Proportions

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Proportion is represented by two equal ratios. There is direct and indirect proportion. With direct proportion, the two variables change at the same rate.

Direct Proportion
With direct proportion, the two variable change at the same time. In direct proportion, as the first variable increases (decreases), the second variable also increases (decreases). In mathematical statements, it can be expressed as y = kx. This reads as “y varies directly as x” or “y is directly proportional as x” where k is constant in the equation.

Example:
y is directly proportional to x, when x = 15, y = 30. Find y when x = 40.

Solution 1
In solving proportion, it’s usually solved using the two equal ratios then cross multiplying.
15 : 30 = 40 : y
\(\frac{15}{30}\)  = \(\frac{40}{ y}\)            Cross multiply the given proportion.
15y = 40 (30)
y =\( \frac{40(30)}{15}\)
y = \(\frac{1200}{15}\)
y = 80

Solution 2
There is an easier way to solve direct proportion and still continue to solve the change of value.  Use x = ky, where k stands as a constant.
x = ky                          Substitute the given to get the value of constant (k).
15 = k (30)
k = \(\frac{15}{30}\)
k = 0.5
Rewrite the equation with the value of constant to solve for the change in y.
x = ky
40 = (0.5)y
y = \(\frac{40}{0.5}\)
y = 80

Example 2:
The distance fell (d) of a freely falling body varies directly as the time (t) the object hits the ground.
When d = 3 and t = 1.2.
a. Find d when t = 2
b. Find t when d = 7

Let’s express it in an equation where distance is directly proportional the square of time or
d = kt .   To solve the problem, first solve for k.
d =kt        Substitute the given
3 = k (1.2)
\(\frac{3}{1.2}\) =k
2.5 =k

a. d = ?        t = 2        From the formula, substitute the value of time and k.
d = kt
d = 2.5 (2)
d = 5

b. t = ?        d = 7
d = kt
7 = 2.5t
\(\frac{7}{2.5}\) = t
t = 2.8

Indirect Proportion
An inverse variation occurs if one of the variables increases or decreases and the other variable decreases or increases. It can be read as “varies inversely” and “inverse proportion”. Inverse variation exists if the relationship exists between the two variables whose product is constant (k).

Example:
The number of hours (h) it takes a block of ice to melt varies inversely with the temperature (t)
when h = 1 and t = 35 degree Celsius.

a. Find h when t = 20 degree Celsius.
Solve for the constant
k = ht
k= 1(35)
k = 35

b. h = ?           t = 20
h = \(\frac{k}{t}\)
h = \(\frac{35}{20}\)
h = 1.75 hours