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Proportion

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Proportion

Proportion is simply saying we have a relationship between two things. For example, if one variable increases and this causes an equal increase in another variable, this means that these two variables are in proportion. If two things are in proportion, there exists a proportionality constant between the two variables. We will see more on this later.


What symbol do we use for proportion?

To represent that two variables are proportional to one another, we use the symbol ∝. For example, Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Letting voltage be represented by V, and current by I, then we can write .

Whenever we see a proportionality symbol, we can replace this with an equals sign and a proportionality constant. This means we could write Ohm's law as V = kI, where k is our constant.

What are direct proportions?

If two variables are in direct proportion, as one variable increases, then so does the other. Conversely, it means as one variable decreases, then so does the other. Any direct relationship, for variables A and B, can be written as A = kB. This means on a graph, this relationship will be represented as a straight line, passing through the origin. This is shown below.

Directly proportional graph, StudySmarterGraph showing a direct relationship of the form A = kB

The weight of a piece of string is directly proportional to its length. When the piece of string is 30cm long, it weighs 0.2N. Find the weight of the piece of string when the string is 50 cm long.
We know that this is directly proportional, so we know the relationship W∝L, when W represents weight, and L represents length. Let a be our constant of proportionality, so that W = aL. From the first part of the question, we know , so . We can now use this to find the weight when the string is 50cm long. The same relation holds, so . Substituting in our length of 50 cm, we get , so to two decimal places, W = 0.33N.

What are inverse proportions?

Inverse proportion occurs when one variable increasing causes another variable to decrease. If this relationship were to occur between the variables c and d, we would write c∝ . An example of an inverse proportion would be as speed increases, then the time to travel a distance will decrease. Graphically, this means that the shape of the relationship will be represented by y = k / x, with k constant, and x, y variables. This means that the graph will never touch the axis, but it will get very very close as we put a very big x value in, or an x value extremely close to 0. This is shown below.

Inversely proportional graph, StudySmarterGraph showing an inversely proportional relationship

Two variables, b and n are inversely proportional to one another. When b = 6, n = 2. Find the value of n when b is 15.
We know b∝ , so , when k is our constant of proportionality. Filling in values of b and n, we get , so k = 12. This means for all values of b and n, b = 12 / n. We need to find n when b = 15, so we can fill this in, to get 15 = 12 / n. Rearranging this for n, we get n = 12/15 = 0.8.

Proportions and shapes

If two shapes are in proportion, this means that both shapes are the same, with the exception that one of these shapes will have been scaled either up or down. For two shapes to be similar, it is necessary for all the angles in the shape to be the same, and all the sides to be in proportion. Again, here we will have a proportionality constant, which relates the two shapes. In one dimension, this is called a length scale factor, in two dimensions we will call this an area scale factor, and in three dimensions this is called a volume scale factor. We are able to translate between length scale factor and volume or area scale factor. To get the area scale factor, we must multiply the length scale factor in two dimensions, so To get the volume scale factor, we must multiply the length scale factor in three dimensions, so

Two cubes are mathematically similar. The first cube has a face area of 16m². The sides on the second cube are half the length of the sides on the first cube. Find the volume of the second cube. The length scale factor between the shapes is , which implies that the volume scale factor is . If the first cube has a face area of 16m² , this means that it must have side lengths of 4m , which implies it has a volume of 64m³ . As the volume scale factor is , this gives the volume of the second cube as

64 ÷ 8 = 8m³.


The triangles ABE and ACD are similar. Find the length of CD. (All lengths are in cm)

Proportion, Proportion and shapes worked example, StudySmarter

The constant of proportionality, k, between AB and AC is given by (AC) = k (AB), which gives 12 = k 8, so k = 1.5. This means that because the triangles are similar, CD = k (BE), so CD = 1.5 × 10 = 15 cm

Proportion - Key takeaways

  • The symbol for proportion is ∝

  • If two things are in proportion, this means that there is a relationship between them.

  • Direct proportions are of the form y∝x

  • Inverse proportion is of the form y∝

  • If two variables/shapes are in proportion, a proportionality constant exists.

  • (length scale factor) ² = area scale factor.

  • (length scale factor) ³ = volume scale factor .

Frequently Asked Questions about Proportion

Proportion is a measure of how big something is in relation to another object.

Directly proportional means as one variable increases, so does the other. It can be represented by y=kx, when k is a constant.

Inversely proportional means as one variable increases,the other decreases. It can be represented by y=k/x, when k is a constant.

Final Proportion Quiz

Question

A and B are directly proportional. When A=3, B=9. Find A when B=300

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Answer

100

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Question

V and S are directly proportional. When V=7, S=9. Find S when V=-14

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Answer

-18

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Question

Sunflower height and root length are directly proportional. When the flower is 50cm tall, the roots are 15cm in length. Find the height when the roots are 50cm long.

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Answer

50 cm

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Question

The number of prefects is directly proportional to the number of students. At one time, there were 30 prefects to 200 students. Find the number of prefects if there are 350 students.

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Answer

53 (round 52.5 as half a student can’t be a prefect)

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Question

The number of office workers in a building is directly proportional to the number of water coolers. In one building, there are 700 workers and 28 water coolers. Find the number of workers if there are 33 water coolers in a building.

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Answer

825

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Question

G and H are inversely proportional. When G=60, H=21. Find G when H=7

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Answer

180

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R and Y are inversely proportional. When R=14, Y=4. Find Y when R=18

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Answer

28/9

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Question

The waiting time at a dentist is inversely proportional to the number of dentists. When there are two dentists, the waiting time is 34 minutes. Find the waiting time when there are five dentists.

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Answer

13 minutes, 36 seconds

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Question

The amount of goals conceded in a season is inversely proportional to the time spent training per week. One team trains 5 hours per week, and concedes 25 goals in a season. How many goals will be conceded if a team trains for 10 hours?

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Answer

13 (round 12.5 to 13, as can’t have half a goal)

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Question

The number of crampons bought is inversely proportional to the temperature outside. When the temperature is 20°C, 10 crampons are bought. How many crampons are bought if the temperature is 2°C

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Answer

100

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Question

Two rectangles are similar. The sides of one rectangle are of length 6 and 4. One of the sides on the other rectangle is of length 2. Find the two possibilities of the other side length.

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Answer

3 or

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Question

Two prisms are similar. The length of the second prism is 25% shorter. Find the volume scale factor.

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Two spheres are similar. The larger has a volume of 512. The smaller sphere is th of the volume of the larger. Find the surface area of the smaller sphere.

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Question

Three cubes, A, B and C, are similar. B is ⅛  of the volume of A, and C is ⅛ of the volume of B. C has a volume of 125cm³. Find the surface area of A.

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Answer

2400cm³

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Question

The triangles ABE and ACD are similar. Find the length of BE. 

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Answer

9

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60%

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