Ever wondered what someone meant when they say something is “proportionate”? Have you ever needed to adjust a recipe or a problem involving a ratio of numbers? If your recipe was for 4 servings (or people) and you now have to make it for 12, ratios come into play and thus so do proportions!
This topic may seem challenging but resources like online math tutoring can be helpful to understand different concepts in mathematics and dive into them even more. With that in mind, let’s dig into proportions, see how they can help us in our day-to-day lives, and find out the steps we must use to solve problems in math class or individual tutoring sessions.
What is a Proportion?
A proportion is an equation that says that two ratios are equal. A ratio is a comparison of two numbers, usually expressed as a fraction. If you need a refresher on fractions or ratios and percentages, revising these topics will help you understand proportions much better. In a proportion, two ratios are set equal to each other using a colon (:) or the equals sign (=). The typical setup of a proportion is:
a/b = c/d
When something is in proportion, their relative sizes are the same. A dog that is 20 inches long and 10 inches tall is proportional to a dog that is 10 inches long and 5 inches tall. When we understand this, we can properly utilize these relative sized to solve problems.
How to Use Proportions to Solve
Proportions can be used to solve percentages, triangles, and more. Since a percentage is out of 100, it is technically a ratio. 25% as a ratio means 25 per every 100. As mentioned, we can use this knowledge to solve questions involving percentages. Let’s say you want to find out the percentage of a number. You take the percent (25), times it by the whole number (let’s say 50), then divide by 100.
Part/50 = 25/100
Part = (50 × 25)/100
= 1250/100
= 12.5
This method could also be used to find a missing whole number (50), or even a missing percentage (25%). To solve for a missing number, multiply the two known numbers diagonal from one another and divide by the third number. Using proportions can help when looking at the price of an item on sale, or analyzing statistical data.
Here is another example. If 7 markers cost $10, but you only need 4 markers, how much would that cost? This can be set up using the same method. Now you can multiply 4 x 10, and divide by 7 to find how much the 4 markers cost!
7/10 = 4/x
More Than Two
Proportions may not always have only 2 numbers in the ratio. If a proportion has 3 numbers, it will take this form: 1:2:3. This is the same as 10:20:30. Let’s look at another example. You have 1 red marble, 2 blue, and 3 orange. This is proportionate to having 10 red, 20 blue, and 30 orange. If you only had 6 orange marbles and wanted to figure out how many red and blue are needed, you can lay it out like so:
1:2:3
a:b:6
Looking at this, 6 is twice the size of 3, therefore you can double 1 and 2 to find a and b, respectively. The solution would then be 2:4:6.
Directly Proportional
Directly proportional means as one amount increases, the other amount increases at the same rate. The symbol for this is ∝. A great example of this is the number of hours you work is directly proportional to how much money you earn. The more hours you work, the more money you earn. If you make $20 an hour and work 8 hours a day, you would make $160 each day. Earned Pay ∝ Hours Worked.
Within this concept is the idea of “constant proportionality”. This is the value that relates the two amounts. In this example, the hourly pay of $20 is the constant proportionality because it is the constant number that increases the Earned Pay.
You can write this out as an equation, like so: y = kx where y is the Earned Pay, k is the constant proportionality (Hourly Pay), and x is the Hours Worked. Therefore, if you have y and x, and need to solve for k, basic algebra can be used.
In this instance,
y=kx
$160= k (8)
Divide each side by 8, and you will find that k equals 20, the constant proportionality.
Once you find the constant proportionality, you can use it to solve other questions like, How much would you earn if you worked 10 hours instead of 8? This would then look like y = (20) (10), and the answer would be $200.
At Online Math Center
As you can see, proportions are not as difficult as they seem. However, it’s a skill that can only be improved with practice. As middle school students work on proportions, they will become more skilled at identifying where these methods & concepts are useful. Getting your kids the practice they need to excel at proportions, ratios, and fractions doesn’t need to be stressful, either.
At Online Math Center, we have years of experience helping middle school students grasp concepts such as these. In our tailored one-to-one tutoring lessons, students get the attention they deserve and the opportunity to focus on the topics they’re struggling with. We also offer prep classes for the Math SAT and math competitions, as well as classes aimed at high school students
Take the stress out of your child’s math education and contact OMC today to enroll them in a tutoring program tailored to their specific needs.
