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. 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  2581  October 13, 2012 
Checking for proportion – medium  459.4 kB  2598  October 13, 2012 
Checking for proportion – hard  459.6 kB  2005  October 13, 2012 
Solve proportions


Solving proportions – Integers to fractions – easy  461.8 kB  2396  October 13, 2012 
Solving proportions – Integers to fractions – medium  462.2 kB  2092  October 13, 2012 
Solving proportions – Integers to fractions – hard  462.2 kB  1399  October 13, 2012 
Solving proportions – Integers to decimals – easy  461.2 kB  711  October 13, 2012 
Solving proportions – Integers to decimals – medium  461.6 kB  791  October 13, 2012 
Solving proportions – Integers to decimals – hard  465.2 kB  792  October 13, 2012 
Solving proportions – Decimals to decimals – easy  461.4 kB  660  October 13, 2012 
Solving proportions – Decimals to decimals – medium  11.3 kB  646  October 13, 2012 
Solving proportions – Decimals to decimals – hard  461.8 kB  779  October 13, 2012 
Word problems


Proportions – Word problems – easy  458.9 kB  2697  October 13, 2012 
Proportions – Word problems – medium  460.4 kB  3212  October 13, 2012 
Proportions – Word problems – hard  460.5 kB  3425  October 13, 2012 
Similar proportions


Similar figures – very easy  631.1 kB  2548  October 13, 2012 
Similar figures – easy  651.4 kB  3208  October 13, 2012 
Similar figures – medium  710.3 kB  5652  October 13, 2012 
Similar figures – hard  722.1 kB  3355  October 13, 2012 
Similar figures – very hard  705.7 kB  2404  October 13, 2012 
Similar proportions – Word problems


Similar figures – Word problems – easy  459.3 kB  3054  October 13, 2012 
Similar figures – Word problems – medium  459.7 kB  6074  October 13, 2012 
Similar figures – Word problems – hard  460.4 kB  2576  October 13, 2012 
Proportions worksheets for students
Worksheet Name  File Size  Downloads  Upload date 
Checking for a proportion  532.1 kB  1518  October 14, 2012 
Solving proportions of integers  566.6 kB  1605  October 14, 2012 
Solving proportions of decimals  566.8 kB  719  October 14, 2012 
Proportions – Word problems  557.4 kB  2418  October 14, 2012 
Proportions – Similar figures – Integers  3 MB  2343  October 14, 2012 
Proportions – Similar figures – Decimals  545.2 kB  1309  October 14, 2012 
Proportions – Similar figures – Word problems  552.2 kB  3793  October 14, 2012 