In the previous post, we have learned the meaning and notation of ratio. We can write the ratio of 8 girls and 12 boys as 8:12 or as 8/12. However, if we represent this as fraction, we can also reduce the fraction to its lowest terms which is equal to 2/3. Converting to lowest term is dividing the numerator and denominator by the largest possible integer known as the greatest common factor or greatest common divisor. In the example above, the greatest common divisor of 8 and 12 is 4, and 8 divided by 4 is 2, and 12 divided by 4 is 3, so, 8:12 can also be represented as 2:3.
The Meaning of Direct Proportion
Consider the following problem.
A car is traveling at an average speed of 60 kilometers per hour. What is the total distance it traveled after 5 hours?
We can solve the problem above by simply multiplying 5 hours by 60 kilometers per hour giving us 300 kilometers. We can also answer the problem by simply constructing the table below.
Notice from the table that if the number of hours is multiplied by 2, then the distance is also multiplied by 2. For example, from 1 hour to 2 hours, the number of hours is multiplied by 2, and the distance is also multiplied by 2, that is 60 × 2 = 120 hours. From 2 hours to 4 hours, the number of hours is also multiplied by 2 and the distance is also multiplied by 2, that is 120 × 2 = 240 hours. If the number of hours is multiplied by 3, the distance is also multiplied by 3. From 1 hour to 3 hours, the number of hours is multiplied by 3 and the distance is also multiplied by 3, that is 60 × 3 = 180 hours.
Suppose we have two quantities and if we multiply one quantity by a number, then the other quantity is also multiplied the same number, then we say that the two quantities are directly proportional. In the example above, time and distance are the two quantities that are directly proportional.
Representing Direct Proportions
We can represent the problem above in ratio. The first ratio is 60 kilometers and 1 hour. The second ratio is 5 hours and an unknown number of kilometers. If we let the unknown number of hours be n, then the ratios are
1 hr :60 km and 5 hrs :n km
Notice that the number of hours is multiplied by 5 (1 hr to 5 hrs), so the distance should also be multiplied by 5. That is, 60 × 5 = 300.
Now that we found the answer to the problem above, let us represent them in ratios as shown.
1:60 and 5:300
Observe from the representation that the product of the outer terms (1 and 300) is equal to the product of the inner terms (60 and 5). The product are both 300. This property is always true in directly proportional quantities: the product of the outside terms (extremes) is equal to the product of the inside terms (means). In the original representation, we had
1:60 = 5:n
Using the relationship between the means and extremes, we can solve for n algebraically. That is,
1 × n = 60 × 5. So, n = 300.
This can also be represented in fraction as 1/60 = 5/n. Cross multiplying, we have n = (60)(5) = 300.
In a directly proportional relationship, if the ratios are a:b and c:d, then
a: b = c: d
and a × d = b × c.
Three cubes of sugar is needed to make 1 cups of coffee. How many cubes of sugar is needed to make 20 cups of coffee?
Let x = number of cubes of sugar needed to make 20 cups of coffee.
3:1 = x:20
The product of the extremes is equal to the product of the means, so solving algebraically, we have
1(x) = (3)(20)
x = 60.
Therefore, we need 60 cubes of sugar for 20 cups of coffee.
In the next post, we are going to have some practice problems on how to solve direct proportion problems.