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 5007 January 1, 1970
Checking for proportion – medium 0 B 4921 January 1, 1970
Checking for proportion – hard 0 B 3784 January 1, 1970
Solve proportions
Solving proportions – Integers to fractions – easy 0 B 4570 January 1, 1970
Solving proportions – Integers to fractions – medium 0 B 4208 January 1, 1970
Solving proportions – Integers to fractions – hard 0 B 2957 January 1, 1970
Solving proportions – Integers to decimals – easy 0 B 1624 January 1, 1970
Solving proportions – Integers to decimals – medium 0 B 1797 January 1, 1970
Solving proportions – Integers to decimals – hard 0 B 2243 January 1, 1970
Solving proportions – Decimals to decimals – easy 0 B 1521 January 1, 1970
Solving proportions – Decimals to decimals – medium 11.3 kB 1522 January 1, 1970
Solving proportions – Decimals to decimals – hard 0 B 1789 January 1, 1970
Word problems
Proportions – Word problems – easy 0 B 5558 January 1, 1970
Proportions – Word problems – medium 0 B 6145 January 1, 1970
Proportions – Word problems – hard 0 B 8091 January 1, 1970
Similar proportions
Similar figures – very easy 0 B 5153 January 1, 1970
Similar figures – easy 0 B 6758 January 1, 1970
Similar figures – medium 0 B 11178 January 1, 1970
Similar figures – hard 0 B 6880 January 1, 1970
Similar figures – very hard 0 B 4890 January 1, 1970
Similar proportions – Word problems
Similar figures – Word problems – easy 0 B 6827 January 1, 1970
Similar figures – Word problems – medium 0 B 14036 January 1, 1970
Similar figures – Word problems – hard 0 B 5457 January 1, 1970

Proportions worksheets for students

Worksheet Name File Size Downloads Upload date
Checking for a proportion 0 B 3089 January 1, 1970
Solving proportions of integers 0 B 3503 January 1, 1970
Solving proportions of decimals 0 B 1636 January 1, 1970
Proportions – Word problems 0 B 6738 January 1, 1970
Proportions – Similar figures – Integers 0 B 5584 January 1, 1970
Proportions – Similar figures – Decimals 0 B 3201 January 1, 1970
Proportions – Similar figures – Word problems 0 B 10158 January 1, 1970

Proportions and similarity knowledge test