Have you ever wondered why some quantities increase together, while others do the opposite—one goes up while the other comes down?
That’s the idea behind direct and inverse proportion. In this article, you’ll learn how to tell them apart, how to represent them using graphs, and how they apply to real-life situations.
By the end, you’ll not only know the concepts but also how to spot and solve these types of problems with confidence.
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A van uses 1 litre of petrol to travel a maximum distance of 15 km. The table below shows the maximum distance that the van can travel when there is more petrol.
Observe that:
(i) As x increases, y increases proportionally, ie, if x is doubled, then y will be doubled; if x is tripled, then y will be tripled.
(ii) Similarly, as x decreases, y decreases proportionally, ie, if x is halved, then y will be halved; if x is reduced to 13 of its original value, then y will be reduced to 13 of its original value.
The above relationship between x and y is known as direct proportion.We say that the maximum distance which the van can travel, y km, is directly proportional to the amount of petrol, x litres.
Graphical Representations of Direct Proportion
The graph above shows the relationship between x and y when y is plot against x. Note that for a direct proportion relationship,
(i) the graph is a straight line, and
(ii) the graph must pass through the origin (ie when x = 0, y is also 0.).
Using the same context and table as before, complete the row for quotient.
Note that:
(i) In direct proportion, the quotient (y/x) is a constant. In the above context, the constant 15 represents the maximum distance the van can travel for each litre of petrol.
(ii) If two variables, x and y, are directly proportional, then the quotient of all y values and their respective x values must be the same, ie
y1/x1 = y2/x2 = k, where k is a constant.
(iii) Conversely, if the quotient of each y value and its corresponding x value is the same, such that
y1/x1 = y2/x2 = y3/x3 = y4/x4 = … = yn/xn = k , where k is a constant,
then the variables x and y are directly proportional.
(iv) If y is directly proportional to x, then x is also directly proportional to y. Depending on which variable is the divisor, the value of k may change but it must be constant for all pairs of y values and their corresponding x values.
Recall: In the earlier example, we found that each value of y when divided by the corresponding x value gives us the same constant.
If we represent this constant by k, then y/x = k or y = kx, where k is a constant and ≠0.
Hence, for a general case, we have:
If y is directly proportional to x, then
y/x = k or y = kx, where k is a constant and k ≠ 0.
It is known that A is directly proportional to B.
(i) Given that A = 125 when B = 5, find an equation connecting A and B.
(ii) Find the value of A when B = 15.
(iii) Find the value of B when A = 25.
Solution:
(i)
A = kB
125 = k x 5
k = 25
A = 25B
ii)
A = 25 × 15
= 375
(iii)
B = 25 ÷ 25
= 1
Not all quantities x and y are in direct proportion. Study the following example.
It is given that variables x and y are related by the equation y = 5x2. Complete the table below.
Observe that the quotient y/x is not constant for all pairs of y values and their corresponding
x values. ⸫ the variables x and y are not directly proportional.
The sketch of the graph y = 5x2 for y against x is shown below.
Observe that the graph is not a straight line. This also agrees with the conclusion above that the variables x and y are not directly proportional.
Using the same equation, y = 5x2, complete the table below.
Observe that the quotient y/x is constant for all pairs of y values and their corresponding x values.
⸫ the variables x2 and y are directly proportional.
The sketch of the graph y = 5x2 for y against x2 is shown below.
Observe that the graph is a straight line passing through the origin (when x = 0, x2 = 0, y = 0).
This also agrees with the conclusion above that the variables x2 and y are directly proportional.
In conclusion,
(i) the variables x and y are not directly proportional,
(ii) but the variables x2 and y are directly proportional.
In general, two variables X and Y are directly proportional if Y = kX, where
(i) both Y and X are expressions in terms of 1 or 2 variables,
(ii) k is a constant and k ≠ 0.
Y = k X
The variable expression in the left-side is directly proportional to the variable expression in the right-side.
The table below shows some examples where Y and X are directly proportional.
The table below shows the time taken for a van to travel a distance of 180 km at different speeds.
Observe that:
(i) As x increases, y decreases proportionally, ie, if x is doubled, then y will be halved;
if x is tripled, then y will be reduced to 1/3 of its original value.
(ii) Similarly, as x decreases, y increases proportionally, ie, if x is halved, then y will be
doubled; if x is reduced to 1/3 of its original value, then y will be tripled.
The above relationship between x and y is known as inverse proportion. We say that the
speed of the van, x km/h, is inversely proportional to the time taken, y hours.
Graphical Representation of Inverse Proportion
The graph on below shows the relationship between x and y when y is plot against x. Note
that for an inverse proportion relationship, the graph is never a straight line.
Using the same context and table as before, complete the row for product.
Note that:
(i) In inverse proportion, the product (xy) is a constant. In the above context, theconstant 180 represents the distance the van covered at the given speed with the corresponding time.
(ii) If two variables, x and y, are inversely proportional, then the product of all y values and their respective x values must be the same, ie
x1x1 = x2y2 = k, where k is a constant
(iii) Conversely, if the product of each y value and its corresponding x value is the same,
such that
x1x1 = x2y2 = x3y3 = xnyn = k, where k is a constant
then the variables x and y are inversely proportional.
(iv) If x is inversely proportional to y, then y is also inversely proportional to x.
Recall: In the earlier example, we found that the product of each value of y and the corresponding x value gives us the same constant (more specifically, xy = 180). If we represent this constant by k, then xy = k or = k/x, where ≠0.
Hence, if y is inversely proportional to x, then
xy = k or y = k/x , where k is a constant and ≠0.
It is known that A is inversely proportional to B.
(i) Given that A = 8 when B = 1.5, find an equation connecting A and B.
(ii) Find the value of A when B = 3.
(iii) Find the value of B when A = 48.
Solution:
(i) Let = k/B
When A = 8 and B = 1.5,
8 = k/1.5
k = 8 × 1.5 = 12
⸫ = 12/x
(ii) When B = 3,
A =12/3 = 4
(iii) When A = 48,
48 = 12/B
B = 12/48
= 0.25
Recall that if y is inversely proportional to x, then xy = k or y = k/x , where k is a constant and ≠0.
Similarly, if y is inversely proportional to x3, then x3y = k or y = k/x3 , where k is a constant and ≠0.
In general, two variables X and Y are inversely proportional if Y = k/X, where
(i) both Y and X are expressions in terms of 1 or 2 variables,
(ii) k is a constant and k ≠ 0.
The variable expression in the left-side box is inversely proportional to the variable expression in the right-side box.
The table below shows some examples where Y and X are inversely proportional, and Y and 1/x being directly proportional.
10 workers can paint 8 identical houses in 6 days. How long will it take for 5 workers to paint 2 such houses? State the assumption for your answer to be valid.
Solution:
Assumption: All workers paint at the same rate.
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