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.

proportions

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. 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 1763 October 13, 2012
Checking for proportion – medium 459.4 kB 1837 October 13, 2012
Checking for proportion – hard 459.6 kB 1350 October 13, 2012
Solve proportions
Solving proportions – Integers to fractions – easy 461.8 kB 1646 October 13, 2012
Solving proportions – Integers to fractions – medium 462.2 kB 1447 October 13, 2012
Solving proportions – Integers to fractions – hard 462.2 kB 962 October 13, 2012
Solving proportions – Integers to decimals – easy 461.2 kB 466 October 13, 2012
Solving proportions – Integers to decimals – medium 461.6 kB 510 October 13, 2012
Solving proportions – Integers to decimals – hard 465.2 kB 522 October 13, 2012
Solving proportions – Decimals to decimals – easy 461.4 kB 424 October 13, 2012
Solving proportions – Decimals to decimals – medium 11.3 kB 406 October 13, 2012
Solving proportions – Decimals to decimals – hard 461.8 kB 490 October 13, 2012
Word problems
Proportions – Word problems – easy 458.9 kB 1890 October 13, 2012
Proportions – Word problems – medium 460.4 kB 2262 October 13, 2012
Proportions – Word problems – hard 460.5 kB 2005 October 13, 2012
Similar proportions
Similar figures – very easy 631.1 kB 1621 October 13, 2012
Similar figures – easy 651.4 kB 1981 October 13, 2012
Similar figures – medium 710.3 kB 3552 October 13, 2012
Similar figures – hard 722.1 kB 2114 October 13, 2012
Similar figures – very hard 705.7 kB 1513 October 13, 2012
Similar proportions – Word problems
Similar figures – Word problems – easy 459.3 kB 1889 October 13, 2012
Similar figures – Word problems – medium 459.7 kB 3452 October 13, 2012
Similar figures – Word problems – hard 460.4 kB 1718 October 13, 2012


Proportions worksheets for students

Worksheet Name File Size Downloads Upload date
Checking for a proportion 532.1 kB 1035 October 14, 2012
Solving proportions of integers 566.6 kB 1128 October 14, 2012
Solving proportions of decimals 566.8 kB 467 October 14, 2012
Proportions – Word problems 557.4 kB 1487 October 14, 2012
Proportions – Similar figures – Integers 3 MB 1443 October 14, 2012
Proportions – Similar figures – Decimals 545.2 kB 817 October 14, 2012
Proportions – Similar figures – Word problems 552.2 kB 2378 October 14, 2012


Proportions and similarity knowledge test