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 (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 0 B 4655 January 1, 1970
Checking for proportion – medium 0 B 4591 January 1, 1970
Checking for proportion – hard 0 B 3553 January 1, 1970
Solve proportions
Solving proportions – Integers to fractions – easy 0 B 4255 January 1, 1970
Solving proportions – Integers to fractions – medium 0 B 3879 January 1, 1970
Solving proportions – Integers to fractions – hard 0 B 2713 January 1, 1970
Solving proportions – Integers to decimals – easy 0 B 1508 January 1, 1970
Solving proportions – Integers to decimals – medium 0 B 1668 January 1, 1970
Solving proportions – Integers to decimals – hard 0 B 2121 January 1, 1970
Solving proportions – Decimals to decimals – easy 0 B 1402 January 1, 1970
Solving proportions – Decimals to decimals – medium 11.3 kB 1400 January 1, 1970
Solving proportions – Decimals to decimals – hard 0 B 1650 January 1, 1970
Word problems
Proportions – Word problems – easy 0 B 5071 January 1, 1970
Proportions – Word problems – medium 0 B 5777 January 1, 1970
Proportions – Word problems – hard 0 B 7331 January 1, 1970
Similar proportions
Similar figures – very easy 0 B 4830 January 1, 1970
Similar figures – easy 0 B 6335 January 1, 1970
Similar figures – medium 0 B 10602 January 1, 1970
Similar figures – hard 0 B 6503 January 1, 1970
Similar figures – very hard 0 B 4600 January 1, 1970
Similar proportions – Word problems
Similar figures – Word problems – easy 0 B 6436 January 1, 1970
Similar figures – Word problems – medium 0 B 13259 January 1, 1970
Similar figures – Word problems – hard 0 B 5108 January 1, 1970


Proportions worksheets for students

Worksheet Name File Size Downloads Upload date
Checking for a proportion 0 B 2870 January 1, 1970
Solving proportions of integers 0 B 3208 January 1, 1970
Solving proportions of decimals 0 B 1501 January 1, 1970
Proportions – Word problems 0 B 6043 January 1, 1970
Proportions – Similar figures – Integers 0 B 5167 January 1, 1970
Proportions – Similar figures – Decimals 0 B 2907 January 1, 1970
Proportions – Similar figures – Word problems 0 B 9513 January 1, 1970


Proportions and similarity knowledge test