1.8.1 Proportion
Proportion:
Proportion describes how two variables are connected to one another.
Direct proportion:
The relationship between two values in which the ratio of the two equals a constant number is
known as direct proportion or direct relation. Which means if one quantity increases the other increases by the same factor. It is represented by the proportional sign ∝
If \( x \) is directly proportional to \( y \) then
\( y=kx \)
Where \( k \) is a proportionality constant
And its graph is a straight line
Working with direct proportion:
Some problems may involve a variable being directly proportional to a function of another variable.
e.g.
\( y \) is directly proportional to the square root of \( x \), which means \( y=k \sqrt{x} \)
\( y \)is directly proportional to the cube of \( x \), which means \( y=kx^3 \)
To solve such problems
Identify the variables
Replace sign of proportionality with proportionality constant
Find value of constant by using given values of variables
Write formula in terms of the variables
Use the formula to find the required quantity
Worked example:
A ball falls \( d \) metres in \( t \) seconds.
\( d \) is directly proportional to the square of \( t \).
The ball falls \( 44.1 \) m in \( 3 \) seconds.
Find the formula for \( d \) in terms of \( t \).
Identify the two variables 🠦 \( d \), \( t^2 \)
As this is a direct proportion 🠦 \( d=kt^2 \)
Now we can find k by using \( d=44.1 \) and \( t=3 \)
\( 44.1=k\times 3^2 \)
\(k= \frac{44.1}{3^2} \) 🠦 \( k=4.9 \)
Now we can write the formula in \( d \) and \( t \)
\( d=4.9t^2 \)
Calculate the distance the ball falls in \( 2 \) seconds.
Put value of t in \( d=4.9t^2 \)
\( d=4.9(2)^2 \) 🠦 \( d=19.6 \)
Inverse Proportion:
Inverse proportions indicate that as one variable increases, the other decreases by the equal amount. When two quantities are inversely proportional, one is directly proportional to the reciprocal of the other. It is represented by the proportional sign∝, If \( x \) is inversely proportional to \( y \) then
\( y=\frac{k}{x} \)
Where\( k \) is a proportionality constant.
Working with inverse proportion:
Some problems may involve a variable being inversely proportional to a function of another variable.
e.g.
\( y \) is inversely proportional to the square root of \( x \), which means \( y=\frac{k}{ \sqrt{x} } \)
\( y \) is inversely proportional to the cube of \( x \), which means \( y= \frac{k}{x^3} \)
To solve such problems
Identify the variables
Replace sign of proportionality with proportionality constant
Find value of constant by using given values of variables
Write formula in terms of the variables
Use the formula to find the required quantity
Worked example:
\( y \) is inversely proportional to \( x \).
When \( x=9 \), \( y=8 \).
Find \( y \), when \( x=6 \) .
Identify the two variables 🠦 \( y,x \)
As this is an inverse proportion 🠦 \( y=kx \)
Now we find \( k \) by using \( x=9 \) and \( y=8 \)
\( 8=k \times 9 \) 🠦 \( k= \frac{8}{9} \)
Now we can write the formula in \( y \) and \( x \) ⇒ \( y= \frac{8}{9} x \)
Now we calculate required \( y \) by putting \( x=6 \) in \( y= \frac{8}{9} x \)
\( y= \frac{8}{9}(6) \)
Simplify
\( y= \frac{16}{3} \) 🠦 \( y=5 \frac{1}{3} \)
Test yourself
Question 1:
\( y \) is directly proportional to the square root of \( x \) .
When \( x=9 \), \( y=6 \)
Find \( y \) when \( x=25 \)
[3]
Question 2:
\( m \) is inversely proportional to the square of \( (p-1) \).
When \( p=4 \), \( m=5 \).
Find \( m \) when \( p=6 \).
[3]
Question 3:
\( y \) is directly proportional to the cube root of \( (x+3) \).
When \( x=5 \) , \( y= \frac{2}{3} \)
Find \( y \) when \( x=24 \).
[3]
Question 4:
\( y \) is inversely proportional to \( x^3 \).
When \( x=2 \), \( y=0.5 \)
Find \( y \) in terms of \( x \).
[3]