Similarity

Proportions in mathematics can be viewed from a few perspectives. For instance, the proportionality of two variable values is determined by checking if one of the values is the product of the other value and some constant. In other words, two variable values (numbers or quantities) are proportional if their ratio is a constant, called the coefficient of proportionality or the proportionality constant. This is best explained using the linear equation:

y = k*x

If k is a constant quantity, x will always be proportional to y for every possible value. Then k is considered to be the coefficient of proportionality.

Proportion is also the name we use when describing the equality of two ratios. If the ratios in question are equal, we say that they are proportional. For example, we have two ratios here:

5/6 = 15/18

These ratios are proportional because when we multiply both the numerator and the denominator of the ratio 5/6 by 3, we get 15/18 as a result. That is also true for the other way around – if we simplify the second ratio by dividing its numerator and denominator by 3, we get the first ratio as a result. Let us try another example:

2/3 = 8/9

As you can see, this equation is not valid – 8 is the product of 2 times 4 and 9 is the product of 3 times 3. That means that these ratios are not proportional. If we wanted to find the proportional ratio to 2/3 while keeping the denominator of the other ratio, we would have to multiply the numerator 2 with the number 3. So the correct proportion would be:

2/3 = 8/9

Similarity is a form of proportion used to compare sizes of shapes and objects and the same rules apply when solving both similarity and proportion. Knowing your way around similarities is especially useful when working with maps, blueprints and models. In those cases you are often given a ratio. The ratio of 1 : 3 in a model means that 1 cm on the model represents 3 cm on the actual object. The important thing to remember is that for two shapes or objects to be similar, they have to have the same shape and all of their sides have to be proportional (cialis online). That means that if one side of an object has been reduced by a factor of 2, all other sides have to be reduced by the same factor if they are to be similar. Let us try to solve a word problem with similarity.

A map has a scale of 1 cm : 20 km. If Elm Grove and Small Creek are 100 km apart, then they are how far apart on the map?

The first thing we should do is to form a proportion. Since the distance between Elm Grove and Small Creek on the map is unknown, it would look like this:

1/20 = x/100

Now, we just have to get rid of the second denominator to find the value of x. We will do it by multiplying the whole equation by 100:

1/20 = x/100 |*100

100/20 = x

x = 5

Now we know that the distance between Elm Grove and Small Creek on the map is 5 cm.

This approach can be used on various similar examples. If you wish to practice proportions and similarity, feel free to use the worksheets below.

## Proportions exams for teachers

 Worksheet Name File Size Downloads Upload date Check valid proportion Checking for proportion – easy 459.5 kB 4231 October 13, 2012 Checking for proportion – medium 459.4 kB 4237 October 13, 2012 Checking for proportion – hard 459.6 kB 3324 October 13, 2012 Solve proportions Solving proportions – Integers to fractions – easy 461.8 kB 3985 October 13, 2012 Solving proportions – Integers to fractions – medium 462.2 kB 3529 October 13, 2012 Solving proportions – Integers to fractions – hard 462.2 kB 2458 October 13, 2012 Solving proportions – Integers to decimals – easy 461.2 kB 1382 October 13, 2012 Solving proportions – Integers to decimals – medium 461.6 kB 1537 October 13, 2012 Solving proportions – Integers to decimals – hard 465.2 kB 1986 October 13, 2012 Solving proportions – Decimals to decimals – easy 461.4 kB 1279 October 13, 2012 Solving proportions – Decimals to decimals – medium 11.3 kB 1273 October 13, 2012 Solving proportions – Decimals to decimals – hard 461.8 kB 1522 October 13, 2012 Word problems Proportions – Word problems – easy 458.9 kB 4552 October 13, 2012 Proportions – Word problems – medium 460.4 kB 5424 October 13, 2012 Proportions – Word problems – hard 460.5 kB 6514 October 13, 2012 Similar proportions Similar figures – very easy 631.1 kB 4483 October 13, 2012 Similar figures – easy 651.4 kB 5887 October 13, 2012 Similar figures – medium 710.3 kB 9952 October 13, 2012 Similar figures – hard 722.1 kB 6096 October 13, 2012 Similar figures – very hard 705.7 kB 4254 October 13, 2012 Similar proportions – Word problems Similar figures – Word problems – easy 459.3 kB 6010 October 13, 2012 Similar figures – Word problems – medium 459.7 kB 12145 October 13, 2012 Similar figures – Word problems – hard 460.4 kB 4727 October 13, 2012

## Proportions worksheets for students

 Worksheet Name File Size Downloads Upload date Checking for a proportion 532.1 kB 2660 October 14, 2012 Solving proportions of integers 566.6 kB 2960 October 14, 2012 Solving proportions of decimals 566.8 kB 1378 October 14, 2012 Proportions – Word problems 557.4 kB 5311 October 14, 2012 Proportions – Similar figures – Integers 3 MB 4699 October 14, 2012 Proportions – Similar figures – Decimals 545.2 kB 2587 October 14, 2012 Proportions – Similar figures – Word problems 552.2 kB 8569 October 14, 2012